3.29.55 \(\int \frac {1+x^6}{\sqrt [4]{x^3+x^5} (1-x^6)} \, dx\) [2855]

3.29.55.1 Optimal result

Integrand size = 26, antiderivative size = 297 \[ \int \frac {1+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\frac {2}{3} \arctan \left (\frac {x}{\sqrt [4]{x^3+x^5}}\right )+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{3 \sqrt [4]{2}}-\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{x^3+x^5}}{\sqrt {2} x^2-\sqrt {x^3+x^5}}\right )}{3\ 2^{3/4}}+\frac {1}{3} \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{x^3+x^5}}{-x^2+\sqrt {x^3+x^5}}\right )+\frac {2}{3} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^5}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{3 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^3+x^5}}{2^{3/4}}}{x \sqrt [4]{x^3+x^5}}\right )}{3\ 2^{3/4}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^3+x^5}}{\sqrt {2}}}{x \sqrt [4]{x^3+x^5}}\right ) \]

output

2/3*arctan(x/(x^5+x^3)^(1/4))+1/6*arctan(2^(1/4)*x/(x^5+x^3)^(1/4))*2^(3/4 
)-1/6*arctan(2^(3/4)*x*(x^5+x^3)^(1/4)/(2^(1/2)*x^2-(x^5+x^3)^(1/2)))*2^(1 
/4)+1/3*2^(1/2)*arctan(2^(1/2)*x*(x^5+x^3)^(1/4)/(-x^2+(x^5+x^3)^(1/2)))+2 
/3*arctanh(x/(x^5+x^3)^(1/4))+1/6*arctanh(2^(1/4)*x/(x^5+x^3)^(1/4))*2^(3/ 
4)+1/6*arctanh((1/2*x^2*2^(3/4)+1/2*(x^5+x^3)^(1/2)*2^(1/4))/x/(x^5+x^3)^( 
1/4))*2^(1/4)+1/3*2^(1/2)*arctanh((1/2*2^(1/2)*x^2+1/2*(x^5+x^3)^(1/2)*2^( 
1/2))/x/(x^5+x^3)^(1/4))
 

3.29.55.2 Mathematica [A] (verified)

Time = 1.33 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.07 \[ \int \frac {1+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\frac {x^{3/4} \sqrt [4]{1+x^2} \left (4 \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+2^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )-\sqrt [4]{2} \arctan \left (\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {2} \sqrt {x}-\sqrt {1+x^2}}\right )+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{-\sqrt {x}+\sqrt {1+x^2}}\right )+4 \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {x}+\sqrt {1+x^2}}\right )+\sqrt [4]{2} \text {arctanh}\left (\frac {2 \sqrt [4]{2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{2 \sqrt {x}+\sqrt {2} \sqrt {1+x^2}}\right )\right )}{6 \sqrt [4]{x^3+x^5}} \]

input

Integrate[(1 + x^6)/((x^3 + x^5)^(1/4)*(1 - x^6)),x]
 

output

(x^(3/4)*(1 + x^2)^(1/4)*(4*ArcTan[x^(1/4)/(1 + x^2)^(1/4)] + 2^(3/4)*ArcT 
an[(2^(1/4)*x^(1/4))/(1 + x^2)^(1/4)] - 2^(1/4)*ArcTan[(2^(3/4)*x^(1/4)*(1 
 + x^2)^(1/4))/(Sqrt[2]*Sqrt[x] - Sqrt[1 + x^2])] + 2*Sqrt[2]*ArcTan[(Sqrt 
[2]*x^(1/4)*(1 + x^2)^(1/4))/(-Sqrt[x] + Sqrt[1 + x^2])] + 4*ArcTanh[x^(1/ 
4)/(1 + x^2)^(1/4)] + 2^(3/4)*ArcTanh[(2^(1/4)*x^(1/4))/(1 + x^2)^(1/4)] + 
 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x^(1/4)*(1 + x^2)^(1/4))/(Sqrt[x] + Sqrt[1 + x 
^2])] + 2^(1/4)*ArcTanh[(2*2^(1/4)*x^(1/4)*(1 + x^2)^(1/4))/(2*Sqrt[x] + S 
qrt[2]*Sqrt[1 + x^2])]))/(6*(x^3 + x^5)^(1/4))
 

3.29.55.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^6+1}{\sqrt [4]{x^5+x^3} \left (1-x^6\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {x^{3/4} \sqrt [4]{x^2+1} \int \frac {x^6+1}{x^{3/4} \sqrt [4]{x^2+1} \left (1-x^6\right )}dx}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 2019

\(\displaystyle \frac {x^{3/4} \sqrt [4]{x^2+1} \int \frac {\left (x^2+1\right )^{3/4} \left (x^4-x^2+1\right )}{x^{3/4} \left (1-x^6\right )}dx}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \frac {\left (x^2+1\right )^{3/4} \left (x^4-x^2+1\right )}{1-x^6}d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-i \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (i \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [12]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [12]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [6]{-1} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [6]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{-1} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [3]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [3]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/12} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/12} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{7/12} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{7/12} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{2/3} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{2/3} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{3/4} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left ((-1)^{3/4} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/6} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/6} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{11/12} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{11/12} \sqrt [4]{x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-i \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (i \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [12]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [12]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [6]{-1} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [6]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{-1} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [3]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [3]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/12} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/12} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{7/12} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{7/12} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{2/3} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{2/3} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{3/4} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left ((-1)^{3/4} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/6} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/6} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{11/12} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{11/12} \sqrt [4]{x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (-\frac {\left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}-1\right )}-\frac {i \left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}-i\right )}+\frac {i \left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}+i\right )}+\frac {\left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}+1\right )}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [6]{-1} \sqrt {x}-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [6]{-1} \sqrt {x}+1\right )}+\frac {i \left (i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [3]{-1} \sqrt {x}-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [3]{-1} \sqrt {x}+1\right )}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{2/3} \sqrt {x}-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{2/3} \sqrt {x}+1\right )}+\frac {i \left (i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{5/6} \sqrt {x}-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{5/6} \sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \frac {\left (x^2+1\right )^{3/4} \left (x^4-x^2+1\right )}{1-x^6}d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-i \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (i \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [12]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [12]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [6]{-1} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [6]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{-1} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [3]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [3]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/12} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/12} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{7/12} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{7/12} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{2/3} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{2/3} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{3/4} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left ((-1)^{3/4} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/6} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/6} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{11/12} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{11/12} \sqrt [4]{x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-i \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (i \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [12]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [12]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [6]{-1} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [6]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{-1} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [3]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [3]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/12} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/12} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{7/12} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{7/12} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{2/3} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{2/3} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{3/4} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left ((-1)^{3/4} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/6} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/6} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{11/12} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{11/12} \sqrt [4]{x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (-\frac {\left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}-1\right )}-\frac {i \left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}-i\right )}+\frac {i \left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}+i\right )}+\frac {\left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}+1\right )}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [6]{-1} \sqrt {x}-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [6]{-1} \sqrt {x}+1\right )}+\frac {i \left (i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [3]{-1} \sqrt {x}-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [3]{-1} \sqrt {x}+1\right )}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{2/3} \sqrt {x}-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{2/3} \sqrt {x}+1\right )}+\frac {i \left (i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{5/6} \sqrt {x}-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{5/6} \sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \frac {\left (x^2+1\right )^{3/4} \left (x^4-x^2+1\right )}{1-x^6}d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-i \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (i \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [12]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [12]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [6]{-1} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [6]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{-1} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [3]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [3]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/12} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/12} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{7/12} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{7/12} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{2/3} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{2/3} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{3/4} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left ((-1)^{3/4} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/6} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/6} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{11/12} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{11/12} \sqrt [4]{x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-i \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (i \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [12]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [12]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [6]{-1} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [6]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{-1} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [3]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [3]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/12} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/12} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{7/12} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{7/12} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{2/3} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{2/3} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{3/4} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left ((-1)^{3/4} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/6} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/6} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{11/12} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{11/12} \sqrt [4]{x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (-\frac {\left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}-1\right )}-\frac {i \left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}-i\right )}+\frac {i \left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}+i\right )}+\frac {\left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}+1\right )}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [6]{-1} \sqrt {x}-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [6]{-1} \sqrt {x}+1\right )}+\frac {i \left (i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [3]{-1} \sqrt {x}-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [3]{-1} \sqrt {x}+1\right )}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{2/3} \sqrt {x}-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{2/3} \sqrt {x}+1\right )}+\frac {i \left (i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{5/6} \sqrt {x}-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{5/6} \sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \frac {\left (x^2+1\right )^{3/4} \left (x^4-x^2+1\right )}{1-x^6}d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-i \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (i \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [12]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [12]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [6]{-1} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [6]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{-1} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [3]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [3]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/12} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/12} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{7/12} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{7/12} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{2/3} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{2/3} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{3/4} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left ((-1)^{3/4} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/6} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/6} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{11/12} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{11/12} \sqrt [4]{x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-i \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (i \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [12]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [12]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [6]{-1} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [6]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{-1} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [3]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [3]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/12} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/12} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{7/12} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{7/12} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{2/3} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{2/3} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{3/4} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left ((-1)^{3/4} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/6} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/6} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{11/12} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{11/12} \sqrt [4]{x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (-\frac {\left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}-1\right )}-\frac {i \left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}-i\right )}+\frac {i \left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}+i\right )}+\frac {\left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}+1\right )}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [6]{-1} \sqrt {x}-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [6]{-1} \sqrt {x}+1\right )}+\frac {i \left (i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [3]{-1} \sqrt {x}-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [3]{-1} \sqrt {x}+1\right )}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{2/3} \sqrt {x}-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{2/3} \sqrt {x}+1\right )}+\frac {i \left (i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{5/6} \sqrt {x}-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{5/6} \sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \frac {\left (x^2+1\right )^{3/4} \left (x^4-x^2+1\right )}{1-x^6}d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-i \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (i \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [12]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [12]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [6]{-1} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [6]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{-1} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [3]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [3]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/12} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/12} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{7/12} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{7/12} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{2/3} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{2/3} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{3/4} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left ((-1)^{3/4} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/6} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/6} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{11/12} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{11/12} \sqrt [4]{x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-i \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (i \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [12]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [12]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [6]{-1} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [6]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{-1} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [3]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [3]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/12} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/12} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{7/12} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{7/12} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{2/3} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{2/3} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{3/4} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left ((-1)^{3/4} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/6} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/6} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{11/12} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{11/12} \sqrt [4]{x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (-\frac {\left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}-1\right )}-\frac {i \left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}-i\right )}+\frac {i \left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}+i\right )}+\frac {\left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}+1\right )}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [6]{-1} \sqrt {x}-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [6]{-1} \sqrt {x}+1\right )}+\frac {i \left (i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [3]{-1} \sqrt {x}-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [3]{-1} \sqrt {x}+1\right )}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{2/3} \sqrt {x}-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{2/3} \sqrt {x}+1\right )}+\frac {i \left (i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{5/6} \sqrt {x}-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{5/6} \sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \frac {\left (x^2+1\right )^{3/4} \left (x^4-x^2+1\right )}{1-x^6}d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-i \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (i \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [12]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [12]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [6]{-1} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [6]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{-1} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [3]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [3]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/12} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/12} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{7/12} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{7/12} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{2/3} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{2/3} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{3/4} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left ((-1)^{3/4} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/6} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/6} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{11/12} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{11/12} \sqrt [4]{x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-i \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (i \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [12]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [12]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [6]{-1} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [6]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{-1} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [3]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [3]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/12} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/12} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{7/12} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{7/12} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{2/3} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{2/3} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{3/4} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left ((-1)^{3/4} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/6} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/6} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{11/12} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{11/12} \sqrt [4]{x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (-\frac {\left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}-1\right )}-\frac {i \left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}-i\right )}+\frac {i \left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}+i\right )}+\frac {\left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}+1\right )}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [6]{-1} \sqrt {x}-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [6]{-1} \sqrt {x}+1\right )}+\frac {i \left (i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [3]{-1} \sqrt {x}-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [3]{-1} \sqrt {x}+1\right )}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{2/3} \sqrt {x}-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{2/3} \sqrt {x}+1\right )}+\frac {i \left (i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{5/6} \sqrt {x}-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{5/6} \sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \frac {\left (x^2+1\right )^{3/4} \left (x^4-x^2+1\right )}{1-x^6}d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-i \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (i \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [12]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [12]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [6]{-1} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [6]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{-1} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [3]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [3]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/12} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/12} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{7/12} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{7/12} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{2/3} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{2/3} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{3/4} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left ((-1)^{3/4} \sqrt [4]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/6} \sqrt [4]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/6} \sqrt [4]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{11/12} \sqrt [4]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{11/12} \sqrt [4]{x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-i \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (i \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [12]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [12]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [6]{-1} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [6]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [4]{-1} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (\sqrt [4]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-\sqrt [3]{-1} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (\sqrt [3]{-1} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/12} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/12} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{7/12} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{7/12} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{2/3} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{2/3} \sqrt [4]{x}+1\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{3/4} \sqrt [4]{x}\right )}+\frac {\left (x^2+1\right )^{3/4}}{24 \left ((-1)^{3/4} \sqrt [4]{x}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{5/6} \sqrt [4]{x}\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{5/6} \sqrt [4]{x}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left (1-(-1)^{11/12} \sqrt [4]{x}\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{24 \left ((-1)^{11/12} \sqrt [4]{x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2+1} \int \left (-\frac {\left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}-1\right )}-\frac {i \left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}-i\right )}+\frac {i \left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}+i\right )}+\frac {\left (x^2+1\right )^{3/4}}{12 \left (\sqrt {x}+1\right )}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [6]{-1} \sqrt {x}-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [6]{-1} \sqrt {x}+1\right )}+\frac {i \left (i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [3]{-1} \sqrt {x}-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left (\sqrt [3]{-1} \sqrt {x}+1\right )}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{2/3} \sqrt {x}-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{2/3} \sqrt {x}+1\right )}+\frac {i \left (i+\sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{5/6} \sqrt {x}-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right )^{3/4}}{12 \left ((-1)^{5/6} \sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5+x^3}}\)

input

Int[(1 + x^6)/((x^3 + x^5)^(1/4)*(1 - x^6)),x]
 

output

$Aborted
 

3.29.55.3.1 Defintions of rubi rules used

rule 2019

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px 
, Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && 
 EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.29.55.4 Maple [A] (verified)

Time = 20.70 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.34

method	result	size
pseudoelliptic	\(-\frac {2 \arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{3}-\frac {\ln \left (\frac {-\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}{\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}\right ) 2^{\frac {1}{4}}}{12}-\frac {\arctan \left (\frac {2^{\frac {1}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+x}{x}\right ) 2^{\frac {1}{4}}}{6}-\frac {\arctan \left (\frac {2^{\frac {1}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-x}{x}\right ) 2^{\frac {1}{4}}}{6}+\frac {\ln \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+x}{x}\right )}{3}-\frac {\ln \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-x}{x}\right )}{3}-\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right ) 2^{\frac {3}{4}}}{6}+\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{12}-\frac {\ln \left (\frac {-\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}{\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}\right ) \sqrt {2}}{6}-\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}}{3}-\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}}{3}\)	\(399\)
trager	\(\text {Expression too large to display}\)	\(1424\)

input

int((x^6+1)/(x^5+x^3)^(1/4)/(-x^6+1),x,method=_RETURNVERBOSE)
 

output

-2/3*arctan((x^3*(x^2+1))^(1/4)/x)-1/12*ln((-(x^3*(x^2+1))^(1/4)*2^(3/4)*x 
+2^(1/2)*x^2+(x^3*(x^2+1))^(1/2))/((x^3*(x^2+1))^(1/4)*2^(3/4)*x+2^(1/2)*x 
^2+(x^3*(x^2+1))^(1/2)))*2^(1/4)-1/6*arctan((2^(1/4)*(x^3*(x^2+1))^(1/4)+x 
)/x)*2^(1/4)-1/6*arctan((2^(1/4)*(x^3*(x^2+1))^(1/4)-x)/x)*2^(1/4)+1/3*ln( 
((x^3*(x^2+1))^(1/4)+x)/x)-1/3*ln(((x^3*(x^2+1))^(1/4)-x)/x)-1/6*arctan(1/ 
2*(x^3*(x^2+1))^(1/4)/x*2^(3/4))*2^(3/4)+1/12*ln((-2^(1/4)*x-(x^3*(x^2+1)) 
^(1/4))/(2^(1/4)*x-(x^3*(x^2+1))^(1/4)))*2^(3/4)-1/6*ln((-(x^3*(x^2+1))^(1 
/4)*2^(1/2)*x+x^2+(x^3*(x^2+1))^(1/2))/((x^3*(x^2+1))^(1/4)*2^(1/2)*x+x^2+ 
(x^3*(x^2+1))^(1/2)))*2^(1/2)-1/3*arctan(((x^3*(x^2+1))^(1/4)*2^(1/2)+x)/x 
)*2^(1/2)-1/3*arctan(((x^3*(x^2+1))^(1/4)*2^(1/2)-x)/x)*2^(1/2)
 

3.29.55.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 35.51 (sec) , antiderivative size = 1113, normalized size of antiderivative = 3.75 \[ \int \frac {1+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\text {Too large to display} \]

input

integrate((x^6+1)/(x^5+x^3)^(1/4)/(-x^6+1),x, algorithm="fricas")
 

output

1/24*2^(3/4)*log((4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + 2^(3/4)*(x^4 + 2*x^3 + 
 x^2) + 4*2^(1/4)*sqrt(x^5 + x^3)*x + 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + 
x^2)) - 1/24*2^(3/4)*log((4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 - 2^(3/4)*(x^4 + 
 2*x^3 + x^2) - 4*2^(1/4)*sqrt(x^5 + x^3)*x + 4*(x^5 + x^3)^(3/4))/(x^4 - 
2*x^3 + x^2)) + 1/24*I*2^(3/4)*log(-(4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 - 2^( 
3/4)*(I*x^4 + 2*I*x^3 + I*x^2) + 4*I*2^(1/4)*sqrt(x^5 + x^3)*x - 4*(x^5 + 
x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) - 1/24*I*2^(3/4)*log(-(4*sqrt(2)*(x^5 + x 
^3)^(1/4)*x^2 - 2^(3/4)*(-I*x^4 - 2*I*x^3 - I*x^2) - 4*I*2^(1/4)*sqrt(x^5 
+ x^3)*x - 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) - (1/12*I - 1/12)*sqr 
t(2)*log((4*I*(x^5 + x^3)^(1/4)*x^2 + (2*I + 2)*sqrt(2)*sqrt(x^5 + x^3)*x 
+ sqrt(2)*(-(I - 1)*x^4 + (I - 1)*x^3 - (I - 1)*x^2) + 4*(x^5 + x^3)^(3/4) 
)/(x^4 + x^3 + x^2)) + (1/12*I - 1/12)*sqrt(2)*log((4*I*(x^5 + x^3)^(1/4)* 
x^2 - (2*I + 2)*sqrt(2)*sqrt(x^5 + x^3)*x + sqrt(2)*((I - 1)*x^4 - (I - 1) 
*x^3 + (I - 1)*x^2) + 4*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)) + (1/12*I + 
1/12)*sqrt(2)*log((-4*I*(x^5 + x^3)^(1/4)*x^2 - (2*I - 2)*sqrt(2)*sqrt(x^5 
 + x^3)*x + sqrt(2)*((I + 1)*x^4 - (I + 1)*x^3 + (I + 1)*x^2) + 4*(x^5 + x 
^3)^(3/4))/(x^4 + x^3 + x^2)) - (1/12*I + 1/12)*sqrt(2)*log((-4*I*(x^5 + x 
^3)^(1/4)*x^2 + (2*I - 2)*sqrt(2)*sqrt(x^5 + x^3)*x + sqrt(2)*(-(I + 1)*x^ 
4 + (I + 1)*x^3 - (I + 1)*x^2) + 4*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)) - 
 (1/24*I - 1/24)*2^(1/4)*log(-2*(4*I*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + (2...
 

3.29.55.6 Sympy [F]

\[ \int \frac {1+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=- \int \frac {x^{6}}{x^{6} \sqrt [4]{x^{5} + x^{3}} - \sqrt [4]{x^{5} + x^{3}}}\, dx - \int \frac {1}{x^{6} \sqrt [4]{x^{5} + x^{3}} - \sqrt [4]{x^{5} + x^{3}}}\, dx \]

input

integrate((x**6+1)/(x**5+x**3)**(1/4)/(-x**6+1),x)
 

output

-Integral(x**6/(x**6*(x**5 + x**3)**(1/4) - (x**5 + x**3)**(1/4)), x) - In 
tegral(1/(x**6*(x**5 + x**3)**(1/4) - (x**5 + x**3)**(1/4)), x)
 

3.29.55.7 Maxima [F]

\[ \int \frac {1+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}}} \,d x } \]

input

integrate((x^6+1)/(x^5+x^3)^(1/4)/(-x^6+1),x, algorithm="maxima")
 

output

-integrate((x^6 + 1)/((x^6 - 1)*(x^5 + x^3)^(1/4)), x)
 

3.29.55.8 Giac [F]

\[ \int \frac {1+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}}} \,d x } \]

input

integrate((x^6+1)/(x^5+x^3)^(1/4)/(-x^6+1),x, algorithm="giac")
 

output

integrate(-(x^6 + 1)/((x^6 - 1)*(x^5 + x^3)^(1/4)), x)
 

3.29.55.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\int -\frac {x^6+1}{{\left (x^5+x^3\right )}^{1/4}\,\left (x^6-1\right )} \,d x \]

input

int(-(x^6 + 1)/((x^3 + x^5)^(1/4)*(x^6 - 1)),x)
 

output

int(-(x^6 + 1)/((x^3 + x^5)^(1/4)*(x^6 - 1)), x)