\(\int \frac {1}{b+a x^{12}} \, dx\) [104]

Optimal result

Integrand size = 9, antiderivative size = 831 \[ \int \frac {1}{b+a x^{12}} \, dx=-\frac {\arctan \left (\frac {\sqrt [12]{b}-\sqrt {2} \sqrt [12]{a} x}{\sqrt [12]{b}}\right )}{6 \sqrt {2} \sqrt [12]{a} b^{11/12}}+\frac {\arctan \left (\frac {\sqrt [12]{b}+\sqrt {2} \sqrt [12]{a} x}{\sqrt [12]{b}}\right )}{6 \sqrt {2} \sqrt [12]{a} b^{11/12}}-\frac {\sqrt {2+\sqrt {3}} \arctan \left (\frac {2 \sqrt [12]{b}-\sqrt {3} \sqrt [12]{b}-2 \sqrt {2-\sqrt {3}} \sqrt [12]{a} x}{\sqrt [12]{b}}\right )}{12 \sqrt [12]{a} b^{11/12}}+\frac {\sqrt {2+\sqrt {3}} \arctan \left (\frac {2 \sqrt [12]{b}-\sqrt {3} \sqrt [12]{b}+2 \sqrt {2-\sqrt {3}} \sqrt [12]{a} x}{\sqrt [12]{b}}\right )}{12 \sqrt [12]{a} b^{11/12}}-\frac {\sqrt {2-\sqrt {3}} \arctan \left (\frac {2 \sqrt [12]{b}+\sqrt {3} \sqrt [12]{b}-2 \sqrt {2+\sqrt {3}} \sqrt [12]{a} x}{\sqrt [12]{b}}\right )}{12 \sqrt [12]{a} b^{11/12}}+\frac {\sqrt {2-\sqrt {3}} \arctan \left (\frac {2 \sqrt [12]{b}+\sqrt {3} \sqrt [12]{b}+2 \sqrt {2+\sqrt {3}} \sqrt [12]{a} x}{\sqrt [12]{b}}\right )}{12 \sqrt [12]{a} b^{11/12}}-\frac {\log \left (\sqrt [6]{b}-\sqrt {2} \sqrt [12]{a} \sqrt [12]{b} x+\sqrt [6]{a} x^2\right )}{12 \sqrt {2} \sqrt [12]{a} b^{11/12}}+\frac {\log \left (\sqrt [6]{b}+\sqrt {2} \sqrt [12]{a} \sqrt [12]{b} x+\sqrt [6]{a} x^2\right )}{12 \sqrt {2} \sqrt [12]{a} b^{11/12}}-\frac {\log \left (\sqrt [3]{b}-\sqrt {2} \sqrt [12]{a} \sqrt [4]{b} x+\sqrt [6]{a} \sqrt [6]{b} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [12]{b} x^3+\sqrt [3]{a} x^4\right )}{24 \sqrt {2} \sqrt [12]{a} b^{11/12}}+\frac {\log \left (\sqrt [3]{b}+\sqrt {2} \sqrt [12]{a} \sqrt [4]{b} x+\sqrt [6]{a} \sqrt [6]{b} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [12]{b} x^3+\sqrt [3]{a} x^4\right )}{24 \sqrt {2} \sqrt [12]{a} b^{11/12}}-\frac {\log \left (\sqrt [3]{b}-\sqrt {6} \sqrt [12]{a} \sqrt [4]{b} x+3 \sqrt [6]{a} \sqrt [6]{b} x^2-\sqrt {6} \sqrt [4]{a} \sqrt [12]{b} x^3+\sqrt [3]{a} x^4\right )}{8 \sqrt {6} \sqrt [12]{a} b^{11/12}}+\frac {\log \left (\sqrt [3]{b}+\sqrt {6} \sqrt [12]{a} \sqrt [4]{b} x+3 \sqrt [6]{a} \sqrt [6]{b} x^2+\sqrt {6} \sqrt [4]{a} \sqrt [12]{b} x^3+\sqrt [3]{a} x^4\right )}{8 \sqrt {6} \sqrt [12]{a} b^{11/12}} \] Output:

-1/12*arctan((b^(1/12)-2^(1/2)*a^(1/12)*x)/b^(1/12))*2^(1/2)/a^(1/12)/b^(1 
1/12)+1/12*arctan((b^(1/12)+2^(1/2)*a^(1/12)*x)/b^(1/12))*2^(1/2)/a^(1/12) 
/b^(11/12)-1/12*(1/2*6^(1/2)+1/2*2^(1/2))*arctan((2*b^(1/12)-3^(1/2)*b^(1/ 
12)-2*(1/2*6^(1/2)-1/2*2^(1/2))*a^(1/12)*x)/b^(1/12))/a^(1/12)/b^(11/12)+1 
/12*(1/2*6^(1/2)+1/2*2^(1/2))*arctan((2*b^(1/12)-3^(1/2)*b^(1/12)+2*(1/2*6 
^(1/2)-1/2*2^(1/2))*a^(1/12)*x)/b^(1/12))/a^(1/12)/b^(11/12)-1/12*(1/2*6^( 
1/2)-1/2*2^(1/2))*arctan((2*b^(1/12)+3^(1/2)*b^(1/12)-2*(1/2*6^(1/2)+1/2*2 
^(1/2))*a^(1/12)*x)/b^(1/12))/a^(1/12)/b^(11/12)+1/12*(1/2*6^(1/2)-1/2*2^( 
1/2))*arctan((2*b^(1/12)+3^(1/2)*b^(1/12)+2*(1/2*6^(1/2)+1/2*2^(1/2))*a^(1 
/12)*x)/b^(1/12))/a^(1/12)/b^(11/12)-1/24*ln(b^(1/6)-2^(1/2)*a^(1/12)*b^(1 
/12)*x+a^(1/6)*x^2)*2^(1/2)/a^(1/12)/b^(11/12)+1/24*ln(b^(1/6)+2^(1/2)*a^( 
1/12)*b^(1/12)*x+a^(1/6)*x^2)*2^(1/2)/a^(1/12)/b^(11/12)-1/48*ln(b^(1/3)-2 
^(1/2)*a^(1/12)*b^(1/4)*x+a^(1/6)*b^(1/6)*x^2-2^(1/2)*a^(1/4)*b^(1/12)*x^3 
+a^(1/3)*x^4)*2^(1/2)/a^(1/12)/b^(11/12)+1/48*ln(b^(1/3)+2^(1/2)*a^(1/12)* 
b^(1/4)*x+a^(1/6)*b^(1/6)*x^2+2^(1/2)*a^(1/4)*b^(1/12)*x^3+a^(1/3)*x^4)*2^ 
(1/2)/a^(1/12)/b^(11/12)-1/48*ln(b^(1/3)-6^(1/2)*a^(1/12)*b^(1/4)*x+3*a^(1 
/6)*b^(1/6)*x^2-6^(1/2)*a^(1/4)*b^(1/12)*x^3+a^(1/3)*x^4)*6^(1/2)/a^(1/12) 
/b^(11/12)+1/48*ln(b^(1/3)+6^(1/2)*a^(1/12)*b^(1/4)*x+3*a^(1/6)*b^(1/6)*x^ 
2+6^(1/2)*a^(1/4)*b^(1/12)*x^3+a^(1/3)*x^4)*6^(1/2)/a^(1/12)/b^(11/12)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 778, normalized size of antiderivative = 0.94 \[ \int \frac {1}{b+a x^{12}} \, dx=\frac {-2 \left (1+\sqrt {3}\right ) \arctan \left (\frac {1-\sqrt {3}-\frac {2 \sqrt {2} \sqrt [12]{a} x}{\sqrt [12]{b}}}{1+\sqrt {3}}\right )+2 \left (-1+\sqrt {3}\right ) \arctan \left (\frac {1+\sqrt {3}-\frac {2 \sqrt {2} \sqrt [12]{a} x}{\sqrt [12]{b}}}{1-\sqrt {3}}\right )-4 \arctan \left (1-\frac {\sqrt {2} \sqrt [12]{a} x}{\sqrt [12]{b}}\right )+4 \arctan \left (1+\frac {\sqrt {2} \sqrt [12]{a} x}{\sqrt [12]{b}}\right )+2 \arctan \left (\frac {1-\sqrt {3}+\frac {2 \sqrt {2} \sqrt [12]{a} x}{\sqrt [12]{b}}}{1+\sqrt {3}}\right )+2 \sqrt {3} \arctan \left (\frac {1-\sqrt {3}+\frac {2 \sqrt {2} \sqrt [12]{a} x}{\sqrt [12]{b}}}{1+\sqrt {3}}\right )+2 \arctan \left (\frac {1+\sqrt {3}+\frac {2 \sqrt {2} \sqrt [12]{a} x}{\sqrt [12]{b}}}{1-\sqrt {3}}\right )-2 \sqrt {3} \arctan \left (\frac {1+\sqrt {3}+\frac {2 \sqrt {2} \sqrt [12]{a} x}{\sqrt [12]{b}}}{1-\sqrt {3}}\right )-2 \log \left (\sqrt [6]{b}-\sqrt {2} \sqrt [12]{a} \sqrt [12]{b} x+\sqrt [6]{a} x^2\right )+2 \log \left (\sqrt [6]{b}+\sqrt {2} \sqrt [12]{a} \sqrt [12]{b} x+\sqrt [6]{a} x^2\right )+\log \left (2 \sqrt [6]{b}-\sqrt {2} \left (-1+\sqrt {3}\right ) \sqrt [12]{a} \sqrt [12]{b} x+2 \sqrt [6]{a} x^2\right )-\sqrt {3} \log \left (2 \sqrt [6]{b}-\sqrt {2} \left (-1+\sqrt {3}\right ) \sqrt [12]{a} \sqrt [12]{b} x+2 \sqrt [6]{a} x^2\right )-\log \left (2 \sqrt [6]{b}+\sqrt {2} \left (-1+\sqrt {3}\right ) \sqrt [12]{a} \sqrt [12]{b} x+2 \sqrt [6]{a} x^2\right )+\sqrt {3} \log \left (2 \sqrt [6]{b}+\sqrt {2} \left (-1+\sqrt {3}\right ) \sqrt [12]{a} \sqrt [12]{b} x+2 \sqrt [6]{a} x^2\right )-\log \left (2 \sqrt [6]{b}-\sqrt {2} \left (1+\sqrt {3}\right ) \sqrt [12]{a} \sqrt [12]{b} x+2 \sqrt [6]{a} x^2\right )-\sqrt {3} \log \left (2 \sqrt [6]{b}-\sqrt {2} \left (1+\sqrt {3}\right ) \sqrt [12]{a} \sqrt [12]{b} x+2 \sqrt [6]{a} x^2\right )+\log \left (2 \sqrt [6]{b}+\sqrt {2} \left (1+\sqrt {3}\right ) \sqrt [12]{a} \sqrt [12]{b} x+2 \sqrt [6]{a} x^2\right )+\sqrt {3} \log \left (2 \sqrt [6]{b}+\sqrt {2} \left (1+\sqrt {3}\right ) \sqrt [12]{a} \sqrt [12]{b} x+2 \sqrt [6]{a} x^2\right )}{24 \sqrt {2} \sqrt [12]{a} b^{11/12}} \] Input:

Integrate[(b + a*x^12)^(-1),x]
 

Output:

(-2*(1 + Sqrt[3])*ArcTan[(1 - Sqrt[3] - (2*Sqrt[2]*a^(1/12)*x)/b^(1/12))/( 
1 + Sqrt[3])] + 2*(-1 + Sqrt[3])*ArcTan[(1 + Sqrt[3] - (2*Sqrt[2]*a^(1/12) 
*x)/b^(1/12))/(1 - Sqrt[3])] - 4*ArcTan[1 - (Sqrt[2]*a^(1/12)*x)/b^(1/12)] 
 + 4*ArcTan[1 + (Sqrt[2]*a^(1/12)*x)/b^(1/12)] + 2*ArcTan[(1 - Sqrt[3] + ( 
2*Sqrt[2]*a^(1/12)*x)/b^(1/12))/(1 + Sqrt[3])] + 2*Sqrt[3]*ArcTan[(1 - Sqr 
t[3] + (2*Sqrt[2]*a^(1/12)*x)/b^(1/12))/(1 + Sqrt[3])] + 2*ArcTan[(1 + Sqr 
t[3] + (2*Sqrt[2]*a^(1/12)*x)/b^(1/12))/(1 - Sqrt[3])] - 2*Sqrt[3]*ArcTan[ 
(1 + Sqrt[3] + (2*Sqrt[2]*a^(1/12)*x)/b^(1/12))/(1 - Sqrt[3])] - 2*Log[b^( 
1/6) - Sqrt[2]*a^(1/12)*b^(1/12)*x + a^(1/6)*x^2] + 2*Log[b^(1/6) + Sqrt[2 
]*a^(1/12)*b^(1/12)*x + a^(1/6)*x^2] + Log[2*b^(1/6) - Sqrt[2]*(-1 + Sqrt[ 
3])*a^(1/12)*b^(1/12)*x + 2*a^(1/6)*x^2] - Sqrt[3]*Log[2*b^(1/6) - Sqrt[2] 
*(-1 + Sqrt[3])*a^(1/12)*b^(1/12)*x + 2*a^(1/6)*x^2] - Log[2*b^(1/6) + Sqr 
t[2]*(-1 + Sqrt[3])*a^(1/12)*b^(1/12)*x + 2*a^(1/6)*x^2] + Sqrt[3]*Log[2*b 
^(1/6) + Sqrt[2]*(-1 + Sqrt[3])*a^(1/12)*b^(1/12)*x + 2*a^(1/6)*x^2] - Log 
[2*b^(1/6) - Sqrt[2]*(1 + Sqrt[3])*a^(1/12)*b^(1/12)*x + 2*a^(1/6)*x^2] - 
Sqrt[3]*Log[2*b^(1/6) - Sqrt[2]*(1 + Sqrt[3])*a^(1/12)*b^(1/12)*x + 2*a^(1 
/6)*x^2] + Log[2*b^(1/6) + Sqrt[2]*(1 + Sqrt[3])*a^(1/12)*b^(1/12)*x + 2*a 
^(1/6)*x^2] + Sqrt[3]*Log[2*b^(1/6) + Sqrt[2]*(1 + Sqrt[3])*a^(1/12)*b^(1/ 
12)*x + 2*a^(1/6)*x^2])/(24*Sqrt[2]*a^(1/12)*b^(11/12))
 

Rubi [A] (verified)

Time = 1.43 (sec) , antiderivative size = 467, normalized size of antiderivative = 0.56, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.444, Rules used = {758, 753, 27, 218, 754, 27, 221, 1142, 25, 27, 1082, 217, 1103}

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.

\(\displaystyle \int \frac {1}{a x^{12}+b} \, dx\)

\(\Big \downarrow \) 758

\(\displaystyle \frac {\int \frac {1}{\sqrt {b}-\sqrt {-a} x^6}dx}{2 \sqrt {b}}+\frac {\int \frac {1}{\sqrt {-a} x^6+\sqrt {b}}dx}{2 \sqrt {b}}\)

\(\Big \downarrow \) 753

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {2 \sqrt [12]{b}-\sqrt {3} \sqrt [12]{-a} x}{\sqrt [6]{-a} x^2-\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\int \frac {\sqrt {3} \sqrt [12]{-a} x+2 \sqrt [12]{b}}{\sqrt [6]{-a} x^2+\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\int \frac {1}{\sqrt [6]{-a} x^2+\sqrt [6]{b}}dx}{3 \sqrt [3]{b}}}{2 \sqrt {b}}+\frac {\int \frac {1}{\sqrt {b}-\sqrt {-a} x^6}dx}{2 \sqrt {b}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\int \frac {2 \sqrt [12]{b}-\sqrt {3} \sqrt [12]{-a} x}{\sqrt [6]{-a} x^2-\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\int \frac {\sqrt {3} \sqrt [12]{-a} x+2 \sqrt [12]{b}}{\sqrt [6]{-a} x^2+\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\arctan \left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}+\frac {\int \frac {1}{\sqrt {b}-\sqrt {-a} x^6}dx}{2 \sqrt {b}}\)

\(\Big \downarrow \) 754

\(\displaystyle \frac {\frac {\int \frac {2 \sqrt [12]{b}-\sqrt {3} \sqrt [12]{-a} x}{\sqrt [6]{-a} x^2-\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\int \frac {\sqrt {3} \sqrt [12]{-a} x+2 \sqrt [12]{b}}{\sqrt [6]{-a} x^2+\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\arctan \left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}+\frac {\frac {\int \frac {2 \sqrt [12]{b}-\sqrt [12]{-a} x}{2 \left (\sqrt [6]{-a} x^2-\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}\right )}dx}{3 b^{5/12}}+\frac {\int \frac {\sqrt [12]{-a} x+2 \sqrt [12]{b}}{2 \left (\sqrt [6]{-a} x^2+\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}\right )}dx}{3 b^{5/12}}+\frac {\int \frac {1}{\sqrt [6]{b}-\sqrt [6]{-a} x^2}dx}{3 \sqrt [3]{b}}}{2 \sqrt {b}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {2 \sqrt [12]{b}-\sqrt {3} \sqrt [12]{-a} x}{\sqrt [6]{-a} x^2-\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\int \frac {\sqrt {3} \sqrt [12]{-a} x+2 \sqrt [12]{b}}{\sqrt [6]{-a} x^2+\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\arctan \left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}+\frac {\frac {\int \frac {2 \sqrt [12]{b}-\sqrt [12]{-a} x}{\sqrt [6]{-a} x^2-\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\int \frac {\sqrt [12]{-a} x+2 \sqrt [12]{b}}{\sqrt [6]{-a} x^2+\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\int \frac {1}{\sqrt [6]{b}-\sqrt [6]{-a} x^2}dx}{3 \sqrt [3]{b}}}{2 \sqrt {b}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\int \frac {2 \sqrt [12]{b}-\sqrt {3} \sqrt [12]{-a} x}{\sqrt [6]{-a} x^2-\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\int \frac {\sqrt {3} \sqrt [12]{-a} x+2 \sqrt [12]{b}}{\sqrt [6]{-a} x^2+\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\arctan \left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}+\frac {\frac {\int \frac {2 \sqrt [12]{b}-\sqrt [12]{-a} x}{\sqrt [6]{-a} x^2-\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\int \frac {\sqrt [12]{-a} x+2 \sqrt [12]{b}}{\sqrt [6]{-a} x^2+\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\text {arctanh}\left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\frac {\frac {1}{2} \sqrt [12]{b} \int \frac {1}{\sqrt [6]{-a} x^2-\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx-\frac {\sqrt {3} \int -\frac {\sqrt [12]{-a} \left (\sqrt {3} \sqrt [12]{b}-2 \sqrt [12]{-a} x\right )}{\sqrt [6]{-a} x^2-\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{2 \sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\frac {1}{2} \sqrt [12]{b} \int \frac {1}{\sqrt [6]{-a} x^2+\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx+\frac {\sqrt {3} \int \frac {\sqrt [12]{-a} \left (2 \sqrt [12]{-a} x+\sqrt {3} \sqrt [12]{b}\right )}{\sqrt [6]{-a} x^2+\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{2 \sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\arctan \left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}+\frac {\frac {\frac {3}{2} \sqrt [12]{b} \int \frac {1}{\sqrt [6]{-a} x^2-\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx-\frac {\int -\frac {\sqrt [12]{-a} \left (\sqrt [12]{b}-2 \sqrt [12]{-a} x\right )}{\sqrt [6]{-a} x^2-\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{2 \sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\frac {3}{2} \sqrt [12]{b} \int \frac {1}{\sqrt [6]{-a} x^2+\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx+\frac {\int \frac {\sqrt [12]{-a} \left (2 \sqrt [12]{-a} x+\sqrt [12]{b}\right )}{\sqrt [6]{-a} x^2+\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{2 \sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\text {arctanh}\left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {1}{2} \sqrt [12]{b} \int \frac {1}{\sqrt [6]{-a} x^2-\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx+\frac {\sqrt {3} \int \frac {\sqrt [12]{-a} \left (\sqrt {3} \sqrt [12]{b}-2 \sqrt [12]{-a} x\right )}{\sqrt [6]{-a} x^2-\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{2 \sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\frac {1}{2} \sqrt [12]{b} \int \frac {1}{\sqrt [6]{-a} x^2+\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx+\frac {\sqrt {3} \int \frac {\sqrt [12]{-a} \left (2 \sqrt [12]{-a} x+\sqrt {3} \sqrt [12]{b}\right )}{\sqrt [6]{-a} x^2+\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{2 \sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\arctan \left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}+\frac {\frac {\frac {3}{2} \sqrt [12]{b} \int \frac {1}{\sqrt [6]{-a} x^2-\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx+\frac {\int \frac {\sqrt [12]{-a} \left (\sqrt [12]{b}-2 \sqrt [12]{-a} x\right )}{\sqrt [6]{-a} x^2-\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{2 \sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\frac {3}{2} \sqrt [12]{b} \int \frac {1}{\sqrt [6]{-a} x^2+\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx+\frac {\int \frac {\sqrt [12]{-a} \left (2 \sqrt [12]{-a} x+\sqrt [12]{b}\right )}{\sqrt [6]{-a} x^2+\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{2 \sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\text {arctanh}\left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {1}{2} \sqrt [12]{b} \int \frac {1}{\sqrt [6]{-a} x^2-\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [12]{b}-2 \sqrt [12]{-a} x}{\sqrt [6]{-a} x^2-\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\frac {1}{2} \sqrt [12]{b} \int \frac {1}{\sqrt [6]{-a} x^2+\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [12]{-a} x+\sqrt {3} \sqrt [12]{b}}{\sqrt [6]{-a} x^2+\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\arctan \left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}+\frac {\frac {\frac {3}{2} \sqrt [12]{b} \int \frac {1}{\sqrt [6]{-a} x^2-\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx+\frac {1}{2} \int \frac {\sqrt [12]{b}-2 \sqrt [12]{-a} x}{\sqrt [6]{-a} x^2-\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\frac {3}{2} \sqrt [12]{b} \int \frac {1}{\sqrt [6]{-a} x^2+\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx+\frac {1}{2} \int \frac {2 \sqrt [12]{-a} x+\sqrt [12]{b}}{\sqrt [6]{-a} x^2+\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx}{6 b^{5/12}}+\frac {\text {arctanh}\left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [12]{b}-2 \sqrt [12]{-a} x}{\sqrt [6]{-a} x^2-\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx+\frac {\int \frac {1}{-\left (1-\frac {2 \sqrt [12]{-a} x}{\sqrt {3} \sqrt [12]{b}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [12]{-a} x}{\sqrt {3} \sqrt [12]{b}}\right )}{\sqrt {3} \sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [12]{-a} x+\sqrt {3} \sqrt [12]{b}}{\sqrt [6]{-a} x^2+\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx-\frac {\int \frac {1}{-\left (\frac {2 \sqrt [12]{-a} x}{\sqrt {3} \sqrt [12]{b}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [12]{-a} x}{\sqrt {3} \sqrt [12]{b}}+1\right )}{\sqrt {3} \sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\arctan \left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}+\frac {\frac {\frac {1}{2} \int \frac {\sqrt [12]{b}-2 \sqrt [12]{-a} x}{\sqrt [6]{-a} x^2-\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [12]{-a} x}{\sqrt [12]{b}}\right )^2-3}d\left (1-\frac {2 \sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{\sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\frac {1}{2} \int \frac {2 \sqrt [12]{-a} x+\sqrt [12]{b}}{\sqrt [6]{-a} x^2+\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx-\frac {3 \int \frac {1}{-\left (\frac {2 \sqrt [12]{-a} x}{\sqrt [12]{b}}+1\right )^2-3}d\left (\frac {2 \sqrt [12]{-a} x}{\sqrt [12]{b}}+1\right )}{\sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\text {arctanh}\left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {\frac {1}{2} \int \frac {\sqrt [12]{b}-2 \sqrt [12]{-a} x}{\sqrt [6]{-a} x^2-\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [12]{-a} x}{\sqrt [12]{b}}}{\sqrt {3}}\right )}{\sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\frac {1}{2} \int \frac {2 \sqrt [12]{-a} x+\sqrt [12]{b}}{\sqrt [6]{-a} x^2+\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [12]{-a} x}{\sqrt [12]{b}}+1}{\sqrt {3}}\right )}{\sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\text {arctanh}\left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}+\frac {\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [12]{b}-2 \sqrt [12]{-a} x}{\sqrt [6]{-a} x^2-\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [12]{-a} x}{\sqrt {3} \sqrt [12]{b}}\right )\right )}{\sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [12]{-a} x+\sqrt {3} \sqrt [12]{b}}{\sqrt [6]{-a} x^2+\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{b}}dx+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [12]{-a} x}{\sqrt {3} \sqrt [12]{b}}+1\right )\right )}{\sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\arctan \left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [12]{-a} x}{\sqrt [12]{b}}}{\sqrt {3}}\right )}{\sqrt [12]{-a}}-\frac {\log \left (-\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{-a} x^2+\sqrt [6]{b}\right )}{2 \sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [12]{-a} x}{\sqrt [12]{b}}+1}{\sqrt {3}}\right )}{\sqrt [12]{-a}}+\frac {\log \left (\sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{-a} x^2+\sqrt [6]{b}\right )}{2 \sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\text {arctanh}\left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}+\frac {\frac {-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [12]{-a} x}{\sqrt {3} \sqrt [12]{b}}\right )\right )}{\sqrt [12]{-a}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{-a} x^2+\sqrt [6]{b}\right )}{2 \sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [12]{-a} x}{\sqrt {3} \sqrt [12]{b}}+1\right )\right )}{\sqrt [12]{-a}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [12]{-a} \sqrt [12]{b} x+\sqrt [6]{-a} x^2+\sqrt [6]{b}\right )}{2 \sqrt [12]{-a}}}{6 b^{5/12}}+\frac {\arctan \left (\frac {\sqrt [12]{-a} x}{\sqrt [12]{b}}\right )}{3 \sqrt [12]{-a} b^{5/12}}}{2 \sqrt {b}}\)

Input:

Int[(b + a*x^12)^(-1),x]
 

Output:

(ArcTanh[((-a)^(1/12)*x)/b^(1/12)]/(3*(-a)^(1/12)*b^(5/12)) + (-((Sqrt[3]* 
ArcTan[(1 - (2*(-a)^(1/12)*x)/b^(1/12))/Sqrt[3]])/(-a)^(1/12)) - Log[b^(1/ 
6) - (-a)^(1/12)*b^(1/12)*x + (-a)^(1/6)*x^2]/(2*(-a)^(1/12)))/(6*b^(5/12) 
) + ((Sqrt[3]*ArcTan[(1 + (2*(-a)^(1/12)*x)/b^(1/12))/Sqrt[3]])/(-a)^(1/12 
) + Log[b^(1/6) + (-a)^(1/12)*b^(1/12)*x + (-a)^(1/6)*x^2]/(2*(-a)^(1/12)) 
)/(6*b^(5/12)))/(2*Sqrt[b]) + (ArcTan[((-a)^(1/12)*x)/b^(1/12)]/(3*(-a)^(1 
/12)*b^(5/12)) + (-(ArcTan[Sqrt[3]*(1 - (2*(-a)^(1/12)*x)/(Sqrt[3]*b^(1/12 
)))]/(-a)^(1/12)) - (Sqrt[3]*Log[b^(1/6) - Sqrt[3]*(-a)^(1/12)*b^(1/12)*x 
+ (-a)^(1/6)*x^2])/(2*(-a)^(1/12)))/(6*b^(5/12)) + (ArcTan[Sqrt[3]*(1 + (2 
*(-a)^(1/12)*x)/(Sqrt[3]*b^(1/12)))]/(-a)^(1/12) + (Sqrt[3]*Log[b^(1/6) + 
Sqrt[3]*(-a)^(1/12)*b^(1/12)*x + (-a)^(1/6)*x^2])/(2*(-a)^(1/12)))/(6*b^(5 
/12)))/(2*Sqrt[b])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 753

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/ 
b, n]], s = Denominator[Rt[a/b, n]], k, u, v}, Simp[u = Int[(r - s*Cos[(2*k 
 - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[ 
(r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2* 
x^2), x]; 2*(r^2/(a*n))   Int[1/(r^2 + s^2*x^2), x] + 2*(r/(a*n))   Sum[u, 
{k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && PosQ[a 
/b]
 

rule 754

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a 
/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k* 
Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2 
*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n)) 
 Int[1/(r^2 - s^2*x^2), x] + 2*(r/(a*n))   Sum[u, {k, 1, (n - 2)/4}], x]] / 
; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
 

rule 758

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b 
, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^(n/2)), 
 x], x] + Simp[r/(2*a)   Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] 
 && IGtQ[n/4, 1] &&  !GtQ[a/b, 0]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.03

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{12}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{11}}}{12 a}\)	\(27\)
risch	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{12}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{11}}}{12 a}\)	\(27\)

Input:

int(1/(a*x^12+b),x,method=_RETURNVERBOSE)
 

Output:

1/12/a*sum(1/_R^11*ln(x-_R),_R=RootOf(_Z^12*a+b))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.67 (sec) , antiderivative size = 639, normalized size of antiderivative = 0.77 \[ \int \frac {1}{b+a x^{12}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a*x^12+b),x, algorithm="fricas")
 

Output:

1/12*(1/2*I*sqrt(3) + 1/2)^(3/2)*(-1/(a*b^11))^(1/12)*log(b*(1/2*I*sqrt(3) 
 + 1/2)^(3/2)*(-1/(a*b^11))^(1/12) + x) - 1/12*(1/2*I*sqrt(3) + 1/2)^(3/2) 
*(-1/(a*b^11))^(1/12)*log(-b*(1/2*I*sqrt(3) + 1/2)^(3/2)*(-1/(a*b^11))^(1/ 
12) + x) + 1/24*(-I*sqrt(3) - 1)*(-1/(a*b^11))^(1/12)*log(1/2*b*(-I*sqrt(3 
) - 1)*(-1/(a*b^11))^(1/12) + x) - 1/24*(-I*sqrt(3) - 1)*(-1/(a*b^11))^(1/ 
12)*log(-1/2*b*(-I*sqrt(3) - 1)*(-1/(a*b^11))^(1/12) + x) + 1/12*((1/2*I*s 
qrt(3) + 1/2)^(3/2)*(-1/(a*b^11))^(1/12) - sqrt(1/2*I*sqrt(3) + 1/2)*(-1/( 
a*b^11))^(1/12))*log(b*(1/2*I*sqrt(3) + 1/2)^(3/2)*(-1/(a*b^11))^(1/12) - 
b*sqrt(1/2*I*sqrt(3) + 1/2)*(-1/(a*b^11))^(1/12) + x) - 1/12*((1/2*I*sqrt( 
3) + 1/2)^(3/2)*(-1/(a*b^11))^(1/12) - sqrt(1/2*I*sqrt(3) + 1/2)*(-1/(a*b^ 
11))^(1/12))*log(-b*(1/2*I*sqrt(3) + 1/2)^(3/2)*(-1/(a*b^11))^(1/12) + b*s 
qrt(1/2*I*sqrt(3) + 1/2)*(-1/(a*b^11))^(1/12) + x) + 1/24*((-I*sqrt(3) - 1 
)*(-1/(a*b^11))^(1/12) + 2*(-1/(a*b^11))^(1/12))*log(1/2*b*(-I*sqrt(3) - 1 
)*(-1/(a*b^11))^(1/12) + b*(-1/(a*b^11))^(1/12) + x) - 1/24*((-I*sqrt(3) - 
 1)*(-1/(a*b^11))^(1/12) + 2*(-1/(a*b^11))^(1/12))*log(-1/2*b*(-I*sqrt(3) 
- 1)*(-1/(a*b^11))^(1/12) - b*(-1/(a*b^11))^(1/12) + x) + 1/12*sqrt(1/2*I* 
sqrt(3) + 1/2)*(-1/(a*b^11))^(1/12)*log(b*sqrt(1/2*I*sqrt(3) + 1/2)*(-1/(a 
*b^11))^(1/12) + x) - 1/12*sqrt(1/2*I*sqrt(3) + 1/2)*(-1/(a*b^11))^(1/12)* 
log(-b*sqrt(1/2*I*sqrt(3) + 1/2)*(-1/(a*b^11))^(1/12) + x) + 1/12*(-1/(a*b 
^11))^(1/12)*log(b*(-1/(a*b^11))^(1/12) + x) - 1/12*(-1/(a*b^11))^(1/12...
 

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.02 \[ \int \frac {1}{b+a x^{12}} \, dx=\operatorname {RootSum} {\left (8916100448256 t^{12} a b^{11} + 1, \left ( t \mapsto t \log {\left (12 t b + x \right )} \right )\right )} \] Input:

integrate(1/(a*x**12+b),x)
 

Output:

RootSum(8916100448256*_t**12*a*b**11 + 1, Lambda(_t, _t*log(12*_t*b + x)))
 

Maxima [F]

\[ \int \frac {1}{b+a x^{12}} \, dx=\int { \frac {1}{a x^{12} + b} \,d x } \] Input:

integrate(1/(a*x^12+b),x, algorithm="maxima")
 

Output:

integrate(1/(a*x^12 + b), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 647, normalized size of antiderivative = 0.78 \[ \int \frac {1}{b+a x^{12}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a*x^12+b),x, algorithm="giac")
 

Output:

1/24*(sqrt(6) + sqrt(2))*(b/a)^(1/12)*arctan(((sqrt(6) - sqrt(2))*(b/a)^(1 
/12) + 4*x)/((sqrt(6) + sqrt(2))*(b/a)^(1/12)))/b + 1/24*(sqrt(6) + sqrt(2 
))*(b/a)^(1/12)*arctan(-((sqrt(6) - sqrt(2))*(b/a)^(1/12) - 4*x)/((sqrt(6) 
 + sqrt(2))*(b/a)^(1/12)))/b + 1/12*sqrt(2)*(b/a)^(1/12)*arctan(1/2*sqrt(2 
)*(2*x + sqrt(2)*(b/a)^(1/12))/(b/a)^(1/12))/b + 1/12*sqrt(2)*(b/a)^(1/12) 
*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(b/a)^(1/12))/(b/a)^(1/12))/b + 1/24*sq 
rt(2)*(b/a)^(1/12)*log(x^2 + sqrt(2)*x*(b/a)^(1/12) + (b/a)^(1/6))/b - 1/2 
4*sqrt(2)*(b/a)^(1/12)*log(x^2 - sqrt(2)*x*(b/a)^(1/12) + (b/a)^(1/6))/b + 
 1/6*(b/a)^(1/12)*arctan(((sqrt(6) + sqrt(2))*(b/a)^(1/12) + 4*x)/((sqrt(6 
) - sqrt(2))*(b/a)^(1/12)))/(b*(sqrt(6) + sqrt(2))) + 1/6*(b/a)^(1/12)*arc 
tan(-((sqrt(6) + sqrt(2))*(b/a)^(1/12) - 4*x)/((sqrt(6) - sqrt(2))*(b/a)^( 
1/12)))/(b*(sqrt(6) + sqrt(2))) + 1/12*(b/a)^(1/12)*log(x^2 + 1/2*x*(sqrt( 
6)*(b/a)^(1/12) + sqrt(2)*(b/a)^(1/12)) + (b/a)^(1/6))/(b*(sqrt(6) - sqrt( 
2))) - 1/12*(b/a)^(1/12)*log(x^2 - 1/2*x*(sqrt(6)*(b/a)^(1/12) + sqrt(2)*( 
b/a)^(1/12)) + (b/a)^(1/6))/(b*(sqrt(6) - sqrt(2))) + 1/12*(b/a)^(1/12)*lo 
g(x^2 + 1/2*x*(sqrt(6)*(b/a)^(1/12) - sqrt(2)*(b/a)^(1/12)) + (b/a)^(1/6)) 
/(b*(sqrt(6) + sqrt(2))) - 1/12*(b/a)^(1/12)*log(x^2 - 1/2*x*(sqrt(6)*(b/a 
)^(1/12) - sqrt(2)*(b/a)^(1/12)) + (b/a)^(1/6))/(b*(sqrt(6) + sqrt(2)))
 

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 435, normalized size of antiderivative = 0.52 \[ \int \frac {1}{b+a x^{12}} \, dx =\text {Too large to display} \] Input:

int(1/(b + a*x^12),x)
 

Output:

atan(((-a)^(1/12)*x)/b^(1/12))/(6*(-a)^(1/12)*b^(11/12)) - (atan(((-a)^(1/ 
12)*x*1i)/b^(1/12))*1i)/(6*(-a)^(1/12)*b^(11/12)) - (atan(((-a)^(131/12)*x 
*1i)/(b^(11/12)*((-a)^(65/6)/b^(5/6) + (3^(1/2)*(-a)^(65/6)*1i)/b^(5/6))) 
+ (3^(1/2)*(-a)^(131/12)*x)/(b^(11/12)*((-a)^(65/6)/b^(5/6) + (3^(1/2)*(-a 
)^(65/6)*1i)/b^(5/6))))*(3^(1/2)*1i - 1)*1i)/(12*(-a)^(1/12)*b^(11/12)) + 
(atan(((-a)^(131/12)*x*1i)/(b^(11/12)*((-a)^(65/6)/b^(5/6) - (3^(1/2)*(-a) 
^(65/6)*1i)/b^(5/6))) - (3^(1/2)*(-a)^(131/12)*x)/(b^(11/12)*((-a)^(65/6)/ 
b^(5/6) - (3^(1/2)*(-a)^(65/6)*1i)/b^(5/6))))*(3^(1/2)*1i + 1)*1i)/(12*(-a 
)^(1/12)*b^(11/12)) - (atan(((-a)^(131/12)*x)/(b^(11/12)*((-a)^(65/6)/b^(5 
/6) - (3^(1/2)*(-a)^(65/6)*1i)/b^(5/6))) + (3^(1/2)*(-a)^(131/12)*x*1i)/(b 
^(11/12)*((-a)^(65/6)/b^(5/6) - (3^(1/2)*(-a)^(65/6)*1i)/b^(5/6))))*(3^(1/ 
2) - 1i)*1i)/(12*(-a)^(1/12)*b^(11/12)) + (atan(((-a)^(131/12)*x)/(b^(11/1 
2)*((-a)^(65/6)/b^(5/6) + (3^(1/2)*(-a)^(65/6)*1i)/b^(5/6))) - (3^(1/2)*(- 
a)^(131/12)*x*1i)/(b^(11/12)*((-a)^(65/6)/b^(5/6) + (3^(1/2)*(-a)^(65/6)*1 
i)/b^(5/6))))*(3^(1/2) + 1i)*1i)/(12*(-a)^(1/12)*b^(11/12))
 

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 649, normalized size of antiderivative = 0.78 \[ \int \frac {1}{b+a x^{12}} \, dx =\text {Too large to display} \] Input:

int(1/(a*x^12+b),x)
 

Output:

(b**(1/12)*a**(1/4)*( - 4*sqrt( - sqrt(3) + 2)*atan((b**(1/12)*a**(1/12)*s 
qrt(6) + b**(1/12)*a**(1/12)*sqrt(2) - 4*a**(1/6)*x)/(2*b**(1/12)*a**(1/12 
)*sqrt( - sqrt(3) + 2))) + 4*sqrt( - sqrt(3) + 2)*atan((b**(1/12)*a**(1/12 
)*sqrt(6) + b**(1/12)*a**(1/12)*sqrt(2) + 4*a**(1/6)*x)/(2*b**(1/12)*a**(1 
/12)*sqrt( - sqrt(3) + 2))) - 4*sqrt(2)*atan((b**(1/12)*a**(1/12)*sqrt(2) 
- 2*a**(1/6)*x)/(b**(1/12)*a**(1/12)*sqrt(2))) + 4*sqrt(2)*atan((b**(1/12) 
*a**(1/12)*sqrt(2) + 2*a**(1/6)*x)/(b**(1/12)*a**(1/12)*sqrt(2))) - 2*sqrt 
(6)*atan((2*b**(1/12)*a**(1/12)*sqrt( - sqrt(3) + 2) - 4*a**(1/6)*x)/(b**( 
1/12)*a**(1/12)*sqrt(6) + b**(1/12)*a**(1/12)*sqrt(2))) - 2*sqrt(2)*atan(( 
2*b**(1/12)*a**(1/12)*sqrt( - sqrt(3) + 2) - 4*a**(1/6)*x)/(b**(1/12)*a**( 
1/12)*sqrt(6) + b**(1/12)*a**(1/12)*sqrt(2))) + 2*sqrt(6)*atan((2*b**(1/12 
)*a**(1/12)*sqrt( - sqrt(3) + 2) + 4*a**(1/6)*x)/(b**(1/12)*a**(1/12)*sqrt 
(6) + b**(1/12)*a**(1/12)*sqrt(2))) + 2*sqrt(2)*atan((2*b**(1/12)*a**(1/12 
)*sqrt( - sqrt(3) + 2) + 4*a**(1/6)*x)/(b**(1/12)*a**(1/12)*sqrt(6) + b**( 
1/12)*a**(1/12)*sqrt(2))) - 2*sqrt( - sqrt(3) + 2)*log( - b**(1/12)*a**(1/ 
12)*sqrt( - sqrt(3) + 2)*x + a**(1/6)*x**2 + b**(1/6)) + 2*sqrt( - sqrt(3) 
 + 2)*log(b**(1/12)*a**(1/12)*sqrt( - sqrt(3) + 2)*x + a**(1/6)*x**2 + b** 
(1/6)) - sqrt(6)*log(( - b**(1/12)*a**(1/12)*sqrt(6)*x - b**(1/12)*a**(1/1 
2)*sqrt(2)*x + 2*a**(1/6)*x**2 + 2*b**(1/6))/2) + sqrt(6)*log((b**(1/12)*a 
**(1/12)*sqrt(6)*x + b**(1/12)*a**(1/12)*sqrt(2)*x + 2*a**(1/6)*x**2 + ...