\(\int \frac {-b+a x^8}{(b+a x^8) \sqrt [4]{b-c x^4+a x^8}} \, dx\) [2130]

Optimal result

Integrand size = 36, antiderivative size = 154 \[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b-c x^4+a x^8}}{-\sqrt {c} x^2+\sqrt {b-c x^4+a x^8}}\right )}{2 \sqrt {2} \sqrt [4]{c}}-\frac {\text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {b-c x^4+a x^8}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{b-c x^4+a x^8}}\right )}{2 \sqrt {2} \sqrt [4]{c}} \] Output:

-1/4*arctan(2^(1/2)*c^(1/4)*x*(a*x^8-c*x^4+b)^(1/4)/(-c^(1/2)*x^2+(a*x^8-c 
*x^4+b)^(1/2)))*2^(1/2)/c^(1/4)-1/4*arctanh((1/2*c^(1/4)*x^2*2^(1/2)+1/2*( 
a*x^8-c*x^4+b)^(1/2)*2^(1/2)/c^(1/4))/x/(a*x^8-c*x^4+b)^(1/4))*2^(1/2)/c^( 
1/4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.89 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.87 \[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b-c x^4+a x^8}}{-\sqrt {c} x^2+\sqrt {b-c x^4+a x^8}}\right )+\text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {b-c x^4+a x^8}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b-c x^4+a x^8}}\right )}{2 \sqrt {2} \sqrt [4]{c}} \] Input:

Integrate[(-b + a*x^8)/((b + a*x^8)*(b - c*x^4 + a*x^8)^(1/4)),x]
 

Output:

-1/2*(ArcTan[(Sqrt[2]*c^(1/4)*x*(b - c*x^4 + a*x^8)^(1/4))/(-(Sqrt[c]*x^2) 
 + Sqrt[b - c*x^4 + a*x^8])] + ArcTanh[(Sqrt[c]*x^2 + Sqrt[b - c*x^4 + a*x 
^8])/(Sqrt[2]*c^(1/4)*x*(b - c*x^4 + a*x^8)^(1/4))])/(Sqrt[2]*c^(1/4))
 

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.

\(\displaystyle \int \frac {a x^8-b}{\left (a x^8+b\right ) \sqrt [4]{a x^8+b-c x^4}} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {1}{\sqrt [4]{a x^8+b-c x^4}}-\frac {2 b}{\left (a x^8+b\right ) \sqrt [4]{a x^8+b-c x^4}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\sqrt {b} \int \frac {1}{\left (\sqrt {b}-\sqrt {-a} x^4\right ) \sqrt [4]{a x^8-c x^4+b}}dx-\sqrt {b} \int \frac {1}{\left (\sqrt {-a} x^4+\sqrt {b}\right ) \sqrt [4]{a x^8-c x^4+b}}dx+\frac {x \sqrt [4]{1-\frac {2 a x^4}{c-\sqrt {c^2-4 a b}}} \sqrt [4]{1-\frac {2 a x^4}{\sqrt {c^2-4 a b}+c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{4},\frac {1}{4},\frac {5}{4},\frac {2 a x^4}{c+\sqrt {c^2-4 a b}},\frac {2 a x^4}{c-\sqrt {c^2-4 a b}}\right )}{\sqrt [4]{a x^8+b-c x^4}}\)

Input:

Int[(-b + a*x^8)/((b + a*x^8)*(b - c*x^4 + a*x^8)^(1/4)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.89 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.09

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{8}-c \,x^{4}+b}}{\left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{8}-c \,x^{4}+b}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}+1\right )\right )}{8 c^{\frac {1}{4}}}\)	\(168\)

Input:

int((a*x^8-b)/(a*x^8+b)/(a*x^8-c*x^4+b)^(1/4),x,method=_RETURNVERBOSE)
 

Output:

1/8/c^(1/4)*2^(1/2)*(ln((-(a*x^8-c*x^4+b)^(1/4)*x*c^(1/4)*2^(1/2)+c^(1/2)* 
x^2+(a*x^8-c*x^4+b)^(1/2))/((a*x^8-c*x^4+b)^(1/4)*x*c^(1/4)*2^(1/2)+c^(1/2 
)*x^2+(a*x^8-c*x^4+b)^(1/2)))+2*arctan(2^(1/2)/c^(1/4)*(a*x^8-c*x^4+b)^(1/ 
4)/x+1)-2*arctan(-2^(1/2)/c^(1/4)*(a*x^8-c*x^4+b)^(1/4)/x+1))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=\text {Timed out} \] Input:

integrate((a*x^8-b)/(a*x^8+b)/(a*x^8-c*x^4+b)^(1/4),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=\int \frac {a x^{8} - b}{\left (a x^{8} + b\right ) \sqrt [4]{a x^{8} + b - c x^{4}}}\, dx \] Input:

integrate((a*x**8-b)/(a*x**8+b)/(a*x**8-c*x**4+b)**(1/4),x)
 

Output:

Integral((a*x**8 - b)/((a*x**8 + b)*(a*x**8 + b - c*x**4)**(1/4)), x)
 

Maxima [F]

\[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=\int { \frac {a x^{8} - b}{{\left (a x^{8} - c x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{8} + b\right )}} \,d x } \] Input:

integrate((a*x^8-b)/(a*x^8+b)/(a*x^8-c*x^4+b)^(1/4),x, algorithm="maxima")
 

Output:

integrate((a*x^8 - b)/((a*x^8 - c*x^4 + b)^(1/4)*(a*x^8 + b)), x)
 

Giac [F]

\[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=\int { \frac {a x^{8} - b}{{\left (a x^{8} - c x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{8} + b\right )}} \,d x } \] Input:

integrate((a*x^8-b)/(a*x^8+b)/(a*x^8-c*x^4+b)^(1/4),x, algorithm="giac")
 

Output:

integrate((a*x^8 - b)/((a*x^8 - c*x^4 + b)^(1/4)*(a*x^8 + b)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=\int -\frac {b-a\,x^8}{\left (a\,x^8+b\right )\,{\left (a\,x^8-c\,x^4+b\right )}^{1/4}} \,d x \] Input:

int(-(b - a*x^8)/((b + a*x^8)*(b + a*x^8 - c*x^4)^(1/4)),x)
                                                                                    
                                                                                    
 

Output:

int(-(b - a*x^8)/((b + a*x^8)*(b + a*x^8 - c*x^4)^(1/4)), x)
 

Reduce [F]

\[ \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx=\left (\int \frac {x^{8}}{\left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}} a \,x^{8}+\left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}} b}d x \right ) a -\left (\int \frac {1}{\left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}} a \,x^{8}+\left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}} b}d x \right ) b \] Input:

int((a*x^8-b)/(a*x^8+b)/(a*x^8-c*x^4+b)^(1/4),x)
 

Output:

int(x**8/((a*x**8 + b - c*x**4)**(1/4)*a*x**8 + (a*x**8 + b - c*x**4)**(1/ 
4)*b),x)*a - int(1/((a*x**8 + b - c*x**4)**(1/4)*a*x**8 + (a*x**8 + b - c* 
x**4)**(1/4)*b),x)*b