\(\int \frac {-b+a x^4}{(b+a x^4) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx\) [2510]

Optimal result

Integrand size = 39, antiderivative size = 209 \[ \int \frac {-b+a x^4}{\left (b+a x^4\right ) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2 a b-c} x \sqrt [4]{b^2+c x^4+a^2 x^8}}{\sqrt {2 a b-c} x^2-\sqrt {b^2+c x^4+a^2 x^8}}\right )}{2 \sqrt {2} \sqrt [4]{2 a b-c}}-\frac {\text {arctanh}\left (\frac {\frac {\sqrt [4]{2 a b-c} x^2}{\sqrt {2}}+\frac {\sqrt {b^2+c x^4+a^2 x^8}}{\sqrt {2} \sqrt [4]{2 a b-c}}}{x \sqrt [4]{b^2+c x^4+a^2 x^8}}\right )}{2 \sqrt {2} \sqrt [4]{2 a b-c}} \] Output:

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

Mathematica [A] (verified)

Time = 2.13 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.88 \[ \int \frac {-b+a x^4}{\left (b+a x^4\right ) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2 a b-c} x \sqrt [4]{b^2+c x^4+a^2 x^8}}{\sqrt {2 a b-c} x^2-\sqrt {b^2+c x^4+a^2 x^8}}\right )-\text {arctanh}\left (\frac {\sqrt {2 a b-c} x^2+\sqrt {b^2+c x^4+a^2 x^8}}{\sqrt {2} \sqrt [4]{2 a b-c} x \sqrt [4]{b^2+c x^4+a^2 x^8}}\right )}{2 \sqrt {2} \sqrt [4]{2 a b-c}} \] Input:

Integrate[(-b + a*x^4)/((b + a*x^4)*(b^2 + c*x^4 + a^2*x^8)^(1/4)),x]
 

Output:

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

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

Int[(-b + a*x^4)/((b + a*x^4)*(b^2 + c*x^4 + a^2*x^8)^(1/4)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.19 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (a^{2} x^{8}+c \,x^{4}+b^{2}\right )^{\frac {1}{4}} x \left (2 a b -c \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {2 a b -c}\, x^{2}+\sqrt {a^{2} x^{8}+c \,x^{4}+b^{2}}}{\left (a^{2} x^{8}+c \,x^{4}+b^{2}\right )^{\frac {1}{4}} x \left (2 a b -c \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {2 a b -c}\, x^{2}+\sqrt {a^{2} x^{8}+c \,x^{4}+b^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a^{2} x^{8}+c \,x^{4}+b^{2}\right )^{\frac {1}{4}}}{\left (2 a b -c \right )^{\frac {1}{4}} x}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \left (a^{2} x^{8}+c \,x^{4}+b^{2}\right )^{\frac {1}{4}}}{\left (2 a b -c \right )^{\frac {1}{4}} x}+1\right )\right )}{8 \left (2 a b -c \right )^{\frac {1}{4}}}\)	\(235\)

Input:

int((a*x^4-b)/(a*x^4+b)/(a^2*x^8+c*x^4+b^2)^(1/4),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {-b+a x^4}{\left (b+a x^4\right ) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx=\text {Timed out} \] Input:

integrate((a*x^4-b)/(a*x^4+b)/(a^2*x^8+c*x^4+b^2)^(1/4),x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {-b+a x^4}{\left (b+a x^4\right ) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx=\int \frac {a x^{4} - b}{\left (a x^{4} + b\right ) \sqrt [4]{a^{2} x^{8} + b^{2} + c x^{4}}}\, dx \] Input:

integrate((a*x**4-b)/(a*x**4+b)/(a**2*x**8+c*x**4+b**2)**(1/4),x)
 

Output:

Integral((a*x**4 - b)/((a*x**4 + b)*(a**2*x**8 + b**2 + c*x**4)**(1/4)), x 
)
 

Maxima [F]

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

integrate((a*x^4-b)/(a*x^4+b)/(a^2*x^8+c*x^4+b^2)^(1/4),x, algorithm="maxi 
ma")
 

Output:

integrate((a*x^4 - b)/((a^2*x^8 + c*x^4 + b^2)^(1/4)*(a*x^4 + b)), x)
 

Giac [F]

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

integrate((a*x^4-b)/(a*x^4+b)/(a^2*x^8+c*x^4+b^2)^(1/4),x, algorithm="giac 
")
 

Output:

integrate((a*x^4 - b)/((a^2*x^8 + c*x^4 + b^2)^(1/4)*(a*x^4 + b)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-b+a x^4}{\left (b+a x^4\right ) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx=\int -\frac {b-a\,x^4}{\left (a\,x^4+b\right )\,{\left (a^2\,x^8+b^2+c\,x^4\right )}^{1/4}} \,d x \] Input:

int(-(b - a*x^4)/((b + a*x^4)*(c*x^4 + b^2 + a^2*x^8)^(1/4)),x)
                                                                                    
                                                                                    
 

Output:

int(-(b - a*x^4)/((b + a*x^4)*(c*x^4 + b^2 + a^2*x^8)^(1/4)), x)
 

Reduce [F]

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

int((a*x^4-b)/(a*x^4+b)/(a^2*x^8+c*x^4+b^2)^(1/4),x)
 

Output:

int(x**4/((a**2*x**8 + b**2 + c*x**4)**(1/4)*a*x**4 + (a**2*x**8 + b**2 + 
c*x**4)**(1/4)*b),x)*a - int(1/((a**2*x**8 + b**2 + c*x**4)**(1/4)*a*x**4 
+ (a**2*x**8 + b**2 + c*x**4)**(1/4)*b),x)*b