\(\int \frac {-2 b+c x^2}{(-b+c x^2) \sqrt [4]{-b+c x^2+a x^4}} \, dx\) [858]

Optimal result

Integrand size = 39, antiderivative size = 65 \[ \int \frac {-2 b+c x^2}{\left (-b+c x^2\right ) \sqrt [4]{-b+c x^2+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+c x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+c x^2+a x^4}}\right )}{\sqrt [4]{a}} \] Output:

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.91 \[ \int \frac {-2 b+c x^2}{\left (-b+c x^2\right ) \sqrt [4]{-b+c x^2+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+c x^2+a x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+c x^2+a x^4}}\right )}{\sqrt [4]{a}} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 2260

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

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.26

method	result	size
pseudoelliptic	\(\frac {-2 \arctan \left (\frac {\left (a \,x^{4}+c \,x^{2}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+\ln \left (\frac {a^{\frac {1}{4}} x +\left (a \,x^{4}+c \,x^{2}-b \right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (a \,x^{4}+c \,x^{2}-b \right )^{\frac {1}{4}}}\right )}{2 a^{\frac {1}{4}}}\)	\(82\)

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.06 \[ \int \frac {-2 b+c x^2}{\left (-b+c x^2\right ) \sqrt [4]{-b+c x^2+a x^4}} \, dx=\frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {i \, a^{\frac {1}{4}} x + {\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {-i \, a^{\frac {1}{4}} x + {\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} \] Input:

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

Output:

1/2*log((a^(1/4)*x + (a*x^4 + c*x^2 - b)^(1/4))/x)/a^(1/4) - 1/2*log(-(a^( 
1/4)*x - (a*x^4 + c*x^2 - b)^(1/4))/x)/a^(1/4) - 1/2*I*log((I*a^(1/4)*x + 
(a*x^4 + c*x^2 - b)^(1/4))/x)/a^(1/4) + 1/2*I*log((-I*a^(1/4)*x + (a*x^4 + 
 c*x^2 - b)^(1/4))/x)/a^(1/4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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