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

Optimal result

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

[Out]

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)

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-2 b+c x^2}{\left (-b+c x^2\right ) \sqrt [4]{-b+c x^2+a x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

(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)

Maple [A] (verified)

Time = 2.71 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.32

method	result	size
pseudoelliptic	\(\frac {-2 \arctan \left (\frac {\left (a \,x^{4}+c \,x^{2}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+\ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}+c \,x^{2}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}+c \,x^{2}-b \right )^{\frac {1}{4}}}\right )}{2 a^{\frac {1}{4}}}\)	\(86\)

[In]

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

[Out]

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 = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.06 \[ \int \frac {-2 b+c x^2}{\left (-b+c x^2\right ) \sqrt [4]{-b+c x^2+a x^4}} \, dx=\frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {i \, a^{\frac {1}{4}} x + {\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {-i \, a^{\frac {1}{4}} x + {\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} \]

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

[In]

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

[Out]

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