\(\int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} (b-c^4 x^4+a x^6)} \, dx\) [447]

Optimal result

Integrand size = 40, antiderivative size = 35 \[ \int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx=-\arctan \left (\frac {c x}{\sqrt [4]{b+a x^6}}\right )-\text {arctanh}\left (\frac {c x}{\sqrt [4]{b+a x^6}}\right ) \]

[Out]

-arctan(c*x/(a*x^6+b)^(1/4))-arctanh(c*x/(a*x^6+b)^(1/4))

Rubi [F]

\[ \int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx=\int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx \]

[In]

Int[(-2*b*c + a*c*x^6)/((b + a*x^6)^(1/4)*(b - c^4*x^4 + a*x^6)),x]

[Out]

(c*x*(1 + (a*x^6)/b)^(1/4)*Hypergeometric2F1[1/6, 1/4, 7/6, -((a*x^6)/b)])/(b + a*x^6)^(1/4) - c^5*Defer[Int][
x^4/((-b + c^4*x^4 - a*x^6)*(b + a*x^6)^(1/4)), x] - 3*b*c*Defer[Int][1/((b + a*x^6)^(1/4)*(b - c^4*x^4 + a*x^
6)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c}{\sqrt [4]{b+a x^6}}-\frac {3 b c-c^5 x^4}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )}\right ) \, dx \\ & = c \int \frac {1}{\sqrt [4]{b+a x^6}} \, dx-\int \frac {3 b c-c^5 x^4}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx \\ & = \frac {\left (c \sqrt [4]{1+\frac {a x^6}{b}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {a x^6}{b}}} \, dx}{\sqrt [4]{b+a x^6}}-\int \left (\frac {c^5 x^4}{\left (-b+c^4 x^4-a x^6\right ) \sqrt [4]{b+a x^6}}+\frac {3 b c}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )}\right ) \, dx \\ & = \frac {c x \sqrt [4]{1+\frac {a x^6}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},-\frac {a x^6}{b}\right )}{\sqrt [4]{b+a x^6}}-(3 b c) \int \frac {1}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx-c^5 \int \frac {x^4}{\left (-b+c^4 x^4-a x^6\right ) \sqrt [4]{b+a x^6}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.57 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx=-\arctan \left (\frac {c x}{\sqrt [4]{b+a x^6}}\right )-\text {arctanh}\left (\frac {c x}{\sqrt [4]{b+a x^6}}\right ) \]

[In]

Integrate[(-2*b*c + a*c*x^6)/((b + a*x^6)^(1/4)*(b - c^4*x^4 + a*x^6)),x]

[Out]

-ArcTan[(c*x)/(b + a*x^6)^(1/4)] - ArcTanh[(c*x)/(b + a*x^6)^(1/4)]

Maple [A] (verified)

Time = 1.97 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.71

method	result	size
pseudoelliptic	\(-\frac {\ln \left (\frac {c x +\left (a \,x^{6}+b \right )^{\frac {1}{4}}}{x}\right )}{2}+\arctan \left (\frac {\left (a \,x^{6}+b \right )^{\frac {1}{4}}}{c x}\right )+\frac {\ln \left (\frac {-c x +\left (a \,x^{6}+b \right )^{\frac {1}{4}}}{x}\right )}{2}\)	\(60\)

[In]

int((a*c*x^6-2*b*c)/(a*x^6+b)^(1/4)/(-c^4*x^4+a*x^6+b),x,method=_RETURNVERBOSE)

[Out]

-1/2*ln((c*x+(a*x^6+b)^(1/4))/x)+arctan(1/c/x*(a*x^6+b)^(1/4))+1/2*ln((-c*x+(a*x^6+b)^(1/4))/x)

Fricas [F(-1)]

Timed out. \[ \int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx=\text {Timed out} \]

[In]

integrate((a*c*x^6-2*b*c)/(a*x^6+b)^(1/4)/(-c^4*x^4+a*x^6+b),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx=\text {Timed out} \]

[In]

integrate((a*c*x**6-2*b*c)/(a*x**6+b)**(1/4)/(-c**4*x**4+a*x**6+b),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx=\int { -\frac {a c x^{6} - 2 \, b c}{{\left (c^{4} x^{4} - a x^{6} - b\right )} {\left (a x^{6} + b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate((a*c*x^6-2*b*c)/(a*x^6+b)^(1/4)/(-c^4*x^4+a*x^6+b),x, algorithm="maxima")

[Out]

-integrate((a*c*x^6 - 2*b*c)/((c^4*x^4 - a*x^6 - b)*(a*x^6 + b)^(1/4)), x)

Giac [F]

\[ \int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx=\int { -\frac {a c x^{6} - 2 \, b c}{{\left (c^{4} x^{4} - a x^{6} - b\right )} {\left (a x^{6} + b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate((a*c*x^6-2*b*c)/(a*x^6+b)^(1/4)/(-c^4*x^4+a*x^6+b),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx=\int -\frac {2\,b\,c-a\,c\,x^6}{{\left (a\,x^6+b\right )}^{1/4}\,\left (-c^4\,x^4+a\,x^6+b\right )} \,d x \]

[In]

int(-(2*b*c - a*c*x^6)/((b + a*x^6)^(1/4)*(b + a*x^6 - c^4*x^4)),x)

[Out]

int(-(2*b*c - a*c*x^6)/((b + a*x^6)^(1/4)*(b + a*x^6 - c^4*x^4)), x)