3.190 \(\int \frac{\left (1+b x^4\right )^p}{1-x^2} \, dx\)

Optimal. Leaf size=50 \[ x F_1\left (\frac{1}{4};1,-p;\frac{5}{4};x^4,-b x^4\right )+\frac{1}{3} x^3 F_1\left (\frac{3}{4};1,-p;\frac{7}{4};x^4,-b x^4\right ) \]

[Out]

x*AppellF1[1/4, 1, -p, 5/4, x^4, -(b*x^4)] + (x^3*AppellF1[3/4, 1, -p, 7/4, x^4,
-(b*x^4)])/3

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Rubi [A]  time = 0.121223, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ x F_1\left (\frac{1}{4};1,-p;\frac{5}{4};x^4,-b x^4\right )+\frac{1}{3} x^3 F_1\left (\frac{3}{4};1,-p;\frac{7}{4};x^4,-b x^4\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + b*x^4)^p/(1 - x^2),x]

[Out]

x*AppellF1[1/4, 1, -p, 5/4, x^4, -(b*x^4)] + (x^3*AppellF1[3/4, 1, -p, 7/4, x^4,
-(b*x^4)])/3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{\left (b x^{4} + 1\right )^{p}}{x^{2} - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+1)**p/(-x**2+1),x)

[Out]

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

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Mathematica [A]  time = 0.043094, size = 0, normalized size = 0. \[ \int \frac{\left (1+b x^4\right )^p}{1-x^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(1 + b*x^4)^p/(1 - x^2), x]

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{ \left ( b{x}^{4}+1 \right ) ^{p}}{-{x}^{2}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+1)^p/(-x^2+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (b x^{4} + 1\right )}^{p}}{x^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x^4 + 1)^p/(x^2 - 1),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (b x^{4} + 1\right )}^{p}}{x^{2} - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x^4 + 1)^p/(x^2 - 1),x, algorithm="fricas")

[Out]

integral(-(b*x^4 + 1)^p/(x^2 - 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+1)**p/(-x**2+1),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (b x^{4} + 1\right )}^{p}}{x^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x^4 + 1)^p/(x^2 - 1),x, algorithm="giac")

[Out]

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