3.895 \(\int \frac{x^{31} \sqrt{1+x^{16}}}{1-x^{16}} \, dx\)

Optimal. Leaf size=52 \[ -\frac{1}{24} \left (x^{16}+1\right )^{3/2}-\frac{\sqrt{x^{16}+1}}{8}+\frac{\tanh ^{-1}\left (\frac{\sqrt{x^{16}+1}}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

[Out]

-Sqrt[1 + x^16]/8 - (1 + x^16)^(3/2)/24 + ArcTanh[Sqrt[1 + x^16]/Sqrt[2]]/(4*Sqr
t[2])

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Rubi [A]  time = 0.0994686, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{1}{24} \left (x^{16}+1\right )^{3/2}-\frac{\sqrt{x^{16}+1}}{8}+\frac{\tanh ^{-1}\left (\frac{\sqrt{x^{16}+1}}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^31*Sqrt[1 + x^16])/(1 - x^16),x]

[Out]

-Sqrt[1 + x^16]/8 - (1 + x^16)^(3/2)/24 + ArcTanh[Sqrt[1 + x^16]/Sqrt[2]]/(4*Sqr
t[2])

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Rubi in Sympy [A]  time = 10.1705, size = 42, normalized size = 0.81 \[ - \frac{\left (x^{16} + 1\right )^{\frac{3}{2}}}{24} - \frac{\sqrt{x^{16} + 1}}{8} + \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{x^{16} + 1}}{2} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**31*(x**16+1)**(1/2)/(-x**16+1),x)

[Out]

-(x**16 + 1)**(3/2)/24 - sqrt(x**16 + 1)/8 + sqrt(2)*atanh(sqrt(2)*sqrt(x**16 +
1)/2)/8

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Mathematica [A]  time = 0.0403159, size = 44, normalized size = 0.85 \[ \frac{1}{24} \left (3 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{x^{16}+1}}{\sqrt{2}}\right )-\sqrt{x^{16}+1} \left (x^{16}+4\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x^31*Sqrt[1 + x^16])/(1 - x^16),x]

[Out]

(-(Sqrt[1 + x^16]*(4 + x^16)) + 3*Sqrt[2]*ArcTanh[Sqrt[1 + x^16]/Sqrt[2]])/24

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Maple [F]  time = 0.168, size = 0, normalized size = 0. \[ \int{\frac{{x}^{31}}{-{x}^{16}+1}\sqrt{{x}^{16}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^31*(x^16+1)^(1/2)/(-x^16+1),x)

[Out]

int(x^31*(x^16+1)^(1/2)/(-x^16+1),x)

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Maxima [A]  time = 1.52646, size = 74, normalized size = 1.42 \[ -\frac{1}{24} \,{\left (x^{16} + 1\right )}^{\frac{3}{2}} - \frac{1}{16} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \sqrt{x^{16} + 1}\right )}}{2 \, \sqrt{2} + 2 \, \sqrt{x^{16} + 1}}\right ) - \frac{1}{8} \, \sqrt{x^{16} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x^16 + 1)*x^31/(x^16 - 1),x, algorithm="maxima")

[Out]

-1/24*(x^16 + 1)^(3/2) - 1/16*sqrt(2)*log(-2*(sqrt(2) - sqrt(x^16 + 1))/((2*sqrt
(2)) + 2*sqrt(x^16 + 1))) - 1/8*sqrt(x^16 + 1)

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Fricas [A]  time = 0.228864, size = 70, normalized size = 1.35 \[ -\frac{1}{48} \, \sqrt{2}{\left (\sqrt{2}{\left (x^{16} + 4\right )} \sqrt{x^{16} + 1} - 3 \, \log \left (\frac{\sqrt{2}{\left (x^{16} + 3\right )} + 4 \, \sqrt{x^{16} + 1}}{x^{16} - 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x^16 + 1)*x^31/(x^16 - 1),x, algorithm="fricas")

[Out]

-1/48*sqrt(2)*(sqrt(2)*(x^16 + 4)*sqrt(x^16 + 1) - 3*log((sqrt(2)*(x^16 + 3) + 4
*sqrt(x^16 + 1))/(x^16 - 1)))

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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(x**31*(x**16+1)**(1/2)/(-x**16+1),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.211592, size = 76, normalized size = 1.46 \[ -\frac{1}{24} \,{\left (x^{16} + 1\right )}^{\frac{3}{2}} - \frac{1}{16} \, \sqrt{2}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \sqrt{x^{16} + 1} \right |}}{2 \,{\left (\sqrt{2} + \sqrt{x^{16} + 1}\right )}}\right ) - \frac{1}{8} \, \sqrt{x^{16} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x^16 + 1)*x^31/(x^16 - 1),x, algorithm="giac")

[Out]

-1/24*(x^16 + 1)^(3/2) - 1/16*sqrt(2)*ln(1/2*abs(-2*sqrt(2) + 2*sqrt(x^16 + 1))/
(sqrt(2) + sqrt(x^16 + 1))) - 1/8*sqrt(x^16 + 1)