3.588 \(\int \frac{1}{1-\sinh ^4(x)} \, dx\)

Optimal. Leaf size=25 \[ \frac{\tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{2 \sqrt{2}}+\frac{\tanh (x)}{2} \]

[Out]

ArcTanh[Sqrt[2]*Tanh[x]]/(2*Sqrt[2]) + Tanh[x]/2

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Rubi [A]  time = 0.0353424, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{\tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{2 \sqrt{2}}+\frac{\tanh (x)}{2} \]

Antiderivative was successfully verified.

[In]  Int[(1 - Sinh[x]^4)^(-1),x]

[Out]

ArcTanh[Sqrt[2]*Tanh[x]]/(2*Sqrt[2]) + Tanh[x]/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-sinh(x)**4),x)

[Out]

-Integral(1/(-sinh(x) - 1), x)/4 + I*Integral(1/(-sinh(x) + I), x)/4 - Integral(
1/(sinh(x) - 1), x)/4 + I*Integral(1/(sinh(x) + I), x)/4

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Mathematica [A]  time = 0.0555097, size = 24, normalized size = 0.96 \[ \frac{1}{4} \left (\sqrt{2} \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )+2 \tanh (x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - Sinh[x]^4)^(-1),x]

[Out]

(Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]] + 2*Tanh[x])/4

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Maple [B]  time = 0.032, size = 55, normalized size = 2.2 \[{1\tanh \left ({\frac{x}{2}} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1}}+{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) +2 \right ) } \right ) }+{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) -2 \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-sinh(x)^4),x)

[Out]

tanh(1/2*x)/(tanh(1/2*x)^2+1)+1/4*2^(1/2)*arctanh(1/4*(2*tanh(1/2*x)+2)*2^(1/2))
+1/4*2^(1/2)*arctanh(1/4*(2*tanh(1/2*x)-2)*2^(1/2))

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Maxima [A]  time = 1.505, size = 99, normalized size = 3.96 \[ \frac{1}{8} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - e^{\left (-x\right )} + 1\right )}}{2 \, \sqrt{2} + 2 \, e^{\left (-x\right )} - 2}\right ) - \frac{1}{8} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - e^{\left (-x\right )} - 1\right )}}{2 \, \sqrt{2} + 2 \, e^{\left (-x\right )} + 2}\right ) + \frac{1}{e^{\left (-2 \, x\right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(sinh(x)^4 - 1),x, algorithm="maxima")

[Out]

1/8*sqrt(2)*log(-2*(sqrt(2) - e^(-x) + 1)/((2*sqrt(2)) + 2*e^(-x) - 2)) - 1/8*sq
rt(2)*log(-2*(sqrt(2) - e^(-x) - 1)/((2*sqrt(2)) + 2*e^(-x) + 2)) + 1/(e^(-2*x)
+ 1)

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Fricas [A]  time = 0.220252, size = 157, normalized size = 6.28 \[ \frac{{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \log \left (\frac{3 \,{\left (3 \, \sqrt{2} - 4\right )} \cosh \left (x\right )^{2} - 8 \,{\left (2 \, \sqrt{2} - 3\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \,{\left (3 \, \sqrt{2} - 4\right )} \sinh \left (x\right )^{2} - 3 \, \sqrt{2} + 4}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}\right ) - 4 \, \sqrt{2}}{4 \,{\left (\sqrt{2} \cosh \left (x\right )^{2} + 2 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt{2} \sinh \left (x\right )^{2} + \sqrt{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(sinh(x)^4 - 1),x, algorithm="fricas")

[Out]

1/4*((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*log((3*(3*sqrt(2) - 4)*cosh
(x)^2 - 8*(2*sqrt(2) - 3)*cosh(x)*sinh(x) + 3*(3*sqrt(2) - 4)*sinh(x)^2 - 3*sqrt
(2) + 4)/(cosh(x)^2 + sinh(x)^2 - 3)) - 4*sqrt(2))/(sqrt(2)*cosh(x)^2 + 2*sqrt(2
)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 + sqrt(2))

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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(1/(1-sinh(x)**4),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.201693, size = 65, normalized size = 2.6 \[ -\frac{1}{8} \, \sqrt{2}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt{2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) - \frac{1}{e^{\left (2 \, x\right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(sinh(x)^4 - 1),x, algorithm="giac")

[Out]

-1/8*sqrt(2)*ln(abs(-4*sqrt(2) + 2*e^(2*x) - 6)/abs(4*sqrt(2) + 2*e^(2*x) - 6))
- 1/(e^(2*x) + 1)