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