Optimal. Leaf size=25 \[ \frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{2 \sqrt {2}}+\frac {\tanh (x)}{2} \]
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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3288, 396, 212}
\begin {gather*} \frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{2 \sqrt {2}}+\frac {\tanh (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 396
Rule 3288
Rubi steps
\begin {align*} \int \frac {1}{1-\sinh ^4(x)} \, dx &=\text {Subst}\left (\int \frac {1-x^2}{1-2 x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh (x)}{2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{2 \sqrt {2}}+\frac {\tanh (x)}{2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 24, normalized size = 0.96 \begin {gather*} \frac {1}{4} \left (\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )+2 \tanh (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(467\) vs. \(2(25)=50\).
time = 30.11, size = 359, normalized size = 14.36 \begin {gather*} \frac {-2167073 \sqrt {2} \left (-1+\text {Cosh}\left [x\right ]\right ) \left (2167073+1532352 \sqrt {2}+1532352 \sqrt {2} \text {Tanh}\left [\frac {x}{2}\right ]^2+2167073 \text {Tanh}\left [\frac {x}{2}\right ]^2\right ) \left (\text {Log}\left [-1-\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]+\text {Log}\left [1-\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]\right )+4334146 \left (2167073+1532352 \sqrt {2}\right ) \sqrt {2} \text {Cosh}\left [x\right ] \left (\text {Log}\left [-1+\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]+\text {Log}\left [1+\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]-\text {Log}\left [-1-\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]-\text {Log}\left [1-\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]\right )+\left (34673168+24517632 \sqrt {2}\right ) \text {Cosh}\left [x\right ] \left (-383088 \text {Log}\left [-1-\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]-383088 \text {Log}\left [1-\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]+383088 \text {Log}\left [-1+\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]+383088 \text {Log}\left [1+\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]+1532352 \sqrt {2} \text {Tanh}\left [\frac {x}{2}\right ]+2167073 \text {Tanh}\left [\frac {x}{2}\right ]\right )+\left (-1+\text {Cosh}\left [x\right ]\right ) \left (2167073+1532352 \sqrt {2}+1532352 \sqrt {2} \text {Tanh}\left [\frac {x}{2}\right ]^2+2167073 \text {Tanh}\left [\frac {x}{2}\right ]^2\right ) \left (-3064704 \text {Log}\left [-1-\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]-3064704 \text {Log}\left [1-\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]+2167073 \sqrt {2} \text {Log}\left [-1+\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]+2167073 \sqrt {2} \text {Log}\left [1+\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]+3064704 \text {Log}\left [-1+\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]+3064704 \text {Log}\left [1+\sqrt {2}+\text {Tanh}\left [\frac {x}{2}\right ]\right ]\right )}{16 \left (2167073+1532352 \sqrt {2}\right ) \text {Cosh}\left [x\right ] \left (2167073+1532352 \sqrt {2}+1532352 \sqrt {2} \text {Tanh}\left [\frac {x}{2}\right ]^2+2167073 \text {Tanh}\left [\frac {x}{2}\right ]^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs.
\(2(17)=34\).
time = 0.06, size = 55, normalized size = 2.20
method | result | size |
risch | \(-\frac {1}{1+{\mathrm e}^{2 x}}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+2 \sqrt {2}-3\right )}{8}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3-2 \sqrt {2}\right )}{8}\) | \(46\) |
default | \(\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{4}+\frac {\tanh \left (\frac {x}{2}\right )}{\tanh ^{2}\left (\frac {x}{2}\right )+1}+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{4}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs.
\(2 (17) = 34\).
time = 0.35, size = 69, normalized size = 2.76 \begin {gather*} \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} + 1}{\sqrt {2} + e^{\left (-x\right )} - 1}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - 1}{\sqrt {2} + e^{\left (-x\right )} + 1}\right ) + \frac {1}{e^{\left (-2 \, x\right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs.
\(2 (17) = 34\).
time = 0.31, size = 113, normalized size = 4.52 \begin {gather*} \frac {{\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 )} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {2} + 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}\right ) - 8}{8 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 908 vs.
\(2 (20) = 40\)
time = 2.77, size = 908, normalized size = 36.32
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs.
\(2 (17) = 34\).
time = 0.00, size = 59, normalized size = 2.36 \begin {gather*} -8 \left (\frac {1}{8 \left (\left (\mathrm {e}^{x}\right )^{2}+1\right )}+\frac {\ln \left (\frac {\left |2 \left (\mathrm {e}^{x}\right )^{2}-6-4 \sqrt {2}\right |}{\left |2 \left (\mathrm {e}^{x}\right )^{2}-6+4 \sqrt {2}\right |}\right )}{32 \sqrt {2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.40, size = 63, normalized size = 2.52 \begin {gather*} \frac {\sqrt {2}\,\ln \left (2\,{\mathrm {e}}^{2\,x}+\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{8}\right )}{8}-\frac {\sqrt {2}\,\ln \left (2\,{\mathrm {e}}^{2\,x}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{8}\right )}{8}-\frac {1}{{\mathrm {e}}^{2\,x}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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