Optimal. Leaf size=51 \[ \frac {19 \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{32 \sqrt {2}}-\frac {9 \sinh (x) \cosh (x)}{32 \left (\cosh ^2(x)+1\right )}-\frac {\sinh (x) \cosh (x)}{8 \left (\cosh ^2(x)+1\right )^2} \]
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Rubi [A] time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3184, 3173, 12, 3181, 206} \[ \frac {19 \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{32 \sqrt {2}}-\frac {9 \sinh (x) \cosh (x)}{32 \left (\cosh ^2(x)+1\right )}-\frac {\sinh (x) \cosh (x)}{8 \left (\cosh ^2(x)+1\right )^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 3173
Rule 3181
Rule 3184
Rubi steps
\begin {align*} \int \frac {1}{\left (1+\cosh ^2(x)\right )^3} \, dx &=-\frac {\cosh (x) \sinh (x)}{8 \left (1+\cosh ^2(x)\right )^2}-\frac {1}{8} \int \frac {-7+2 \cosh ^2(x)}{\left (1+\cosh ^2(x)\right )^2} \, dx\\ &=-\frac {\cosh (x) \sinh (x)}{8 \left (1+\cosh ^2(x)\right )^2}-\frac {9 \cosh (x) \sinh (x)}{32 \left (1+\cosh ^2(x)\right )}-\frac {1}{32} \int -\frac {19}{1+\cosh ^2(x)} \, dx\\ &=-\frac {\cosh (x) \sinh (x)}{8 \left (1+\cosh ^2(x)\right )^2}-\frac {9 \cosh (x) \sinh (x)}{32 \left (1+\cosh ^2(x)\right )}+\frac {19}{32} \int \frac {1}{1+\cosh ^2(x)} \, dx\\ &=-\frac {\cosh (x) \sinh (x)}{8 \left (1+\cosh ^2(x)\right )^2}-\frac {9 \cosh (x) \sinh (x)}{32 \left (1+\cosh ^2(x)\right )}+\frac {19}{32} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\coth (x)\right )\\ &=\frac {19 \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{32 \sqrt {2}}-\frac {\cosh (x) \sinh (x)}{8 \left (1+\cosh ^2(x)\right )^2}-\frac {9 \cosh (x) \sinh (x)}{32 \left (1+\cosh ^2(x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 51, normalized size = 1.00 \[ \frac {19 \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{32 \sqrt {2}}-\frac {9 \sinh (2 x)}{32 (\cosh (2 x)+3)}-\frac {\sinh (2 x)}{4 (\cosh (2 x)+3)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 575, normalized size = 11.27 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 71, normalized size = 1.39 \[ \frac {19}{128} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) + \frac {19 \, e^{\left (6 \, x\right )} + 171 \, e^{\left (4 \, x\right )} + 89 \, e^{\left (2 \, x\right )} + 9}{16 \, {\left (e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 129, normalized size = 2.53 \[ -\frac {\frac {11 \left (\tanh ^{7}\left (\frac {x}{2}\right )\right )}{8}+\frac {7 \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{8}+\frac {7 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{8}+\frac {11 \tanh \left (\frac {x}{2}\right )}{8}}{4 \left (\tanh ^{4}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {19 \sqrt {2}\, \ln \left (\frac {\tanh ^{2}\left (\frac {x}{2}\right )+\sqrt {2}\, \tanh \left (\frac {x}{2}\right )+1}{\tanh ^{2}\left (\frac {x}{2}\right )-\sqrt {2}\, \tanh \left (\frac {x}{2}\right )+1}\right )}{256}-\frac {19 \sqrt {2}\, \ln \left (\frac {\tanh ^{2}\left (\frac {x}{2}\right )-\sqrt {2}\, \tanh \left (\frac {x}{2}\right )+1}{\tanh ^{2}\left (\frac {x}{2}\right )+\sqrt {2}\, \tanh \left (\frac {x}{2}\right )+1}\right )}{256} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 83, normalized size = 1.63 \[ -\frac {19}{128} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) - \frac {89 \, e^{\left (-2 \, x\right )} + 171 \, e^{\left (-4 \, x\right )} + 19 \, e^{\left (-6 \, x\right )} + 9}{16 \, {\left (12 \, e^{\left (-2 \, x\right )} + 38 \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 112, normalized size = 2.20 \[ \frac {19\,\sqrt {2}\,\ln \left (-\frac {19\,{\mathrm {e}}^{2\,x}}{8}-\frac {19\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{128}\right )}{128}-\frac {17\,{\mathrm {e}}^{2\,x}+3}{12\,{\mathrm {e}}^{2\,x}+38\,{\mathrm {e}}^{4\,x}+12\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {19\,\sqrt {2}\,\ln \left (\frac {19\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{128}-\frac {19\,{\mathrm {e}}^{2\,x}}{8}\right )}{128}+\frac {\frac {19\,{\mathrm {e}}^{2\,x}}{16}+\frac {57}{16}}{6\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 13.57, size = 428, normalized size = 8.39 \[ - \frac {19 \sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )} \tanh ^{8}{\left (\frac {x}{2} \right )}}{128 \tanh ^{8}{\left (\frac {x}{2} \right )} + 256 \tanh ^{4}{\left (\frac {x}{2} \right )} + 128} - \frac {38 \sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )} \tanh ^{4}{\left (\frac {x}{2} \right )}}{128 \tanh ^{8}{\left (\frac {x}{2} \right )} + 256 \tanh ^{4}{\left (\frac {x}{2} \right )} + 128} - \frac {19 \sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )}}{128 \tanh ^{8}{\left (\frac {x}{2} \right )} + 256 \tanh ^{4}{\left (\frac {x}{2} \right )} + 128} + \frac {19 \sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )} \tanh ^{8}{\left (\frac {x}{2} \right )}}{128 \tanh ^{8}{\left (\frac {x}{2} \right )} + 256 \tanh ^{4}{\left (\frac {x}{2} \right )} + 128} + \frac {38 \sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )} \tanh ^{4}{\left (\frac {x}{2} \right )}}{128 \tanh ^{8}{\left (\frac {x}{2} \right )} + 256 \tanh ^{4}{\left (\frac {x}{2} \right )} + 128} + \frac {19 \sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )}}{128 \tanh ^{8}{\left (\frac {x}{2} \right )} + 256 \tanh ^{4}{\left (\frac {x}{2} \right )} + 128} - \frac {44 \tanh ^{7}{\left (\frac {x}{2} \right )}}{128 \tanh ^{8}{\left (\frac {x}{2} \right )} + 256 \tanh ^{4}{\left (\frac {x}{2} \right )} + 128} - \frac {28 \tanh ^{5}{\left (\frac {x}{2} \right )}}{128 \tanh ^{8}{\left (\frac {x}{2} \right )} + 256 \tanh ^{4}{\left (\frac {x}{2} \right )} + 128} - \frac {28 \tanh ^{3}{\left (\frac {x}{2} \right )}}{128 \tanh ^{8}{\left (\frac {x}{2} \right )} + 256 \tanh ^{4}{\left (\frac {x}{2} \right )} + 128} - \frac {44 \tanh {\left (\frac {x}{2} \right )}}{128 \tanh ^{8}{\left (\frac {x}{2} \right )} + 256 \tanh ^{4}{\left (\frac {x}{2} \right )} + 128} \]
Verification of antiderivative is not currently implemented for this CAS.
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