Optimal. Leaf size=11 \[ \tanh (x)-\frac {\tanh ^3(x)}{3} \]
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Rubi [A] time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3175, 3767} \[ \tanh (x)-\frac {\tanh ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3767
Rubi steps
\begin {align*} \int \frac {1}{\left (1+\sinh ^2(x)\right )^2} \, dx &=\int \text {sech}^4(x) \, dx\\ &=i \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \tanh (x)\right )\\ &=\tanh (x)-\frac {\tanh ^3(x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.55 \[ \frac {2 \tanh (x)}{3}+\frac {1}{3} \tanh (x) \text {sech}^2(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.03, size = 84, normalized size = 7.64 \[ -\frac {8 \, {\left (2 \, \cosh \relax (x) + \sinh \relax (x)\right )}}{3 \, {\left (\cosh \relax (x)^{5} + 5 \, \cosh \relax (x) \sinh \relax (x)^{4} + \sinh \relax (x)^{5} + {\left (10 \, \cosh \relax (x)^{2} + 3\right )} \sinh \relax (x)^{3} + 3 \, \cosh \relax (x)^{3} + {\left (10 \, \cosh \relax (x)^{3} + 9 \, \cosh \relax (x)\right )} \sinh \relax (x)^{2} + {\left (5 \, \cosh \relax (x)^{4} + 9 \, \cosh \relax (x)^{2} + 2\right )} \sinh \relax (x) + 4 \, \cosh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 18, normalized size = 1.64 \[ -\frac {4 \, {\left (3 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 36, normalized size = 3.27 \[ -\frac {2 \left (-\left (\tanh ^{5}\left (\frac {x}{2}\right )\right )-\frac {2 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}-\tanh \left (\frac {x}{2}\right )\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 49, normalized size = 4.45 \[ \frac {4 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} + \frac {4}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 18, normalized size = 1.64 \[ -\frac {4\,\left (3\,{\mathrm {e}}^{2\,x}+1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.57, size = 104, normalized size = 9.45 \[ \frac {6 \tanh ^{5}{\left (\frac {x}{2} \right )}}{3 \tanh ^{6}{\left (\frac {x}{2} \right )} + 9 \tanh ^{4}{\left (\frac {x}{2} \right )} + 9 \tanh ^{2}{\left (\frac {x}{2} \right )} + 3} + \frac {4 \tanh ^{3}{\left (\frac {x}{2} \right )}}{3 \tanh ^{6}{\left (\frac {x}{2} \right )} + 9 \tanh ^{4}{\left (\frac {x}{2} \right )} + 9 \tanh ^{2}{\left (\frac {x}{2} \right )} + 3} + \frac {6 \tanh {\left (\frac {x}{2} \right )}}{3 \tanh ^{6}{\left (\frac {x}{2} \right )} + 9 \tanh ^{4}{\left (\frac {x}{2} \right )} + 9 \tanh ^{2}{\left (\frac {x}{2} \right )} + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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