Optimal. Leaf size=19 \[ \frac{\tanh ^5(x)}{5}-\frac{2 \tanh ^3(x)}{3}+\tanh (x) \]
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Rubi [A] time = 0.0201132, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3175, 3767} \[ \frac{\tanh ^5(x)}{5}-\frac{2 \tanh ^3(x)}{3}+\tanh (x) \]
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 )^3} \, dx &=\int \text{sech}^6(x) \, dx\\ &=i \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \tanh (x)\right )\\ &=\tanh (x)-\frac{2 \tanh ^3(x)}{3}+\frac{\tanh ^5(x)}{5}\\ \end{align*}
Mathematica [A] time = 0.0029111, size = 27, normalized size = 1.42 \[ \frac{8 \tanh (x)}{15}+\frac{1}{5} \tanh (x) \text{sech}^4(x)+\frac{4}{15} \tanh (x) \text{sech}^2(x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 52, normalized size = 2.7 \begin{align*} -2\,{\frac{1}{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{5}} \left ( - \left ( \tanh \left ( x/2 \right ) \right ) ^{9}-4/3\, \left ( \tanh \left ( x/2 \right ) \right ) ^{7}-{\frac{58\, \left ( \tanh \left ( x/2 \right ) \right ) ^{5}}{15}}-4/3\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}-\tanh \left ( x/2 \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04492, size = 150, normalized size = 7.89 \begin{align*} \frac{16 \, e^{\left (-2 \, x\right )}}{3 \,{\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} + \frac{32 \, e^{\left (-4 \, x\right )}}{3 \,{\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} + \frac{16}{15 \,{\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.78008, size = 628, normalized size = 33.05 \begin{align*} -\frac{16 \,{\left (11 \, \cosh \left (x\right )^{2} + 18 \, \cosh \left (x\right ) \sinh \left (x\right ) + 11 \, \sinh \left (x\right )^{2} + 5\right )}}{15 \,{\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} +{\left (28 \, \cosh \left (x\right )^{2} + 5\right )} \sinh \left (x\right )^{6} + 5 \, \cosh \left (x\right )^{6} + 2 \,{\left (28 \, \cosh \left (x\right )^{3} + 15 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 5 \,{\left (14 \, \cosh \left (x\right )^{4} + 15 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{4} + 10 \, \cosh \left (x\right )^{4} + 4 \,{\left (14 \, \cosh \left (x\right )^{5} + 25 \, \cosh \left (x\right )^{3} + 10 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} +{\left (28 \, \cosh \left (x\right )^{6} + 75 \, \cosh \left (x\right )^{4} + 60 \, \cosh \left (x\right )^{2} + 11\right )} \sinh \left (x\right )^{2} + 11 \, \cosh \left (x\right )^{2} + 2 \,{\left (4 \, \cosh \left (x\right )^{7} + 15 \, \cosh \left (x\right )^{5} + 20 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 18.1921, size = 260, normalized size = 13.68 \begin{align*} \frac{30 \tanh ^{9}{\left (\frac{x}{2} \right )}}{15 \tanh ^{10}{\left (\frac{x}{2} \right )} + 75 \tanh ^{8}{\left (\frac{x}{2} \right )} + 150 \tanh ^{6}{\left (\frac{x}{2} \right )} + 150 \tanh ^{4}{\left (\frac{x}{2} \right )} + 75 \tanh ^{2}{\left (\frac{x}{2} \right )} + 15} + \frac{40 \tanh ^{7}{\left (\frac{x}{2} \right )}}{15 \tanh ^{10}{\left (\frac{x}{2} \right )} + 75 \tanh ^{8}{\left (\frac{x}{2} \right )} + 150 \tanh ^{6}{\left (\frac{x}{2} \right )} + 150 \tanh ^{4}{\left (\frac{x}{2} \right )} + 75 \tanh ^{2}{\left (\frac{x}{2} \right )} + 15} + \frac{116 \tanh ^{5}{\left (\frac{x}{2} \right )}}{15 \tanh ^{10}{\left (\frac{x}{2} \right )} + 75 \tanh ^{8}{\left (\frac{x}{2} \right )} + 150 \tanh ^{6}{\left (\frac{x}{2} \right )} + 150 \tanh ^{4}{\left (\frac{x}{2} \right )} + 75 \tanh ^{2}{\left (\frac{x}{2} \right )} + 15} + \frac{40 \tanh ^{3}{\left (\frac{x}{2} \right )}}{15 \tanh ^{10}{\left (\frac{x}{2} \right )} + 75 \tanh ^{8}{\left (\frac{x}{2} \right )} + 150 \tanh ^{6}{\left (\frac{x}{2} \right )} + 150 \tanh ^{4}{\left (\frac{x}{2} \right )} + 75 \tanh ^{2}{\left (\frac{x}{2} \right )} + 15} + \frac{30 \tanh{\left (\frac{x}{2} \right )}}{15 \tanh ^{10}{\left (\frac{x}{2} \right )} + 75 \tanh ^{8}{\left (\frac{x}{2} \right )} + 150 \tanh ^{6}{\left (\frac{x}{2} \right )} + 150 \tanh ^{4}{\left (\frac{x}{2} \right )} + 75 \tanh ^{2}{\left (\frac{x}{2} \right )} + 15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27453, size = 32, normalized size = 1.68 \begin{align*} -\frac{16 \,{\left (10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} + 1\right )}}{15 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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