Optimal. Leaf size=11 \[ \tanh (x)-\frac{\tanh ^3(x)}{3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0190848, antiderivative size = 11, 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} \[ \tanh (x)-\frac{\tanh ^3(x)}{3} \]
Antiderivative was successfully verified.
[In]
[Out]
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*}
Mathematica [A] time = 0.0027172, size = 17, normalized size = 1.55 \[ \frac{2 \tanh (x)}{3}+\frac{1}{3} \tanh (x) \text{sech}^2(x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.015, size = 36, normalized size = 3.3 \begin{align*} -2\,{\frac{- \left ( \tanh \left ( x/2 \right ) \right ) ^{5}-2/3\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}-\tanh \left ( x/2 \right ) }{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.03861, size = 66, normalized size = 6. \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.79279, size = 286, normalized size = 26. \begin{align*} -\frac{8 \,{\left (2 \, \cosh \left (x\right ) + \sinh \left (x\right )\right )}}{3 \,{\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} +{\left (10 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{3} +{\left (10 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} +{\left (5 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right ) + 4 \, \cosh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 4.90601, size = 104, normalized size = 9.45 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.27509, size = 24, normalized size = 2.18 \begin{align*} -\frac{4 \,{\left (3 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]