Optimal. Leaf size=36 \[ \frac{2 \tanh (x)}{3 a \sqrt{a \text{sech}^2(x)}}+\frac{\tanh (x)}{3 \left (a \text{sech}^2(x)\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0201417, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4122, 192, 191} \[ \frac{2 \tanh (x)}{3 a \sqrt{a \text{sech}^2(x)}}+\frac{\tanh (x)}{3 \left (a \text{sech}^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4122
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\left (a \text{sech}^2(x)\right )^{3/2}} \, dx &=a \operatorname{Subst}\left (\int \frac{1}{\left (a-a x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh (x)}{3 \left (a \text{sech}^2(x)\right )^{3/2}}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{\left (a-a x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh (x)}{3 \left (a \text{sech}^2(x)\right )^{3/2}}+\frac{2 \tanh (x)}{3 a \sqrt{a \text{sech}^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0211334, size = 27, normalized size = 0.75 \[ \frac{(9 \sinh (x)+\sinh (3 x)) \text{sech}^3(x)}{12 \left (a \text{sech}^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.059, size = 130, normalized size = 3.6 \begin{align*}{\frac{{{\rm e}^{4\,x}}}{24\,a \left ({{\rm e}^{2\,x}}+1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}}+{\frac{3\,{{\rm e}^{2\,x}}}{8\,a \left ({{\rm e}^{2\,x}}+1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}}-{\frac{3}{8\,a \left ({{\rm e}^{2\,x}}+1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}}-{\frac{{{\rm e}^{-2\,x}}}{24\,a \left ({{\rm e}^{2\,x}}+1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.70525, size = 47, normalized size = 1.31 \begin{align*} \frac{e^{\left (3 \, x\right )}}{24 \, a^{\frac{3}{2}}} - \frac{3 \, e^{\left (-x\right )}}{8 \, a^{\frac{3}{2}}} - \frac{e^{\left (-3 \, x\right )}}{24 \, a^{\frac{3}{2}}} + \frac{3 \, e^{x}}{8 \, a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.19873, size = 856, normalized size = 23.78 \begin{align*} \frac{{\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{6} + \cosh \left (x\right )^{6} + 6 \,{\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 3 \,{\left (5 \, \cosh \left (x\right )^{2} +{\left (5 \, \cosh \left (x\right )^{2} + 3\right )} e^{\left (2 \, x\right )} + 3\right )} \sinh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{4} + 4 \,{\left (5 \, \cosh \left (x\right )^{3} +{\left (5 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} + 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \,{\left (5 \, \cosh \left (x\right )^{4} + 18 \, \cosh \left (x\right )^{2} +{\left (5 \, \cosh \left (x\right )^{4} + 18 \, \cosh \left (x\right )^{2} - 3\right )} e^{\left (2 \, x\right )} - 3\right )} \sinh \left (x\right )^{2} - 9 \, \cosh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{6} + 9 \, \cosh \left (x\right )^{4} - 9 \, \cosh \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )} + 6 \,{\left (\cosh \left (x\right )^{5} + 6 \, \cosh \left (x\right )^{3} +{\left (\cosh \left (x\right )^{5} + 6 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )} \sqrt{\frac{a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{24 \,{\left (a^{2} \cosh \left (x\right )^{3} e^{x} + 3 \, a^{2} \cosh \left (x\right )^{2} e^{x} \sinh \left (x\right ) + 3 \, a^{2} \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{2} + a^{2} e^{x} \sinh \left (x\right )^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.53155, size = 37, normalized size = 1.03 \begin{align*} - \frac{2 \tanh ^{3}{\left (x \right )}}{3 a^{\frac{3}{2}} \left (\operatorname{sech}^{2}{\left (x \right )}\right )^{\frac{3}{2}}} + \frac{\tanh{\left (x \right )}}{a^{\frac{3}{2}} \left (\operatorname{sech}^{2}{\left (x \right )}\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13785, size = 39, normalized size = 1.08 \begin{align*} -\frac{{\left (9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} - e^{\left (3 \, x\right )} - 9 \, e^{x}}{24 \, a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]