Optimal. Leaf size=67 \[ \frac{\sinh (x) \cosh (x)}{a \sqrt{a \cosh ^4(x)}}+\frac{\sinh ^2(x) \tanh ^3(x)}{5 a \sqrt{a \cosh ^4(x)}}-\frac{2 \sinh ^2(x) \tanh (x)}{3 a \sqrt{a \cosh ^4(x)}} \]
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Rubi [A] time = 0.025165, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3207, 3767} \[ \frac{\sinh (x) \cosh (x)}{a \sqrt{a \cosh ^4(x)}}+\frac{\sinh ^2(x) \tanh ^3(x)}{5 a \sqrt{a \cosh ^4(x)}}-\frac{2 \sinh ^2(x) \tanh (x)}{3 a \sqrt{a \cosh ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 3767
Rubi steps
\begin{align*} \int \frac{1}{\left (a \cosh ^4(x)\right )^{3/2}} \, dx &=\frac{\cosh ^2(x) \int \text{sech}^6(x) \, dx}{a \sqrt{a \cosh ^4(x)}}\\ &=\frac{\left (i \cosh ^2(x)\right ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \tanh (x)\right )}{a \sqrt{a \cosh ^4(x)}}\\ &=\frac{\cosh (x) \sinh (x)}{a \sqrt{a \cosh ^4(x)}}-\frac{2 \sinh ^2(x) \tanh (x)}{3 a \sqrt{a \cosh ^4(x)}}+\frac{\sinh ^2(x) \tanh ^3(x)}{5 a \sqrt{a \cosh ^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.0267642, size = 30, normalized size = 0.45 \[ \frac{\sinh (x) \cosh (x) (6 \cosh (2 x)+\cosh (4 x)+8)}{15 \left (a \cosh ^4(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.096, size = 80, normalized size = 1.2 \begin{align*}{\frac{\sqrt{8}\sqrt{2} \left ( 2\, \left ( \cosh \left ( 2\,x \right ) \right ) ^{2}+6\,\cosh \left ( 2\,x \right ) +7 \right ) }{15\,{a}^{2} \left ( \cosh \left ( 2\,x \right ) +1 \right ) ^{2}\sinh \left ( 2\,x \right ) }\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a \left ( -1+\cosh \left ( 2\,x \right ) \right ) \left ( \cosh \left ( 2\,x \right ) +1 \right ) }{\frac{1}{\sqrt{a \left ( \cosh \left ( 2\,x \right ) +1 \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.58217, size = 223, normalized size = 3.33 \begin{align*} \frac{16 \, e^{\left (-2 \, x\right )}}{3 \,{\left (5 \, a^{\frac{3}{2}} e^{\left (-2 \, x\right )} + 10 \, a^{\frac{3}{2}} e^{\left (-4 \, x\right )} + 10 \, a^{\frac{3}{2}} e^{\left (-6 \, x\right )} + 5 \, a^{\frac{3}{2}} e^{\left (-8 \, x\right )} + a^{\frac{3}{2}} e^{\left (-10 \, x\right )} + a^{\frac{3}{2}}\right )}} + \frac{32 \, e^{\left (-4 \, x\right )}}{3 \,{\left (5 \, a^{\frac{3}{2}} e^{\left (-2 \, x\right )} + 10 \, a^{\frac{3}{2}} e^{\left (-4 \, x\right )} + 10 \, a^{\frac{3}{2}} e^{\left (-6 \, x\right )} + 5 \, a^{\frac{3}{2}} e^{\left (-8 \, x\right )} + a^{\frac{3}{2}} e^{\left (-10 \, x\right )} + a^{\frac{3}{2}}\right )}} + \frac{16}{15 \,{\left (5 \, a^{\frac{3}{2}} e^{\left (-2 \, x\right )} + 10 \, a^{\frac{3}{2}} e^{\left (-4 \, x\right )} + 10 \, a^{\frac{3}{2}} e^{\left (-6 \, x\right )} + 5 \, a^{\frac{3}{2}} e^{\left (-8 \, x\right )} + a^{\frac{3}{2}} e^{\left (-10 \, x\right )} + a^{\frac{3}{2}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97198, size = 3163, normalized size = 47.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18343, size = 47, normalized size = 0.7 \begin{align*} -\frac{16 \,{\left (10 \, \sqrt{a} e^{\left (4 \, x\right )} + 5 \, \sqrt{a} e^{\left (2 \, x\right )} + \sqrt{a}\right )}}{15 \, a^{2}{\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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