Optimal. Leaf size=62 \[ -\frac{1}{5} a \cosh ^2(x) \coth ^3(x) \sqrt{a \text{csch}^4(x)}+\frac{2}{3} a \cosh ^2(x) \coth (x) \sqrt{a \text{csch}^4(x)}-a \sinh (x) \cosh (x) \sqrt{a \text{csch}^4(x)} \]
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Rubi [A] time = 0.0226303, antiderivative size = 62, 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 = {4123, 3767} \[ -\frac{1}{5} a \cosh ^2(x) \coth ^3(x) \sqrt{a \text{csch}^4(x)}+\frac{2}{3} a \cosh ^2(x) \coth (x) \sqrt{a \text{csch}^4(x)}-a \sinh (x) \cosh (x) \sqrt{a \text{csch}^4(x)} \]
Antiderivative was successfully verified.
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Rule 4123
Rule 3767
Rubi steps
\begin{align*} \int \left (a \text{csch}^4(x)\right )^{3/2} \, dx &=\left (a \sqrt{a \text{csch}^4(x)} \sinh ^2(x)\right ) \int \text{csch}^6(x) \, dx\\ &=-\left (\left (i a \sqrt{a \text{csch}^4(x)} \sinh ^2(x)\right ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \coth (x)\right )\right )\\ &=\frac{2}{3} a \cosh ^2(x) \coth (x) \sqrt{a \text{csch}^4(x)}-\frac{1}{5} a \cosh ^2(x) \coth ^3(x) \sqrt{a \text{csch}^4(x)}-a \cosh (x) \sqrt{a \text{csch}^4(x)} \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0181743, size = 33, normalized size = 0.53 \[ -\frac{1}{15} a \sinh (x) \cosh (x) \left (3 \text{csch}^4(x)-4 \text{csch}^2(x)+8\right ) \sqrt{a \text{csch}^4(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 46, normalized size = 0.7 \begin{align*} -{\frac{16\,a{{\rm e}^{-2\,x}} \left ( 10\,{{\rm e}^{4\,x}}-5\,{{\rm e}^{2\,x}}+1 \right ) }{15\, \left ({{\rm e}^{2\,x}}-1 \right ) ^{3}}\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{4}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.70253, size = 162, normalized size = 2.61 \begin{align*} -\frac{16 \, a^{\frac{3}{2}} 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 \, a^{\frac{3}{2}} 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 \, a^{\frac{3}{2}}}{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 = 1.65655, size = 1623, normalized size = 26.18 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \operatorname{csch}^{4}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16261, size = 36, normalized size = 0.58 \begin{align*} -\frac{16 \, a^{\frac{3}{2}}{\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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