Optimal. Leaf size=65 \[ -\frac {3}{8} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right )+\frac {3}{8} a^2 \coth (x) \sqrt {a \text {csch}^2(x)}-\frac {1}{4} a \coth (x) \left (a \text {csch}^2(x)\right )^{3/2} \]
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Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4122, 195, 217, 206} \[ \frac {3}{8} a^2 \coth (x) \sqrt {a \text {csch}^2(x)}-\frac {3}{8} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right )-\frac {1}{4} a \coth (x) \left (a \text {csch}^2(x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 4122
Rubi steps
\begin {align*} \int \left (a \text {csch}^2(x)\right )^{5/2} \, dx &=-\left (a \operatorname {Subst}\left (\int \left (-a+a x^2\right )^{3/2} \, dx,x,\coth (x)\right )\right )\\ &=-\frac {1}{4} a \coth (x) \left (a \text {csch}^2(x)\right )^{3/2}+\frac {1}{4} \left (3 a^2\right ) \operatorname {Subst}\left (\int \sqrt {-a+a x^2} \, dx,x,\coth (x)\right )\\ &=\frac {3}{8} a^2 \coth (x) \sqrt {a \text {csch}^2(x)}-\frac {1}{4} a \coth (x) \left (a \text {csch}^2(x)\right )^{3/2}-\frac {1}{8} \left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+a x^2}} \, dx,x,\coth (x)\right )\\ &=\frac {3}{8} a^2 \coth (x) \sqrt {a \text {csch}^2(x)}-\frac {1}{4} a \coth (x) \left (a \text {csch}^2(x)\right )^{3/2}-\frac {1}{8} \left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a \text {csch}^2(x)}}\right )\\ &=-\frac {3}{8} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right )+\frac {3}{8} a^2 \coth (x) \sqrt {a \text {csch}^2(x)}-\frac {1}{4} a \coth (x) \left (a \text {csch}^2(x)\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 41, normalized size = 0.63 \[ \frac {1}{64} \sinh (x) \left (a \text {csch}^2(x)\right )^{5/2} \left (6 \left (\cosh (3 x)+4 \sinh ^4(x) \log \left (\tanh \left (\frac {x}{2}\right )\right )\right )-22 \cosh (x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 1128, normalized size = 17.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 75, normalized size = 1.15 \[ \frac {1}{16} \, a^{\frac {5}{2}} {\left (\frac {4 \, {\left (3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 20 \, e^{\left (-x\right )} - 20 \, e^{x}\right )}}{{\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{2}} - 3 \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + 3 \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.25, size = 123, normalized size = 1.89 \[ \frac {a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left (3 \,{\mathrm e}^{6 x}-11 \,{\mathrm e}^{4 x}-11 \,{\mathrm e}^{2 x}+3\right )}{4 \left ({\mathrm e}^{2 x}-1\right )^{3}}+\frac {3 a^{2} {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}-1\right )}{8}-\frac {3 a^{2} {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}+1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 92, normalized size = 1.42 \[ \frac {3}{8} \, a^{\frac {5}{2}} \log \left (e^{\left (-x\right )} + 1\right ) - \frac {3}{8} \, a^{\frac {5}{2}} \log \left (e^{\left (-x\right )} - 1\right ) + \frac {3 \, a^{\frac {5}{2}} e^{\left (-x\right )} - 11 \, a^{\frac {5}{2}} e^{\left (-3 \, x\right )} - 11 \, a^{\frac {5}{2}} e^{\left (-5 \, x\right )} + 3 \, a^{\frac {5}{2}} e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\frac {a}{{\mathrm {sinh}\relax (x)}^2}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \operatorname {csch}^{2}{\relax (x )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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