Optimal. Leaf size=16 \[ \sinh (x) (-\cosh (x)) \sqrt {a \text {csch}^4(x)} \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4123, 3767, 8} \[ \sinh (x) (-\cosh (x)) \sqrt {a \text {csch}^4(x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 4123
Rubi steps
\begin {align*} \int \sqrt {a \text {csch}^4(x)} \, dx &=\left (\sqrt {a \text {csch}^4(x)} \sinh ^2(x)\right ) \int \text {csch}^2(x) \, dx\\ &=-\left (\left (i \sqrt {a \text {csch}^4(x)} \sinh ^2(x)\right ) \operatorname {Subst}(\int 1 \, dx,x,-i \coth (x))\right )\\ &=-\cosh (x) \sqrt {a \text {csch}^4(x)} \sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \sinh (x) (-\cosh (x)) \sqrt {a \text {csch}^4(x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 81, normalized size = 5.06 \[ -\frac {2 \, \sqrt {\frac {a}{e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (2 \, x\right )}}{2 \, \cosh \relax (x) e^{\left (2 \, x\right )} \sinh \relax (x) + e^{\left (2 \, x\right )} \sinh \relax (x)^{2} + {\left (\cosh \relax (x)^{2} - 1\right )} e^{\left (2 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 13, normalized size = 0.81 \[ -\frac {2 \, \sqrt {a}}{e^{\left (2 \, x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 29, normalized size = 1.81 \[ -2 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}\, {\mathrm e}^{-2 x} \left ({\mathrm e}^{2 x}-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 13, normalized size = 0.81 \[ \frac {2 \, \sqrt {a}}{e^{\left (-2 \, x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 71, normalized size = 4.44 \[ -\frac {\sqrt {a}\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (3\,{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{6\,x}+\frac {{\mathrm {e}}^{8\,x}}{2}+\frac {1}{2}\right )}{\left ({\mathrm {e}}^{2\,x}-1\right )\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \]
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 \[ \int \sqrt {a \operatorname {csch}^{4}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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