Optimal. Leaf size=20 \[ \frac {2 x}{\sqrt {\text {csch}(x)}}-\frac {4 \text {sech}(x)}{\text {csch}^{\frac {3}{2}}(x)} \]
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Rubi [A] time = 0.17, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6742, 5445, 3771, 2639, 2626} \[ \frac {2 x}{\sqrt {\text {csch}(x)}}-\frac {4 \text {sech}(x)}{\text {csch}^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 2626
Rule 2639
Rule 3771
Rule 5445
Rule 6742
Rubi steps
\begin {align*} \int \sqrt {\text {csch}(x)} (x \cosh (x)-4 \text {sech}(x) \tanh (x)) \, dx &=\int \left (x \cosh (x) \sqrt {\text {csch}(x)}-\frac {4 \text {sech}^2(x)}{\sqrt {\text {csch}(x)}}\right ) \, dx\\ &=-\left (4 \int \frac {\text {sech}^2(x)}{\sqrt {\text {csch}(x)}} \, dx\right )+\int x \cosh (x) \sqrt {\text {csch}(x)} \, dx\\ &=\frac {2 x}{\sqrt {\text {csch}(x)}}-\frac {4 \text {sech}(x)}{\text {csch}^{\frac {3}{2}}(x)}\\ \end {align*}
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Mathematica [A] time = 1.14, size = 17, normalized size = 0.85 \[ \frac {2 (x \text {csch}(x)-2 \text {sech}(x))}{\text {csch}^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x \cosh \relax (x) - 4 \, \operatorname {sech}\relax (x) \tanh \relax (x)\right )} \sqrt {\operatorname {csch}\relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.84, size = 0, normalized size = 0.00 \[ \int \sqrt {\mathrm {csch}\relax (x )}\, \left (x \cosh \relax (x )-4 \,\mathrm {sech}\relax (x ) \tanh \relax (x )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x \cosh \relax (x) - 4 \, \operatorname {sech}\relax (x) \tanh \relax (x)\right )} \sqrt {\operatorname {csch}\relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.90, size = 51, normalized size = 2.55 \[ \frac {{\mathrm {e}}^{-x}\,\sqrt {-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}}\,\left ({\mathrm {e}}^{2\,x}-1\right )\,\left (x-2\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{2\,x}+2\right )}{{\mathrm {e}}^{2\,x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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