Optimal. Leaf size=71 \[ -\frac {2 x}{b \sqrt {\sinh (a+b x)}}-\frac {4 i \sqrt {i \sinh (a+b x)} F\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b^2 \sqrt {\sinh (a+b x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5372, 2642, 2641} \[ -\frac {2 x}{b \sqrt {\sinh (a+b x)}}-\frac {4 i \sqrt {i \sinh (a+b x)} F\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b^2 \sqrt {\sinh (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 2642
Rule 5372
Rubi steps
\begin {align*} \int \frac {x \cosh (a+b x)}{\sinh ^{\frac {3}{2}}(a+b x)} \, dx &=-\frac {2 x}{b \sqrt {\sinh (a+b x)}}+\frac {2 \int \frac {1}{\sqrt {\sinh (a+b x)}} \, dx}{b}\\ &=-\frac {2 x}{b \sqrt {\sinh (a+b x)}}+\frac {\left (2 \sqrt {i \sinh (a+b x)}\right ) \int \frac {1}{\sqrt {i \sinh (a+b x)}} \, dx}{b \sqrt {\sinh (a+b x)}}\\ &=-\frac {2 x}{b \sqrt {\sinh (a+b x)}}-\frac {4 i F\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {i \sinh (a+b x)}}{b^2 \sqrt {\sinh (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 56, normalized size = 0.79 \[ \frac {-2 b x+4 i \sqrt {i \sinh (a+b x)} F\left (\left .\frac {1}{4} (-2 i a-2 i b x+\pi )\right |2\right )}{b^2 \sqrt {\sinh (a+b x)}} \]
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 \[ \int \frac {x \cosh \left (b x + a\right )}{\sinh \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {x \cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x \cosh \left (b x + a\right )}{\sinh \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\mathrm {cosh}\left (a+b\,x\right )}{{\mathrm {sinh}\left (a+b\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \cosh {\left (a + b x \right )}}{\sinh ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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