Optimal. Leaf size=37 \[ -\frac {2 x}{b \sqrt {\cosh (a+b x)}}-\frac {4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5373, 2641} \[ -\frac {2 x}{b \sqrt {\cosh (a+b x)}}-\frac {4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 5373
Rubi steps
\begin {align*} \int \frac {x \sinh (a+b x)}{\cosh ^{\frac {3}{2}}(a+b x)} \, dx &=-\frac {2 x}{b \sqrt {\cosh (a+b x)}}+\frac {2 \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx}{b}\\ &=-\frac {2 x}{b \sqrt {\cosh (a+b x)}}-\frac {4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 37, normalized size = 1.00 \[ -\frac {2 x}{b \sqrt {\cosh (a+b x)}}-\frac {4 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \]
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 \[ \int \frac {x \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {x \sinh \left (b x +a \right )}{\cosh \left (b x +a \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x\,\mathrm {sinh}\left (a+b\,x\right )}{{\mathrm {cosh}\left (a+b\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \sinh {\left (a + b x \right )}}{\cosh ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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