Optimal. Leaf size=121 \[ \frac{20 i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right ),2\right )}{147 b^2 \sqrt{\sinh (a+b x)}}-\frac{4 \sinh ^{\frac{5}{2}}(a+b x) \cosh (a+b x)}{49 b^2}+\frac{20 \sqrt{\sinh (a+b x)} \cosh (a+b x)}{147 b^2}+\frac{2 x \sinh ^{\frac{7}{2}}(a+b x)}{7 b} \]
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Rubi [A] time = 0.0702647, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5372, 2635, 2642, 2641} \[ -\frac{4 \sinh ^{\frac{5}{2}}(a+b x) \cosh (a+b x)}{49 b^2}+\frac{20 \sqrt{\sinh (a+b x)} \cosh (a+b x)}{147 b^2}+\frac{20 i \sqrt{i \sinh (a+b x)} F\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{147 b^2 \sqrt{\sinh (a+b x)}}+\frac{2 x \sinh ^{\frac{7}{2}}(a+b x)}{7 b} \]
Antiderivative was successfully verified.
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Rule 5372
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int x \cosh (a+b x) \sinh ^{\frac{5}{2}}(a+b x) \, dx &=\frac{2 x \sinh ^{\frac{7}{2}}(a+b x)}{7 b}-\frac{2 \int \sinh ^{\frac{7}{2}}(a+b x) \, dx}{7 b}\\ &=-\frac{4 \cosh (a+b x) \sinh ^{\frac{5}{2}}(a+b x)}{49 b^2}+\frac{2 x \sinh ^{\frac{7}{2}}(a+b x)}{7 b}+\frac{10 \int \sinh ^{\frac{3}{2}}(a+b x) \, dx}{49 b}\\ &=\frac{20 \cosh (a+b x) \sqrt{\sinh (a+b x)}}{147 b^2}-\frac{4 \cosh (a+b x) \sinh ^{\frac{5}{2}}(a+b x)}{49 b^2}+\frac{2 x \sinh ^{\frac{7}{2}}(a+b x)}{7 b}-\frac{10 \int \frac{1}{\sqrt{\sinh (a+b x)}} \, dx}{147 b}\\ &=\frac{20 \cosh (a+b x) \sqrt{\sinh (a+b x)}}{147 b^2}-\frac{4 \cosh (a+b x) \sinh ^{\frac{5}{2}}(a+b x)}{49 b^2}+\frac{2 x \sinh ^{\frac{7}{2}}(a+b x)}{7 b}-\frac{\left (10 \sqrt{i \sinh (a+b x)}\right ) \int \frac{1}{\sqrt{i \sinh (a+b x)}} \, dx}{147 b \sqrt{\sinh (a+b x)}}\\ &=\frac{20 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)}}{147 b^2 \sqrt{\sinh (a+b x)}}+\frac{20 \cosh (a+b x) \sqrt{\sinh (a+b x)}}{147 b^2}-\frac{4 \cosh (a+b x) \sinh ^{\frac{5}{2}}(a+b x)}{49 b^2}+\frac{2 x \sinh ^{\frac{7}{2}}(a+b x)}{7 b}\\ \end{align*}
Mathematica [A] time = 0.310207, size = 103, normalized size = 0.85 \[ \frac{-80 i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 i a-2 i b x+\pi ),2\right )+52 \sinh (2 (a+b x))-6 \sinh (4 (a+b x))-84 b x \cosh (2 (a+b x))+21 b x \cosh (4 (a+b x))+63 b x}{588 b^2 \sqrt{\sinh (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int x\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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