Optimal. Leaf size=88 \[ \frac {6 i b^2 E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right ) \sqrt {b \sinh (c+d x)}}{5 d \sqrt {i \sinh (c+d x)}}+\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{3/2}}{5 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 2721,
2719} \begin {gather*} \frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{3/2}}{5 d}+\frac {6 i b^2 E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {b \sinh (c+d x)}}{5 d \sqrt {i \sinh (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2719
Rule 2721
Rubi steps
\begin {align*} \int (b \sinh (c+d x))^{5/2} \, dx &=\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{3/2}}{5 d}-\frac {1}{5} \left (3 b^2\right ) \int \sqrt {b \sinh (c+d x)} \, dx\\ &=\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{3/2}}{5 d}-\frac {\left (3 b^2 \sqrt {b \sinh (c+d x)}\right ) \int \sqrt {i \sinh (c+d x)} \, dx}{5 \sqrt {i \sinh (c+d x)}}\\ &=\frac {6 i b^2 E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right ) \sqrt {b \sinh (c+d x)}}{5 d \sqrt {i \sinh (c+d x)}}+\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{3/2}}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 68, normalized size = 0.77 \begin {gather*} \frac {b^2 \sqrt {b \sinh (c+d x)} \left (-\frac {6 i E\left (\left .\frac {1}{4} (-2 i c+\pi -2 i d x)\right |2\right )}{\sqrt {i \sinh (c+d x)}}+\sinh (2 (c+d x))\right )}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.71, size = 170, normalized size = 1.93
method | result | size |
default | \(-\frac {b^{3} \left (6 \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticE \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cosh ^{4}\left (d x +c \right )\right )+2 \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{5 \cosh \left (d x +c \right ) \sqrt {b \sinh \left (d x +c \right )}\, d}\) | \(170\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.10, size = 249, normalized size = 2.83 \begin {gather*} \frac {12 \, {\left (\sqrt {2} b^{2} \cosh \left (d x + c\right )^{2} + 2 \, \sqrt {2} b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sqrt {2} b^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {b} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )\right ) + {\left (b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} + 12 \, b^{2} \cosh \left (d x + c\right )^{2} + 6 \, {\left (b^{2} \cosh \left (d x + c\right )^{2} + 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} - b^{2} + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} + 6 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {b \sinh \left (d x + c\right )}}{10 \, {\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2}\right )}} \end {gather*}
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 \begin {gather*} \int \left (b \sinh {\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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