Optimal. Leaf size=88 \[ \frac {2 i b^2 F\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right ) \sqrt {i \sinh (c+d x)}}{3 d \sqrt {b \sinh (c+d x)}}+\frac {2 b \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{3 d} \]
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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,
2720} \begin {gather*} \frac {2 b \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{3 d}+\frac {2 i b^2 \sqrt {i \sinh (c+d x)} F\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{3 d \sqrt {b \sinh (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2720
Rule 2721
Rubi steps
\begin {align*} \int (b \sinh (c+d x))^{3/2} \, dx &=\frac {2 b \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{3 d}-\frac {1}{3} b^2 \int \frac {1}{\sqrt {b \sinh (c+d x)}} \, dx\\ &=\frac {2 b \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{3 d}-\frac {\left (b^2 \sqrt {i \sinh (c+d x)}\right ) \int \frac {1}{\sqrt {i \sinh (c+d x)}} \, dx}{3 \sqrt {b \sinh (c+d x)}}\\ &=\frac {2 i b^2 F\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right ) \sqrt {i \sinh (c+d x)}}{3 d \sqrt {b \sinh (c+d x)}}+\frac {2 b \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{3 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.09, size = 88, normalized size = 1.00 \begin {gather*} \frac {b^2 \left (\sinh (2 (c+d x))-2 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\cosh (2 (c+d x))+\sinh (2 (c+d x))\right ) \sqrt {1-\cosh (2 c+2 d x)-\sinh (2 c+2 d x)}\right )}{3 d \sqrt {b \sinh (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.67, size = 106, normalized size = 1.20
method | result | size |
default | \(-\frac {b^{2} \left (i \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cosh ^{2}\left (d x +c \right )\right ) \sinh \left (d x +c \right )\right )}{3 \cosh \left (d x +c \right ) \sqrt {b \sinh \left (d x +c \right )}\, d}\) | \(106\) |
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 = 0.13, size = 115, normalized size = 1.31 \begin {gather*} -\frac {2 \, {\left (\sqrt {2} b \cosh \left (d x + c\right ) + \sqrt {2} b \sinh \left (d x + c\right )\right )} \sqrt {b} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - {\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + b\right )} \sqrt {b \sinh \left (d x + c\right )}}{3 \, {\left (d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right )\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 {3}{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 )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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