Optimal. Leaf size=90 \[ -\frac {2 \cosh (c+d x)}{3 b d (b \sinh (c+d x))^{3/2}}+\frac {2 i F\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right ) \sqrt {i \sinh (c+d x)}}{3 b^2 d \sqrt {b \sinh (c+d x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 90, 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 = {2716, 2721,
2720} \begin {gather*} -\frac {2 \cosh (c+d x)}{3 b d (b \sinh (c+d x))^{3/2}}+\frac {2 i \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 b^2 d \sqrt {b \sinh (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2716
Rule 2720
Rule 2721
Rubi steps
\begin {align*} \int \frac {1}{(b \sinh (c+d x))^{5/2}} \, dx &=-\frac {2 \cosh (c+d x)}{3 b d (b \sinh (c+d x))^{3/2}}-\frac {\int \frac {1}{\sqrt {b \sinh (c+d x)}} \, dx}{3 b^2}\\ &=-\frac {2 \cosh (c+d x)}{3 b d (b \sinh (c+d x))^{3/2}}-\frac {\sqrt {i \sinh (c+d x)} \int \frac {1}{\sqrt {i \sinh (c+d x)}} \, dx}{3 b^2 \sqrt {b \sinh (c+d x)}}\\ &=-\frac {2 \cosh (c+d x)}{3 b d (b \sinh (c+d x))^{3/2}}+\frac {2 i 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 b^2 d \sqrt {b \sinh (c+d x)}}\\ \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 = 84, normalized size = 0.93 \begin {gather*} -\frac {2 \left (\coth (c+d x)+\sqrt {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 {-\left ((1+\coth (c+d x)) \sinh ^2(c+d x)\right )}\right )}{3 b^2 d \sqrt {b \sinh (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.67, size = 114, normalized size = 1.27
method | result | size |
default | \(-\frac {i \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \sinh \left (d x +c \right )+2 \left (\cosh ^{2}\left (d x +c \right )\right )}{3 b^{2} \sinh \left (d x +c \right ) \cosh \left (d x +c \right ) \sqrt {b \sinh \left (d x +c \right )}\, d}\) | \(114\) |
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 = 0.09, size = 347, normalized size = 3.86 \begin {gather*} -\frac {2 \, {\left ({\left (\sqrt {2} \cosh \left (d x + c\right )^{4} + 4 \, \sqrt {2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + \sqrt {2} \sinh \left (d x + c\right )^{4} + 2 \, {\left (3 \, \sqrt {2} \cosh \left (d x + c\right )^{2} - \sqrt {2}\right )} \sinh \left (d x + c\right )^{2} - 2 \, \sqrt {2} \cosh \left (d x + c\right )^{2} + 4 \, {\left (\sqrt {2} \cosh \left (d x + c\right )^{3} - \sqrt {2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + \sqrt {2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + 2 \, {\left (\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right )^{2} + 1\right )} \sinh \left (d x + c\right ) + \cosh \left (d x + c\right )\right )} \sqrt {b \sinh \left (d x + c\right )}\right )}}{3 \, {\left (b^{3} d \cosh \left (d x + c\right )^{4} + 4 \, b^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{3} d \sinh \left (d x + c\right )^{4} - 2 \, b^{3} d \cosh \left (d x + c\right )^{2} + b^{3} d + 2 \, {\left (3 \, b^{3} d \cosh \left (d x + c\right )^{2} - b^{3} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b^{3} d \cosh \left (d x + c\right )^{3} - b^{3} d \cosh \left (d x + c\right )\right )} \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 \frac {1}{\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 \frac {1}{{\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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