Optimal. Leaf size=116 \[ -\frac {10 i b^4 F\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right ) \sqrt {i \sinh (c+d x)}}{21 d \sqrt {b \sinh (c+d x)}}-\frac {10 b^3 \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{21 d}+\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{5/2}}{7 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {10 i b^4 \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 )}{21 d \sqrt {b \sinh (c+d x)}}-\frac {10 b^3 \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{21 d}+\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{5/2}}{7 d} \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))^{7/2} \, dx &=\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{5/2}}{7 d}-\frac {1}{7} \left (5 b^2\right ) \int (b \sinh (c+d x))^{3/2} \, dx\\ &=-\frac {10 b^3 \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{21 d}+\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{5/2}}{7 d}+\frac {1}{21} \left (5 b^4\right ) \int \frac {1}{\sqrt {b \sinh (c+d x)}} \, dx\\ &=-\frac {10 b^3 \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{21 d}+\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{5/2}}{7 d}+\frac {\left (5 b^4 \sqrt {i \sinh (c+d x)}\right ) \int \frac {1}{\sqrt {i \sinh (c+d x)}} \, dx}{21 \sqrt {b \sinh (c+d x)}}\\ &=-\frac {10 i b^4 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)}}{21 d \sqrt {b \sinh (c+d x)}}-\frac {10 b^3 \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{21 d}+\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{5/2}}{7 d}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 76, normalized size = 0.66 \begin {gather*} \frac {b^3 \left (-23 \cosh (c+d x)+3 \cosh (3 (c+d x))-\frac {20 F\left (\left .\frac {1}{4} (-2 i c+\pi -2 i d x)\right |2\right )}{\sqrt {i \sinh (c+d x)}}\right ) \sqrt {b \sinh (c+d x)}}{42 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.77, size = 122, normalized size = 1.05
method | result | size |
default | \(\frac {b^{4} \left (5 i \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+6 \left (\cosh ^{4}\left (d x +c \right )\right ) \sinh \left (d x +c \right )-16 \left (\cosh ^{2}\left (d x +c \right )\right ) \sinh \left (d x +c \right )\right )}{21 \cosh \left (d x +c \right ) \sqrt {b \sinh \left (d x +c \right )}\, d}\) | \(122\) |
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 = 394, normalized size = 3.40 \begin {gather*} \frac {40 \, {\left (\sqrt {2} b^{3} \cosh \left (d x + c\right )^{3} + 3 \, \sqrt {2} b^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, \sqrt {2} b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sqrt {2} b^{3} \sinh \left (d x + c\right )^{3}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + {\left (3 \, b^{3} \cosh \left (d x + c\right )^{6} + 18 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 3 \, b^{3} \sinh \left (d x + c\right )^{6} - 23 \, b^{3} \cosh \left (d x + c\right )^{4} - 23 \, b^{3} \cosh \left (d x + c\right )^{2} + {\left (45 \, b^{3} \cosh \left (d x + c\right )^{2} - 23 \, b^{3}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (15 \, b^{3} \cosh \left (d x + c\right )^{3} - 23 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, b^{3} + {\left (45 \, b^{3} \cosh \left (d x + c\right )^{4} - 138 \, b^{3} \cosh \left (d x + c\right )^{2} - 23 \, b^{3}\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (9 \, b^{3} \cosh \left (d x + c\right )^{5} - 46 \, b^{3} \cosh \left (d x + c\right )^{3} - 23 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {b \sinh \left (d x + c\right )}}{84 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 )}^{7/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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