Optimal. Leaf size=91 \[ -\frac {10 i F\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{21 d}+\frac {2 i (i \sinh (c+d x))^{5/2} \cosh (c+d x)}{7 d}+\frac {10 i \sqrt {i \sinh (c+d x)} \cosh (c+d x)}{21 d} \]
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Rubi [A] time = 0.03, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2635, 2641} \[ -\frac {10 i F\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{21 d}+\frac {2 i (i \sinh (c+d x))^{5/2} \cosh (c+d x)}{7 d}+\frac {10 i \sqrt {i \sinh (c+d x)} \cosh (c+d x)}{21 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rubi steps
\begin {align*} \int (i \sinh (c+d x))^{7/2} \, dx &=\frac {2 i \cosh (c+d x) (i \sinh (c+d x))^{5/2}}{7 d}+\frac {5}{7} \int (i \sinh (c+d x))^{3/2} \, dx\\ &=\frac {10 i \cosh (c+d x) \sqrt {i \sinh (c+d x)}}{21 d}+\frac {2 i \cosh (c+d x) (i \sinh (c+d x))^{5/2}}{7 d}+\frac {5}{21} \int \frac {1}{\sqrt {i \sinh (c+d x)}} \, dx\\ &=-\frac {10 i F\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right )}{21 d}+\frac {10 i \cosh (c+d x) \sqrt {i \sinh (c+d x)}}{21 d}+\frac {2 i \cosh (c+d x) (i \sinh (c+d x))^{5/2}}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 65, normalized size = 0.71 \[ \frac {i \left (20 F\left (\left .\frac {1}{4} (-2 i c-2 i d x+\pi )\right |2\right )+\sqrt {i \sinh (c+d x)} (23 \cosh (c+d x)-3 \cosh (3 (c+d x)))\right )}{42 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ \frac {{\left (\sqrt {\frac {1}{2}} {\left (-3 i \, e^{\left (6 \, d x + 6 \, c\right )} + 23 i \, e^{\left (4 \, d x + 4 \, c\right )} + 23 i \, e^{\left (2 \, d x + 2 \, c\right )} - 3 i\right )} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} + 84 \, d e^{\left (3 \, d x + 3 \, c\right )} {\rm integral}\left (-\frac {10 i \, \sqrt {\frac {1}{2}} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{21 \, {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )}}, x\right )\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{84 \, d} \]
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 \[ \int \left (i \, \sinh \left (d x + c\right )\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 122, normalized size = 1.34 \[ -\frac {i \left (6 i \sinh \left (d x +c \right ) \left (\cosh ^{4}\left (d x +c \right )\right )-5 \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-16 i \sinh \left (d x +c \right ) \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{21 \cosh \left (d x +c \right ) \sqrt {i \sinh \left (d x +c \right )}\, d} \]
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 \[ \int \left (i \, \sinh \left (d x + c\right )\right )^{\frac {7}{2}}\,{d x} \]
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 \[ \int {\left (\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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