Optimal. Leaf size=91 \[ -\frac{10 i \text{EllipticF}\left (\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right ),2\right )}{21 d}+\frac{2 i (i \sinh (c+d x))^{5/2} \cosh (c+d x)}{7 d}+\frac{10 i \sqrt{i \sinh (c+d x)} \cosh (c+d x)}{21 d} \]
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Rubi [A] time = 0.0343493, antiderivative size = 91, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.164432, size = 65, normalized size = 0.71 \[ \frac{i \left (20 \text{EllipticF}\left (\frac{1}{4} (-2 i c-2 i d x+\pi ),2\right )+\sqrt{i \sinh (c+d x)} (23 \cosh (c+d x)-3 \cosh (3 (c+d x)))\right )}{42 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 122, normalized size = 1.3 \begin{align*}{\frac{-{\frac{i}{21}}}{d\cosh \left ( dx+c \right ) } \left ( 6\,i\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{4}-5\,\sqrt{1-i\sinh \left ( dx+c \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( dx+c \right ) }\sqrt{i\sinh \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) -16\,i \left ( \cosh \left ( dx+c \right ) \right ) ^{2}\sinh \left ( dx+c \right ) \right ){\frac{1}{\sqrt{i\sinh \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (i \, \sinh \left (d x + c\right )\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (i \, \sinh \left (d x + c\right )\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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