Optimal. Leaf size=62 \[ \frac {2 i \sqrt {i \sinh (c+d x)} \cosh (c+d x)}{3 d}-\frac {2 i F\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{3 d} \]
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Rubi [A] time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2635, 2641} \[ \frac {2 i \sqrt {i \sinh (c+d x)} \cosh (c+d x)}{3 d}-\frac {2 i F\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{3 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rubi steps
\begin {align*} \int (i \sinh (c+d x))^{3/2} \, dx &=\frac {2 i \cosh (c+d x) \sqrt {i \sinh (c+d x)}}{3 d}+\frac {1}{3} \int \frac {1}{\sqrt {i \sinh (c+d x)}} \, dx\\ &=-\frac {2 i F\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right )}{3 d}+\frac {2 i \cosh (c+d x) \sqrt {i \sinh (c+d x)}}{3 d}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 94, normalized size = 1.52 \[ -\frac {2 i \sqrt {i \sinh (c+d x)} \left (\text {csch}(c+d x) \sqrt {-\sinh (2 c+2 d x)-\cosh (2 c+2 d x)+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\cosh (2 (c+d x))+\sinh (2 (c+d x))\right )-\cosh (c+d x)\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ \frac {{\left (\sqrt {\frac {1}{2}} {\left (i \, e^{\left (2 \, d x + 2 \, c\right )} + i\right )} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} + 3 \, d e^{\left (d x + c\right )} {\rm integral}\left (-\frac {2 i \, \sqrt {\frac {1}{2}} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{3 \, {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )}}, x\right )\right )} e^{\left (-d x - c\right )}}{3 \, d} \]
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 \[ \int \left (i \, \sinh \left (d x + c\right )\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 104, normalized size = 1.68 \[ \frac {i \left (\sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+2 i \sinh \left (d x +c \right ) \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{3 \cosh \left (d x +c \right ) \sqrt {i \sinh \left (d x +c \right )}\, d} \]
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 \[ \int \left (i \, \sinh \left (d x + c\right )\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2} \,d x \]
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 \[ \int \left (i \sinh {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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