Optimal. Leaf size=72 \[ \frac {i \cosh (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};-\sinh ^2(c+d x)\right ) (-i \sinh (c+d x))^{1+n}}{d (1+n) \sqrt {\cosh ^2(c+d x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2722}
\begin {gather*} \frac {i \cosh (c+d x) (-i \sinh (c+d x))^{n+1} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};-\sinh ^2(c+d x)\right )}{d (n+1) \sqrt {\cosh ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rubi steps
\begin {align*} \int (-i \sinh (c+d x))^n \, dx &=\frac {i \cosh (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};-\sinh ^2(c+d x)\right ) (-i \sinh (c+d x))^{1+n}}{d (1+n) \sqrt {\cosh ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 67, normalized size = 0.93 \begin {gather*} \frac {\sqrt {\cosh ^2(c+d x)} \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};-\sinh ^2(c+d x)\right ) (-i \sinh (c+d x))^n \tanh (c+d x)}{d (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.27, size = 0, normalized size = 0.00 \[\int \left (-i \sinh \left (d x +c \right )\right )^{n}\, dx\]
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- i \sinh {\left (c + d x \right )}\right )^{n}\, 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 {\left (-\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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