Integrand size = 12, antiderivative size = 72 \[ \int (i \sinh (c+d x))^n \, dx=-\frac {i \cosh (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},-\sinh ^2(c+d x)\right ) (i \sinh (c+d x))^{1+n}}{d (1+n) \sqrt {\cosh ^2(c+d x)}} \]
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Time = 0.01 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2722} \[ \int (i \sinh (c+d x))^n \, dx=-\frac {i \cosh (c+d x) (i \sinh (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},-\sinh ^2(c+d x)\right )}{d (n+1) \sqrt {\cosh ^2(c+d x)}} \]
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Rule 2722
Rubi steps \begin{align*} \text {integral}& = -\frac {i \cosh (c+d x) \operatorname {Hypergeometric2F1}\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*}
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.93 \[ \int (i \sinh (c+d x))^n \, dx=\frac {\sqrt {\cosh ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},-\sinh ^2(c+d x)\right ) (i \sinh (c+d x))^n \tanh (c+d x)}{d (1+n)} \]
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\[\int \left (i \sinh \left (d x +c \right )\right )^{n}d x\]
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\[ \int (i \sinh (c+d x))^n \, dx=\int { \left (i \, \sinh \left (d x + c\right )\right )^{n} \,d x } \]
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\[ \int (i \sinh (c+d x))^n \, dx=\int \left (i \sinh {\left (c + d x \right )}\right )^{n}\, dx \]
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\[ \int (i \sinh (c+d x))^n \, dx=\int { \left (i \, \sinh \left (d x + c\right )\right )^{n} \,d x } \]
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\[ \int (i \sinh (c+d x))^n \, dx=\int { \left (i \, \sinh \left (d x + c\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (i \sinh (c+d x))^n \, dx=\int {\left (\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]
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