Integrand size = 22, antiderivative size = 150 \[ \int (e x)^{-1+n} \left (a+b \sinh \left (c+d x^n\right )\right )^p \, dx=\frac {i \sqrt {2} x^{-n} (e x)^n \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-p,\frac {3}{2},\frac {1}{2} \left (1-i \sinh \left (c+d x^n\right )\right ),\frac {b \left (1-i \sinh \left (c+d x^n\right )\right )}{i a+b}\right ) \cosh \left (c+d x^n\right ) \left (a+b \sinh \left (c+d x^n\right )\right )^p \left (\frac {a+b \sinh \left (c+d x^n\right )}{a-i b}\right )^{-p}}{d e n \sqrt {1+i \sinh \left (c+d x^n\right )}} \]
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Time = 0.15 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5430, 5428, 2744, 144, 143} \[ \int (e x)^{-1+n} \left (a+b \sinh \left (c+d x^n\right )\right )^p \, dx=\frac {i \sqrt {2} x^{-n} (e x)^n \cosh \left (c+d x^n\right ) \left (a+b \sinh \left (c+d x^n\right )\right )^p \left (\frac {a+b \sinh \left (c+d x^n\right )}{a-i b}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-p,\frac {3}{2},\frac {1}{2} \left (1-i \sinh \left (d x^n+c\right )\right ),\frac {b \left (1-i \sinh \left (d x^n+c\right )\right )}{i a+b}\right )}{d e n \sqrt {1+i \sinh \left (c+d x^n\right )}} \]
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Rule 143
Rule 144
Rule 2744
Rule 5428
Rule 5430
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-n} (e x)^n\right ) \int x^{-1+n} \left (a+b \sinh \left (c+d x^n\right )\right )^p \, dx}{e} \\ & = \frac {\left (x^{-n} (e x)^n\right ) \text {Subst}\left (\int (a+b \sinh (c+d x))^p \, dx,x,x^n\right )}{e n} \\ & = -\frac {\left (i x^{-n} (e x)^n \cosh \left (c+d x^n\right )\right ) \text {Subst}\left (\int \frac {(a-i b x)^p}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,i \sinh \left (c+d x^n\right )\right )}{d e n \sqrt {1-i \sinh \left (c+d x^n\right )} \sqrt {1+i \sinh \left (c+d x^n\right )}} \\ & = -\frac {\left (i x^{-n} (e x)^n \cosh \left (c+d x^n\right ) \left (a+b \sinh \left (c+d x^n\right )\right )^p \left (-\frac {a+b \sinh \left (c+d x^n\right )}{-a+i b}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a+i b}+\frac {i b x}{-a+i b}\right )^p}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,i \sinh \left (c+d x^n\right )\right )}{d e n \sqrt {1-i \sinh \left (c+d x^n\right )} \sqrt {1+i \sinh \left (c+d x^n\right )}} \\ & = \frac {i \sqrt {2} x^{-n} (e x)^n \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-p,\frac {3}{2},\frac {1}{2} \left (1-i \sinh \left (c+d x^n\right )\right ),\frac {b \left (1-i \sinh \left (c+d x^n\right )\right )}{i a+b}\right ) \cosh \left (c+d x^n\right ) \left (a+b \sinh \left (c+d x^n\right )\right )^p \left (\frac {a+b \sinh \left (c+d x^n\right )}{a-i b}\right )^{-p}}{d e n \sqrt {1+i \sinh \left (c+d x^n\right )}} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.11 \[ \int (e x)^{-1+n} \left (a+b \sinh \left (c+d x^n\right )\right )^p \, dx=\frac {x^{-n} (e x)^n \operatorname {AppellF1}\left (1+p,\frac {1}{2},\frac {1}{2},2+p,\frac {a+b \sinh \left (c+d x^n\right )}{a+i b},\frac {a+b \sinh \left (c+d x^n\right )}{a-i b}\right ) \text {sech}\left (c+d x^n\right ) \sqrt {\frac {b \left (1-i \sinh \left (c+d x^n\right )\right )}{i a+b}} \sqrt {\frac {b \left (1+i \sinh \left (c+d x^n\right )\right )}{-i a+b}} \left (a+b \sinh \left (c+d x^n\right )\right )^{1+p}}{b d e n (1+p)} \]
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\[\int \left (e x \right )^{-1+n} {\left (a +b \sinh \left (c +d \,x^{n}\right )\right )}^{p}d x\]
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\[ \int (e x)^{-1+n} \left (a+b \sinh \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{n - 1} {\left (b \sinh \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]
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\[ \int (e x)^{-1+n} \left (a+b \sinh \left (c+d x^n\right )\right )^p \, dx=\int \left (e x\right )^{n - 1} \left (a + b \sinh {\left (c + d x^{n} \right )}\right )^{p}\, dx \]
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\[ \int (e x)^{-1+n} \left (a+b \sinh \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{n - 1} {\left (b \sinh \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]
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\[ \int (e x)^{-1+n} \left (a+b \sinh \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{n - 1} {\left (b \sinh \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]
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Timed out. \[ \int (e x)^{-1+n} \left (a+b \sinh \left (c+d x^n\right )\right )^p \, dx=\int {\left (e\,x\right )}^{n-1}\,{\left (a+b\,\mathrm {sinh}\left (c+d\,x^n\right )\right )}^p \,d x \]
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