Optimal. Leaf size=139 \[ \frac{(e x)^{m+1} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \sec ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \text{Hypergeometric2F1}\left (p,-\frac{-b d n p+i m+i}{2 b d n},\frac{1}{2} \left (-\frac{i (m+1)}{b d n}+p+2\right ),-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (i b d n p+m+1)} \]
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Rubi [A] time = 0.121491, antiderivative size = 133, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4509, 4507, 364} \[ \frac{(e x)^{m+1} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \, _2F_1\left (p,\frac{1}{2} \left (p-\frac{i (m+1)}{b d n}\right );\frac{1}{2} \left (-\frac{i (m+1)}{b d n}+p+2\right );-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sec ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (i b d n p+m+1)} \]
Antiderivative was successfully verified.
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Rule 4509
Rule 4507
Rule 364
Rubi steps
\begin{align*} \int (e x)^m \sec ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac{\left ((e x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1+m}{n}} \sec ^p(d (a+b \log (x))) \, dx,x,c x^n\right )}{e n}\\ &=\frac{\left ((e x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}-i b d p} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \sec ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1+m}{n}+i b d p} \left (1+e^{2 i a d} x^{2 i b d}\right )^{-p} \, dx,x,c x^n\right )}{e n}\\ &=\frac{(e x)^{1+m} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \, _2F_1\left (p,\frac{1}{2} \left (-\frac{i (1+m)}{b d n}+p\right );\frac{1}{2} \left (2-\frac{i (1+m)}{b d n}+p\right );-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sec ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m+i b d n p)}\\ \end{align*}
Mathematica [A] time = 1.55888, size = 169, normalized size = 1.22 \[ \frac{2^p x (e x)^m \left (\frac{e^{i a d} \left (c x^n\right )^{i b d}}{1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \text{Hypergeometric2F1}\left (p,-\frac{i (i b d n p+m+1)}{2 b d n},\frac{1}{2} \left (-\frac{i (m+1)}{b d n}+p+2\right ),-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{i b d n p+m+1} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.27, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( \sec \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{p}\, dx \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 (e x\right )^{m} \sec \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p}\,{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*}{\rm integral}\left (\left (e x\right )^{m} \sec \left (b d \log \left (c x^{n}\right ) + a d\right )^{p}, x\right ) \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 (e x\right )^{m} \sec \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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