Optimal. Leaf size=139 \[ \frac {(e x)^{m+1} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \, _2F_1\left (p,-\frac {i m-b d n p+i}{2 b d n};\frac {1}{2} \left (-\frac {i (m+1)}{b d n}+p+2\right );-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sec ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (i b d n p+m+1)} \]
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Rubi [A] time = 0.12, 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 364
Rule 4507
Rule 4509
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*}
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Mathematica [A] time = 1.63, size = 169, normalized size = 1.22 \[ \frac {2^p x (e x)^m \left (\frac {e^{i a d} \left (c x^n\right )^{i b d}}{1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \, _2F_1\left (p,-\frac {i (m+i b d n p+1)}{2 b d n};\frac {1}{2} \left (-\frac {i (m+1)}{b d n}+p+2\right );-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{i b d n p+m+1} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \sec \left (b d \log \left (c x^{n}\right ) + a d\right )^{p}, x\right ) \]
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 \[ \int \left (e x\right )^{m} \sec \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\sec ^{p}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx \]
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 \[ \int \left (e x\right )^{m} \sec \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (e\,x\right )}^m\,{\left (\frac {1}{\cos \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}\right )}^p \,d x \]
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 \[ \int \left (e x\right )^{m} \sec ^{p}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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