Optimal. Leaf size=106 \[ \frac {x^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \, _2F_1\left (p,\frac {1}{2} \left (-\frac {2 i}{b n}+p\right );\frac {1}{2} \left (2-\frac {2 i}{b n}+p\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sec ^p\left (a+b \log \left (c x^n\right )\right )}{2+i b n p} \]
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Rubi [A]
time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4605, 4603,
371} \begin {gather*} \frac {x^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \, _2F_1\left (p,\frac {1}{2} \left (p-\frac {2 i}{b n}\right );\frac {1}{2} \left (p-\frac {2 i}{b n}+2\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sec ^p\left (a+b \log \left (c x^n\right )\right )}{2+i b n p} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 4603
Rule 4605
Rubi steps
\begin {align*} \int x \sec ^p\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int x^{-1+\frac {2}{n}} \sec ^p(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x^2 \left (c x^n\right )^{-\frac {2}{n}-i b p} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \sec ^p\left (a+b \log \left (c x^n\right )\right )\right ) \text {Subst}\left (\int x^{-1+\frac {2}{n}+i b p} \left (1+e^{2 i a} x^{2 i b}\right )^{-p} \, dx,x,c x^n\right )}{n}\\ &=\frac {x^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \, _2F_1\left (p,\frac {1}{2} \left (-\frac {2 i}{b n}+p\right );\frac {1}{2} \left (2-\frac {2 i}{b n}+p\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sec ^p\left (a+b \log \left (c x^n\right )\right )}{2+i b n p}\\ \end {align*}
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Mathematica [A]
time = 1.08, size = 142, normalized size = 1.34 \begin {gather*} -\frac {i 2^p x^2 \left (\frac {e^{i a} \left (c x^n\right )^{i b}}{1+e^{2 i a} \left (c x^n\right )^{2 i b}}\right )^p \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \, _2F_1\left (-\frac {i}{b n}+\frac {p}{2},p;1-\frac {i}{b n}+\frac {p}{2};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-2 i+b n p} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int x \left (\sec ^{p}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
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 \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 x \sec ^{p}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, 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 x\,{\left (\frac {1}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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