Optimal. Leaf size=102 \[ \frac {4 e^{2 i a} x^{m+1} \left (c x^n\right )^{2 i b} \, _2F_1\left (2,-\frac {i (m+1)-2 b n}{2 b n};-\frac {i (m+1)-4 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2 i b n+m+1} \]
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Rubi [A] time = 0.08, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4509, 4505, 364} \[ \frac {4 e^{2 i a} x^{m+1} \left (c x^n\right )^{2 i b} \, _2F_1\left (2,-\frac {i (m+1)-2 b n}{2 b n};-\frac {i (m+1)-4 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2 i b n+m+1} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4505
Rule 4509
Rubi steps
\begin {align*} \int x^m \sec ^2\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {1+m}{n}} \sec ^2(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (4 e^{2 i a} x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+2 i b+\frac {1+m}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^2} \, dx,x,c x^n\right )}{n}\\ &=\frac {4 e^{2 i a} x^{1+m} \left (c x^n\right )^{2 i b} \, _2F_1\left (2,-\frac {i (1+m)-2 b n}{2 b n};-\frac {i (1+m)-4 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+m+2 i b n}\\ \end {align*}
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Mathematica [B] time = 17.18, size = 482, normalized size = 4.73 \[ \frac {x^{m+1} \sin (b n \log (x)) \sec \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \sec \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )+b n \log (x)\right )}{b n}-\frac {(m+1) \sec \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \left (\frac {x^{m+1} \sin (b n \log (x)) \sec \left (a+b \log \left (c x^n\right )\right )}{m+1}-\frac {i \cos \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \exp \left (-\frac {(2 m+1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \left ((2 i b n+m+1) \left (-\exp \left (\frac {2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right )\right ) \, _2F_1\left (1,-\frac {i (m+1)}{2 b n};1-\frac {i (m+1)}{2 b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+(m+1) \exp \left (\frac {a (2 i b n+2 m+1)}{b n}+\frac {(2 i b n+2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )}{n}+\log (x) (2 i b n+m+1)\right ) \, _2F_1\left (1,-\frac {i (m+2 i b n+1)}{2 b n};-\frac {i (m+4 i b n+1)}{2 b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )-i (2 i b n+m+1) \tan \left (a+b \log \left (c x^n\right )\right ) \exp \left (\frac {2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right )\right )}{(m+1) (2 i b n+m+1)}\right )}{b n} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \sec \left (b \log \left (c x^{n}\right ) + a\right )^{2}, x\right ) \]
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 \[ \int x^{m} \sec \left (b \log \left (c x^{n}\right ) + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.53, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\sec ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {x^m}{{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
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 \[ \int x^{m} \sec ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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