Optimal. Leaf size=103 \[ \frac {2 e^{i a} x^{m+1} \left (c x^n\right )^{i b} \, _2F_1\left (1,-\frac {i m-b n+i}{2 b n};-\frac {i (m+1)-3 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{i b n+m+1} \]
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Rubi [A] time = 0.07, antiderivative size = 99, normalized size of antiderivative = 0.96, 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 = {4509, 4505, 364} \[ \frac {2 e^{i a} x^{m+1} \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {i (m+1)}{b n}\right );-\frac {i (m+1)-3 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{i b n+m+1} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4505
Rule 4509
Rubi steps
\begin {align*} \int x^m \sec \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 (a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (2 e^{i a} x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+i b+\frac {1+m}{n}}}{1+e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n}\\ &=\frac {2 e^{i a} x^{1+m} \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {i (1+m)}{b n}\right );-\frac {i (1+m)-3 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+m+i b n}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 94, normalized size = 0.91 \[ \frac {2 e^{i a} x^{m+1} \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2}-\frac {i (m+1)}{2 b n};\frac {3}{2}-\frac {i (m+1)}{2 b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{i b n+m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \sec \left (b \log \left (c x^{n}\right ) + a\right ), x\right ) \]
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 \[ \int x^{m} \sec \left (b \log \left (c x^{n}\right ) + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int x^{m} \sec \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx \]
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 \[ \int x^{m} \sec \left (b \log \left (c x^{n}\right ) + a\right )\,{d x} \]
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 \[ \int \frac {x^m}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
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 \[ \int x^{m} \sec {\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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