Optimal. Leaf size=102 \[ \frac {8 e^{3 i a} x^{m+1} \left (c x^n\right )^{3 i b} \, _2F_1\left (3,-\frac {i (m+1)-3 b n}{2 b n};-\frac {i (m+1)-5 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3 i b n+m+1} \]
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Rubi [A] time = 0.09, 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} \[ \frac {8 e^{3 i a} x^{m+1} \left (c x^n\right )^{3 i b} \, _2F_1\left (3,-\frac {i (m+1)-3 b n}{2 b n};-\frac {i (m+1)-5 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3 i b n+m+1} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4505
Rule 4509
Rubi steps
\begin {align*} \int x^m \sec ^3\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 ^3(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (8 e^{3 i a} x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+3 i b+\frac {1+m}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^3} \, dx,x,c x^n\right )}{n}\\ &=\frac {8 e^{3 i a} x^{1+m} \left (c x^n\right )^{3 i b} \, _2F_1\left (3,-\frac {i (1+m)-3 b n}{2 b n};-\frac {i (1+m)-5 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+m+3 i b n}\\ \end {align*}
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Mathematica [A] time = 5.60, size = 134, normalized size = 1.31 \[ \frac {x^{m+1} \left (-2 \sec \left (a+b \log \left (c x^n\right )\right ) \left (-b n \tan \left (a+b \log \left (c x^n\right )\right )+m+1\right )+4 e^{i a} (-i b n+m+1) \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2}-\frac {i (m+1)}{2 b n};\frac {3}{2}-\frac {i (m+1)}{2 b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{4 b^2 n^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \sec \left (b \log \left (c x^{n}\right ) + a\right )^{3}, x\right ) \]
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 \[ \int x^{m} \sec \left (b \log \left (c x^{n}\right ) + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.75, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\sec ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
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 \[ \int \frac {x^m}{{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]
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 \[ \int x^{m} \sec ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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