Optimal. Leaf size=79 \[ \frac {2 e^{2 i a} x^2 \left (c x^n\right )^{2 i b} \, _2F_1\left (2,1-\frac {i}{b n};2-\frac {i}{b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+i b n} \]
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Rubi [A] time = 0.07, antiderivative size = 79, 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 = {4509, 4505, 364} \[ \frac {2 e^{2 i a} x^2 \left (c x^n\right )^{2 i b} \, _2F_1\left (2,1-\frac {i}{b n};2-\frac {i}{b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+i b n} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4505
Rule 4509
Rubi steps
\begin {align*} \int x \sec ^2\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {2}{n}} \sec ^2(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (4 e^{2 i a} x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+2 i b+\frac {2}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^2} \, dx,x,c x^n\right )}{n}\\ &=\frac {2 e^{2 i a} x^2 \left (c x^n\right )^{2 i b} \, _2F_1\left (2,1-\frac {i}{b n};2-\frac {i}{b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+i b n}\\ \end {align*}
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Mathematica [A] time = 5.20, size = 149, normalized size = 1.89 \[ \frac {x^2 \left (e^{2 i a} \left (c x^n\right )^{2 i b} \, _2F_1\left (1,1-\frac {i}{b n};2-\frac {i}{b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+(b n-i) \left (\tan \left (a+b \log \left (c x^n\right )\right )-i \, _2F_1\left (1,-\frac {i}{b n};1-\frac {i}{b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )\right )\right )}{b n (b n-i)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \sec \left (b \log \left (c x^{n}\right ) + a\right )^{2}, x\right ) \]
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 \[ \int x \sec \left (b \log \left (c x^{n}\right ) + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.21, size = 0, normalized size = 0.00 \[ \int x \left (\sec ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
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 \[ \int \frac {x}{{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
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 \[ \int x \sec ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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