Optimal. Leaf size=87 \[ \frac {4 e^{2 i a} x^3 \left (c x^n\right )^{2 i b} \, _2F_1\left (2,\frac {1}{2} \left (2-\frac {3 i}{b n}\right );\frac {1}{2} \left (4-\frac {3 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3+2 i b n} \]
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Rubi [A] time = 0.08, antiderivative size = 87, 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} \[ \frac {4 e^{2 i a} x^3 \left (c x^n\right )^{2 i b} \, _2F_1\left (2,\frac {1}{2} \left (2-\frac {3 i}{b n}\right );\frac {1}{2} \left (4-\frac {3 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3+2 i b n} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4505
Rule 4509
Rubi steps
\begin {align*} \int x^2 \sec ^2\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {3}{n}} \sec ^2(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (4 e^{2 i a} x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+2 i b+\frac {3}{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^3 \left (c x^n\right )^{2 i b} \, _2F_1\left (2,\frac {1}{2} \left (2-\frac {3 i}{b n}\right );\frac {1}{2} \left (4-\frac {3 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3+2 i b n}\\ \end {align*}
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Mathematica [A] time = 5.41, size = 160, normalized size = 1.84 \[ \frac {x^3 \left (3 e^{2 i a} \left (c x^n\right )^{2 i b} \, _2F_1\left (1,1-\frac {3 i}{2 b n};2-\frac {3 i}{2 b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b n-3 i) \left (\tan \left (a+b \log \left (c x^n\right )\right )-i \, _2F_1\left (1,-\frac {3 i}{2 b n};1-\frac {3 i}{2 b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )\right )\right )}{b n (2 b n-3 i)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.36, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \sec \left (b \log \left (c x^{n}\right ) + a\right )^{2}, x\right ) \]
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 \[ \int x^{2} \sec \left (b \log \left (c x^{n}\right ) + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.31, size = 0, normalized size = 0.00 \[ \int x^{2} \left (\sec ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
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 \[ \int \frac {x^2}{{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
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 \[ \int x^{2} \sec ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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