Optimal. Leaf size=87 \[ -\frac {2 e^{i a} \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1+\frac {2 i}{b n}\right );\frac {1}{2} \left (3+\frac {2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-i b n) x^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 87, 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 = {4605, 4601,
371} \begin {gather*} -\frac {2 e^{i a} \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1+\frac {2 i}{b n}\right );\frac {1}{2} \left (3+\frac {2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x^2 (2-i b n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 4601
Rule 4605
Rubi steps
\begin {align*} \int \frac {\sec \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=\frac {\left (c x^n\right )^{2/n} \text {Subst}\left (\int x^{-1-\frac {2}{n}} \sec (a+b \log (x)) \, dx,x,c x^n\right )}{n x^2}\\ &=\frac {\left (2 e^{i a} \left (c x^n\right )^{2/n}\right ) \text {Subst}\left (\int \frac {x^{-1+i b-\frac {2}{n}}}{1+e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n x^2}\\ &=-\frac {2 e^{i a} \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1+\frac {2 i}{b n}\right );\frac {1}{2} \left (3+\frac {2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-i b n) x^2}\\ \end {align*}
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Mathematica [A]
time = 0.76, size = 81, normalized size = 0.93 \begin {gather*} \frac {2 e^{i a} \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2}+\frac {i}{b n};\frac {3}{2}+\frac {i}{b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{(-2+i b n) x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sec \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{3}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sec {\left (a + b \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^3\,\cos \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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