Optimal. Leaf size=130 \[ \frac {2 x^{1+m} \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac {1}{2},-\frac {2 i+2 i m-b n}{4 b n};-\frac {2 i+2 i m-5 b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{2+2 m+i b n} \]
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Rubi [A]
time = 0.06, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4605, 4603,
371} \begin {gather*} \frac {2 x^{m+1} \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac {1}{2},-\frac {2 i m-b n+2 i}{4 b n};-\frac {2 i m-5 b n+2 i}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{i b n+2 m+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 4603
Rule 4605
Rubi steps
\begin {align*} \int x^m \sqrt {\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 ) \text {Subst}\left (\int x^{-1+\frac {1+m}{n}} \sqrt {\sec (a+b \log (x))} \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {i b}{2}-\frac {1+m}{n}} \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {i b}{2}+\frac {1+m}{n}}}{\sqrt {1+e^{2 i a} x^{2 i b}}} \, dx,x,c x^n\right )}{n}\\ &=\frac {2 x^{1+m} \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac {1}{2},-\frac {2 i+2 i m-b n}{4 b n};-\frac {2 i+2 i m-5 b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{2+2 m+i b n}\\ \end {align*}
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Mathematica [A]
time = 5.68, size = 165, normalized size = 1.27 \begin {gather*} \frac {2 \sqrt {2} x^{1+m} \sqrt {\frac {e^{i a} \left (c x^n\right )^{i b}}{1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac {1}{2},\frac {-2 i-2 i m+b n}{4 b n};-\frac {2 i+2 i m-5 b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+2 m+i b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int x^{m} \left (\sqrt {\sec }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 x^{m} \sqrt {\sec {\left (a + b \log {\left (c x^{n} \right )} \right )}}\, 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 x^m\,\sqrt {\frac {1}{\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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