Optimal. Leaf size=111 \[ -\frac {2 \, _2F_1\left (-\frac {1}{2},\frac {1}{4} \left (-1+\frac {2 i}{b n}\right );\frac {1}{4} \left (3+\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{(2+i b n) x \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \]
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Rubi [A]
time = 0.06, antiderivative size = 111, 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 = {4581, 4579,
371} \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {1}{2},\frac {1}{4} \left (\frac {2 i}{b n}-1\right );\frac {1}{4} \left (3+\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{x (2+i b n) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 4579
Rule 4581
Rubi steps
\begin {align*} \int \frac {\sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{x^2} \, dx &=\frac {\left (c x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int x^{-1-\frac {1}{n}} \sqrt {\sin (a+b \log (x))} \, dx,x,c x^n\right )}{n x}\\ &=\frac {\left (\left (c x^n\right )^{\frac {i b}{2}+\frac {1}{n}} \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}\right ) \text {Subst}\left (\int x^{-1-\frac {i b}{2}-\frac {1}{n}} \sqrt {1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n x \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}}\\ &=-\frac {2 \, _2F_1\left (-\frac {1}{2},\frac {1}{4} \left (-1+\frac {2 i}{b n}\right );\frac {1}{4} \left (3+\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{(2+i b n) x \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}}\\ \end {align*}
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Mathematica [A]
time = 6.79, size = 149, normalized size = 1.34 \begin {gather*} \frac {i \sqrt {2} \sqrt {-i e^{-i a} \left (c x^n\right )^{-i b} \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )} \, _2F_1\left (-\frac {1}{2},-\frac {1}{4}+\frac {i}{2 b n};\frac {3}{4}+\frac {i}{2 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(-2 i+b n) x \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\sin }\left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{2}}\, dx\]
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 \frac {\sqrt {\sin {\left (a + b \log {\left (c x^{n} \right )} \right )}}}{x^{2}}\, 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 {\sqrt {\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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