Optimal. Leaf size=129 \[ \frac {2 x^{m+1} \, _2F_1\left (-\frac {1}{2},-\frac {2 i m+b n+2 i}{4 b n};-\frac {2 i m-3 b n+2 i}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(-i b n+2 m+2) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \]
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Rubi [A] time = 0.09, antiderivative size = 126, normalized size of antiderivative = 0.98, 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 = {4510, 4508, 364} \[ \frac {2 x^{m+1} \, _2F_1\left (-\frac {1}{2},\frac {1}{4} \left (-\frac {2 i (m+1)}{b n}-1\right );-\frac {2 i m-3 b n+2 i}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(-i b n+2 m+2) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4508
Rule 4510
Rubi steps
\begin {align*} \int \frac {x^m}{\sqrt {\csc \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 ) \operatorname {Subst}\left (\int \frac {x^{-1+\frac {1+m}{n}}}{\sqrt {\csc (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}}\right ) \operatorname {Subst}\left (\int 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 \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}\\ &=\frac {2 x^{1+m} \, _2F_1\left (-\frac {1}{2},\frac {1}{4} \left (-1-\frac {2 i (1+m)}{b n}\right );-\frac {2 i+2 i m-3 b n}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m-i b n) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}\\ \end {align*}
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Mathematica [B] time = 7.23, size = 441, normalized size = 3.42 \[ \frac {2 x^{m+1} \sin \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \left (2 (m+1) \sin \left (a+b \log \left (c x^n\right )-b n \log (x)\right )+b n \cos \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}-\frac {2 e^{i a} b n x^{m+1} \left (c x^n\right )^{i b} \sqrt {2-2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\frac {i e^{i a} \left (c x^n\right )^{i b}}{-1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \left ((3 b n-2 i m-2 i) \, _2F_1\left (\frac {1}{2},-\frac {2 i m+b n+2 i}{4 b n};-\frac {2 i m-3 b n+2 i}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right )+(b n+2 i m+2 i) x^{2 i b n} \, _2F_1\left (\frac {1}{2},-\frac {i \left (m+\frac {3 i b n}{2}+1\right )}{2 b n};-\frac {2 i m-7 b n+2 i}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )}{(-i b n+2 m+2) (3 i b n+2 m+2) \left (e^{2 i a} (b n-2 i m-2 i) \left (c x^n\right )^{2 i b}+(b n+2 i m+2 i) x^{2 i b n}\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {x^{m}}{\sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\sqrt {\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}}\, dx \]
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 \[ \int \frac {x^{m}}{\sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \]
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 \[ \int \frac {x^m}{\sqrt {\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}}} \,d x \]
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 \[ \int \frac {x^{m}}{\sqrt {\csc {\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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