Optimal. Leaf size=86 \[ \frac {2 e^{i a} x^2 \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {2 i}{b n}\right );\frac {1}{2} \left (3-\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-b n+2 i} \]
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Rubi [A] time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4510, 4506, 364} \[ \frac {2 e^{i a} x^2 \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {2 i}{b n}\right );\frac {1}{2} \left (3-\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-b n+2 i} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4506
Rule 4510
Rubi steps
\begin {align*} \int x \csc \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {2}{n}} \csc (a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=-\frac {\left (2 i e^{i a} x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname {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}\\ &=\frac {2 e^{i a} x^2 \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}\\ \end {align*}
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Mathematica [A] time = 1.51, size = 78, normalized size = 0.91 \[ -\frac {2 e^{i a} x^2 \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2}-\frac {i}{b n};\frac {3}{2}-\frac {i}{b n};e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{b n-2 i} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \csc \left (b \log \left (c x^{n}\right ) + a\right ), x\right ) \]
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 \[ \int x \csc \left (b \log \left (c x^{n}\right ) + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int x \csc \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx \]
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 \[ \int x \csc \left (b \log \left (c x^{n}\right ) + a\right )\,{d x} \]
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 \[ \int \frac {x}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
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 \[ \int x \csc {\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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