Optimal. Leaf size=109 \[ \frac {2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i}{b n}\right );\frac {1}{4} \left (7-\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+3 i b n} \]
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Rubi [A]
time = 0.05, antiderivative size = 109, 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 = {4600, 4604,
371} \begin {gather*} \frac {2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \, _2F_1\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i}{b n}\right );\frac {1}{4} \left (7-\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{2+3 i b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 4600
Rule 4604
Rubi steps
\begin {align*} \int \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \csc ^{\frac {3}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x \left (c x^n\right )^{-\frac {3 i b}{2}-\frac {1}{n}} \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {3 i b}{2}+\frac {1}{n}}}{\left (1-e^{2 i a} x^{2 i b}\right )^{3/2}} \, dx,x,c x^n\right )}{n}\\ &=\frac {2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i}{b n}\right );\frac {1}{4} \left (7-\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+3 i b n}\\ \end {align*}
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Mathematica [A]
time = 7.43, size = 127, normalized size = 1.17 \begin {gather*} \frac {2 e^{-i a} x \left (c x^n\right )^{-i b} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \left (-1+\sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac {1}{2},-\frac {2 i+b n}{4 b n};\frac {3}{4}-\frac {i}{2 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \csc ^{\frac {3}{2}}\left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]
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 \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 \csc ^{\frac {3}{2}}{\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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\left (\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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