Optimal. Leaf size=109 \[ \frac {2 x \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac {1}{2},\frac {1}{4} \left (1-\frac {2 i}{b n}\right );\frac {1}{4} \left (5-\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{2+i b n} \]
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Rubi [A] time = 0.07, 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 = {4504, 4508, 364} \[ \frac {2 x \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac {1}{2},\frac {1}{4} \left (1-\frac {2 i}{b n}\right );\frac {1}{4} \left (5-\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{2+i b n} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4504
Rule 4508
Rubi steps
\begin {align*} \int \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {1}{n}} \sqrt {\csc (a+b \log (x))} \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x \left (c x^n\right )^{-\frac {i b}{2}-\frac {1}{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 )}\right ) \operatorname {Subst}\left (\int \frac {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}\\ &=\frac {2 x \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 )} \, _2F_1\left (\frac {1}{2},\frac {1}{4} \left (1-\frac {2 i}{b n}\right );\frac {1}{4} \left (5-\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 = 0.63, size = 115, normalized size = 1.06 \[ \frac {2 i e^{-2 i a} x \left (c x^n\right )^{-2 i b} \left (-1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \, _2F_1\left (1,\frac {3}{4}+\frac {i}{2 b n};\frac {5}{4}+\frac {i}{2 b n};e^{-2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{b n+2 i} \]
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 \[ \int \sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \sqrt {\csc }\left (a +b \ln \left (c \,x^{n}\right )\right )\, dx \]
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 \[ \int \sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )}\,{d x} \]
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 \[ \int \sqrt {\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}} \,d x \]
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 \[ \int \sqrt {\csc {\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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