Optimal. Leaf size=150 \[ \frac {2 (e x)^{m+1} \sqrt {1-e^{2 i a d} \left (c x^n\right )^{2 i b d}} \, _2F_1\left (\frac {1}{2},-\frac {2 i m-b d n+2 i}{4 b d n};-\frac {2 i m-5 b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (i b d n+2 m+2) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \]
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Rubi [A] time = 0.11, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4493, 4491, 364} \[ \frac {2 (e x)^{m+1} \sqrt {1-e^{2 i a d} \left (c x^n\right )^{2 i b d}} \, _2F_1\left (\frac {1}{2},-\frac {2 i m-b d n+2 i}{4 b d n};-\frac {2 i m-5 b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (i b d n+2 m+2) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4491
Rule 4493
Rubi steps
\begin {align*} \int \frac {(e x)^m}{\sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \, dx &=\frac {\left ((e 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 {\sin (d (a+b \log (x)))}} \, dx,x,c x^n\right )}{e n}\\ &=\frac {\left ((e x)^{1+m} \left (c x^n\right )^{-\frac {1}{2} i b d-\frac {1+m}{n}} \sqrt {1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+\frac {i b d}{2}+\frac {1+m}{n}}}{\sqrt {1-e^{2 i a d} x^{2 i b d}}} \, dx,x,c x^n\right )}{e n \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}}\\ &=\frac {2 (e x)^{1+m} \sqrt {1-e^{2 i a d} \left (c x^n\right )^{2 i b d}} \, _2F_1\left (\frac {1}{2},-\frac {2 i+2 i m-b d n}{4 b d n};-\frac {2 i+2 i m-5 b d n}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2+2 m+i b d n) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 131, normalized size = 0.87 \[ -\frac {2 x (e x)^m \left (-1+e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right ) \, _2F_1\left (1,-\frac {2 i m-3 b d n+2 i}{4 b d n};-\frac {2 i m-5 b d n+2 i}{4 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{(i b d n+2 m+2) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \]
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 \[ \int \frac {\left (e x\right )^{m}}{\sqrt {\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{m}}{\sqrt {\sin \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}}\, dx \]
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 \[ \int \frac {\left (e x\right )^{m}}{\sqrt {\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^m}{\sqrt {\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}} \,d x \]
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 \[ \int \frac {\left (e x\right )^{m}}{\sqrt {\sin {\left (a d + b d \log {\left (c x^{n} \right )} \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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