Optimal. Leaf size=119 \[ -\frac {4 e^{2 i a d} (e x)^{m+1} \left (c x^n\right )^{2 i b d} \, _2F_1\left (2,-\frac {i (m+1)-2 b d n}{2 b d n};-\frac {i (m+1)-4 b d n}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2 i b d n+m+1)} \]
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Rubi [A] time = 0.10, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4510, 4506, 364} \[ -\frac {4 e^{2 i a d} (e x)^{m+1} \left (c x^n\right )^{2 i b d} \, _2F_1\left (2,-\frac {i (m+1)-2 b d n}{2 b d n};-\frac {i (m+1)-4 b d n}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2 i b d n+m+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4506
Rule 4510
Rubi steps
\begin {align*} \int (e x)^m \csc ^2\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 x^{-1+\frac {1+m}{n}} \csc ^2(d (a+b \log (x))) \, dx,x,c x^n\right )}{e n}\\ &=-\frac {\left (4 e^{2 i a d} (e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+2 i b d+\frac {1+m}{n}}}{\left (1-e^{2 i a d} x^{2 i b d}\right )^2} \, dx,x,c x^n\right )}{e n}\\ &=-\frac {4 e^{2 i a d} (e x)^{1+m} \left (c x^n\right )^{2 i b d} \, _2F_1\left (2,-\frac {i (1+m)-2 b d n}{2 b d n};-\frac {i (1+m)-4 b d n}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (1+m+2 i b d n)}\\ \end {align*}
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Mathematica [B] time = 6.53, size = 534, normalized size = 4.49 \[ \frac {x (e x)^m \sin (b d n \log (x)) \csc \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \csc \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b d n \log (x)\right )}{b d n}-\frac {(m+1) x^{-m} (e x)^m \csc \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \left (\frac {x^{m+1} \sin (b d n \log (x)) \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{m+1}-\frac {i \sin \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \exp \left (-\frac {(2 m+1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \left (-(2 i b d n+m+1) \exp \left (\frac {2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right ) \, _2F_1\left (1,-\frac {i (m+1)}{2 b d n};1-\frac {i (m+1)}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-(m+1) \exp \left (\frac {a (2 i b d n+2 m+1)}{b n}+\frac {(2 i b d n+2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )}{n}+\log (x) (2 i b d n+m+1)\right ) \, _2F_1\left (1,-\frac {i (m+2 i b d n+1)}{2 b d n};-\frac {i (m+4 i b d n+1)}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+i (2 i b d n+m+1) \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \exp \left (\frac {2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right )\right )}{(m+1) (2 i b d n+m+1)}\right )}{b d n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \csc \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}, x\right ) \]
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 \[ \int \left (e x\right )^{m} \csc \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.18, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\csc ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (e\,x\right )}^m}{{\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2} \,d x \]
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 \[ \int \left (e x\right )^{m} \csc ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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