Optimal. Leaf size=377 \[ -\frac{4 a b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,-e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{4 a b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{i b^2 x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (2,e^{2 i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1-e^{2 i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cot \left (c+d x^n\right )}{d e n}-\frac{i b^2 x^{-n} (e x)^{3 n}}{d e n} \]
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Rubi [A] time = 0.395994, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4209, 4205, 4190, 4183, 2531, 2282, 6589, 4184, 3717, 2190, 2279, 2391} \[ -\frac{4 a b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,-e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{4 a b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{i b^2 x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (2,e^{2 i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1-e^{2 i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cot \left (c+d x^n\right )}{d e n}-\frac{i b^2 x^{-n} (e x)^{3 n}}{d e n} \]
Antiderivative was successfully verified.
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Rule 4209
Rule 4205
Rule 4190
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rule 4184
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \int x^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx}{e}\\ &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x^2 (a+b \csc (c+d x))^2 \, dx,x,x^n\right )}{e n}\\ &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \left (a^2 x^2+2 a b x^2 \csc (c+d x)+b^2 x^2 \csc ^2(c+d x)\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}+\frac{\left (2 a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x^2 \csc (c+d x) \, dx,x,x^n\right )}{e n}+\frac{\left (b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x^2 \csc ^2(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cot \left (c+d x^n\right )}{d e n}-\frac{\left (4 a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d e n}+\frac{\left (4 a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d e n}+\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \cot (c+d x) \, dx,x,x^n\right )}{d e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{i b^2 x^{-n} (e x)^{3 n}}{d e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cot \left (c+d x^n\right )}{d e n}+\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{\left (4 i a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d^2 e n}+\frac{\left (4 i a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d^2 e n}-\frac{\left (4 i b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i (c+d x)} x}{1-e^{2 i (c+d x)}} \, dx,x,x^n\right )}{d e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{i b^2 x^{-n} (e x)^{3 n}}{d e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cot \left (c+d x^n\right )}{d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1-e^{2 i \left (c+d x^n\right )}\right )}{d^2 e n}+\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{\left (4 a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{\left (4 a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{d^3 e n}-\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i (c+d x)}\right ) \, dx,x,x^n\right )}{d^2 e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{i b^2 x^{-n} (e x)^{3 n}}{d e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cot \left (c+d x^n\right )}{d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1-e^{2 i \left (c+d x^n\right )}\right )}{d^2 e n}+\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{4 a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{4 a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{\left (i b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \left (c+d x^n\right )}\right )}{d^3 e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{i b^2 x^{-n} (e x)^{3 n}}{d e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cot \left (c+d x^n\right )}{d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1-e^{2 i \left (c+d x^n\right )}\right )}{d^2 e n}+\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{i b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (e^{2 i \left (c+d x^n\right )}\right )}{d^3 e n}-\frac{4 a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{4 a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (e^{i \left (c+d x^n\right )}\right )}{d^3 e n}\\ \end{align*}
Mathematica [F] time = 12.912, size = 0, normalized size = 0. \[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 1.053, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{-1+3\,n} \left ( a+b\csc \left ( c+d{x}^{n} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.716747, size = 2184, normalized size = 5.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \csc \left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{3 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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