Integrand size = 24, antiderivative size = 377 \[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\frac {a^2 (e x)^{3 n}}{3 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{d e n}-\frac {4 a b x^{-n} (e x)^{3 n} \text {arctanh}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac {b^2 x^{-n} (e x)^{3 n} \cot \left (c+d x^n\right )}{d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1-e^{2 i \left (c+d x^n\right )}\right )}{d^2 e n}+\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {i b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d x^n\right )}\right )}{d^3 e n}-\frac {4 a b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac {4 a b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,e^{i \left (c+d x^n\right )}\right )}{d^3 e n} \]
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Time = 0.48 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4294, 4290, 4275, 4268, 2611, 2320, 6724, 4269, 3798, 2221, 2317, 2438} \[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\frac {a^2 (e x)^{3 n}}{3 e n}-\frac {4 a b x^{-n} (e x)^{3 n} \text {arctanh}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac {4 a b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-e^{i \left (d x^n+c\right )}\right )}{d^3 e n}+\frac {4 a b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,e^{i \left (d x^n+c\right )}\right )}{d^3 e n}+\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {i b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,e^{2 i \left (d x^n+c\right )}\right )}{d^3 e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1-e^{2 i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {b^2 x^{-n} (e x)^{3 n} \cot \left (c+d x^n\right )}{d e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{d e n} \]
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Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3798
Rule 4268
Rule 4269
Rule 4275
Rule 4290
Rule 4294
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \text {arctanh}\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 ) \text {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 ) \text {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 ) \text {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} \text {arctanh}\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} \operatorname {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} \operatorname {PolyLog}\left (2,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 ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-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 ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,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 ) \text {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} \text {arctanh}\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} \operatorname {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} \operatorname {PolyLog}\left (2,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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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 ) \text {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} \text {arctanh}\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} \operatorname {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} \operatorname {PolyLog}\left (2,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {4 a b x^{-3 n} (e x)^{3 n} \operatorname {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} \operatorname {PolyLog}\left (3,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 ) \text {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} \text {arctanh}\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} \operatorname {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} \operatorname {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} \operatorname {PolyLog}\left (2,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} \operatorname {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} \operatorname {PolyLog}\left (3,e^{i \left (c+d x^n\right )}\right )}{d^3 e n} \\ \end{align*}
\[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx \]
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\[\int \left (e x \right )^{-1+3 n} {\left (a +b \csc \left (c +d \,x^{n}\right )\right )}^{2}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 890 vs. \(2 (362) = 724\).
Time = 0.30 (sec) , antiderivative size = 890, normalized size of antiderivative = 2.36 \[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\frac {a^{2} d^{3} e^{3 \, n - 1} x^{3 \, n} \sin \left (d x^{n} + c\right ) - 3 \, b^{2} d^{2} e^{3 \, n - 1} x^{2 \, n} \cos \left (d x^{n} + c\right ) + 6 \, a b e^{3 \, n - 1} {\rm polylog}\left (3, \cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) \sin \left (d x^{n} + c\right ) + 6 \, a b e^{3 \, n - 1} {\rm polylog}\left (3, \cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) \sin \left (d x^{n} + c\right ) - 6 \, a b e^{3 \, n - 1} {\rm polylog}\left (3, -\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) \sin \left (d x^{n} + c\right ) - 6 \, a b e^{3 \, n - 1} {\rm polylog}\left (3, -\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) \sin \left (d x^{n} + c\right ) + 3 \, {\left (a b c^{2} - b^{2} c\right )} e^{3 \, n - 1} \log \left (-\frac {1}{2} \, \cos \left (d x^{n} + c\right ) + \frac {1}{2} i \, \sin \left (d x^{n} + c\right ) + \frac {1}{2}\right ) \sin \left (d x^{n} + c\right ) + 3 \, {\left (a b c^{2} - b^{2} c\right )} e^{3 \, n - 1} \log \left (-\frac {1}{2} \, \cos \left (d x^{n} + c\right ) - \frac {1}{2} i \, \sin \left (d x^{n} + c\right ) + \frac {1}{2}\right ) \sin \left (d x^{n} + c\right ) - 3 \, {\left (2 i \, a b d e^{3 \, n - 1} x^{n} + i \, b^{2} e^{3 \, n - 1}\right )} {\rm Li}_2\left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) \sin \left (d x^{n} + c\right ) - 3 \, {\left (-2 i \, a b d e^{3 \, n - 1} x^{n} - i \, b^{2} e^{3 \, n - 1}\right )} {\rm Li}_2\left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) \sin \left (d x^{n} + c\right ) - 3 \, {\left (2 i \, a b d e^{3 \, n - 1} x^{n} - i \, b^{2} e^{3 \, n - 1}\right )} {\rm Li}_2\left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) \sin \left (d x^{n} + c\right ) - 3 \, {\left (-2 i \, a b d e^{3 \, n - 1} x^{n} + i \, b^{2} e^{3 \, n - 1}\right )} {\rm Li}_2\left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) \sin \left (d x^{n} + c\right ) - 3 \, {\left (a b d^{2} e^{3 \, n - 1} x^{2 \, n} - b^{2} d e^{3 \, n - 1} x^{n}\right )} \log \left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + 1\right ) \sin \left (d x^{n} + c\right ) - 3 \, {\left (a b d^{2} e^{3 \, n - 1} x^{2 \, n} - b^{2} d e^{3 \, n - 1} x^{n}\right )} \log \left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + 1\right ) \sin \left (d x^{n} + c\right ) + 3 \, {\left (a b d^{2} e^{3 \, n - 1} x^{2 \, n} + b^{2} d e^{3 \, n - 1} x^{n} - {\left (a b c^{2} - b^{2} c\right )} e^{3 \, n - 1}\right )} \log \left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + 1\right ) \sin \left (d x^{n} + c\right ) + 3 \, {\left (a b d^{2} e^{3 \, n - 1} x^{2 \, n} + b^{2} d e^{3 \, n - 1} x^{n} - {\left (a b c^{2} - b^{2} c\right )} e^{3 \, n - 1}\right )} \log \left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + 1\right ) \sin \left (d x^{n} + c\right )}{3 \, d^{3} n \sin \left (d x^{n} + c\right )} \]
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\[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\int \left (e x\right )^{3 n - 1} \left (a + b \csc {\left (c + d x^{n} \right )}\right )^{2}\, dx \]
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\[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\int { {\left (b \csc \left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{3 \, n - 1} \,d x } \]
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\[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\int { {\left (b \csc \left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{3 \, n - 1} \,d x } \]
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Timed out. \[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\int {\left (a+\frac {b}{\sin \left (c+d\,x^n\right )}\right )}^2\,{\left (e\,x\right )}^{3\,n-1} \,d x \]
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