Optimal. Leaf size=499 \[ \frac {2 i b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^3 e n \sqrt {b^2-a^2}}-\frac {2 i b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^3 e n \sqrt {b^2-a^2}}+\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}-\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}+\frac {i b x^{-n} (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d e n \sqrt {b^2-a^2}}-\frac {i b x^{-n} (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d e n \sqrt {b^2-a^2}}+\frac {(e x)^{3 n}}{3 a e n} \]
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Rubi [A] time = 0.94, antiderivative size = 499, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4209, 4205, 4191, 3323, 2264, 2190, 2531, 2282, 6589} \[ \frac {2 i b x^{-3 n} (e x)^{3 n} \text {PolyLog}\left (3,\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^3 e n \sqrt {b^2-a^2}}-\frac {2 i b x^{-3 n} (e x)^{3 n} \text {PolyLog}\left (3,\frac {i a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d^3 e n \sqrt {b^2-a^2}}+\frac {2 b x^{-2 n} (e x)^{3 n} \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}-\frac {2 b x^{-2 n} (e x)^{3 n} \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d^2 e n \sqrt {b^2-a^2}}+\frac {i b x^{-n} (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d e n \sqrt {b^2-a^2}}-\frac {i b x^{-n} (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d e n \sqrt {b^2-a^2}}+\frac {(e x)^{3 n}}{3 a e n} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 3323
Rule 4191
Rule 4205
Rule 4209
Rule 6589
Rubi steps
\begin {align*} \int \frac {(e x)^{-1+3 n}}{a+b \csc \left (c+d x^n\right )} \, dx &=\frac {\left (x^{-3 n} (e x)^{3 n}\right ) \int \frac {x^{-1+3 n}}{a+b \csc \left (c+d x^n\right )} \, dx}{e}\\ &=\frac {\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b \csc (c+d x)} \, dx,x,x^n\right )}{e n}\\ &=\frac {\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \left (\frac {x^2}{a}-\frac {b x^2}{a (b+a \sin (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}-\frac {\left (b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {x^2}{b+a \sin (c+d x)} \, dx,x,x^n\right )}{a e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,x^n\right )}{a e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}+\frac {\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b-2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^n\right )}{\sqrt {-a^2+b^2} e n}-\frac {\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b+2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^n\right )}{\sqrt {-a^2+b^2} e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}+\frac {i b x^{-n} (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {i b x^{-n} (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int x \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int x \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} d e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}+\frac {i b x^{-n} (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {i b x^{-n} (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} d^2 e n}+\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} d^2 e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}+\frac {i b x^{-n} (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {i b x^{-n} (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}+\frac {\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i a x}{b-\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a \sqrt {-a^2+b^2} d^3 e n}-\frac {\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a \sqrt {-a^2+b^2} d^3 e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}+\frac {i b x^{-n} (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {i b x^{-n} (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}+\frac {2 i b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3 e n}-\frac {2 i b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3 e n}\\ \end {align*}
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Mathematica [F] time = 1.88, size = 0, normalized size = 0.00 \[ \int \frac {(e x)^{-1+3 n}}{a+b \csc \left (c+d x^n\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.67, size = 1697, normalized size = 3.40 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (e x\right )^{3 \, n - 1}}{b \csc \left (d x^{n} + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.67, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{-1+3 n}}{a +b \csc \left (c +d \,x^{n}\right )}\, dx \]
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 = -0.00 \[ \int \frac {{\left (e\,x\right )}^{3\,n-1}}{a+\frac {b}{\sin \left (c+d\,x^n\right )}} \,d x \]
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 \[ \int \frac {\left (e x\right )^{3 n - 1}}{a + b \csc {\left (c + d x^{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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