Optimal. Leaf size=141 \[ \frac {a (e x)^{2 n}}{2 e n}+\frac {i b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {i b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {2 b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n} \]
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Rubi [A] time = 0.11, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {14, 4209, 4205, 4183, 2279, 2391} \[ \frac {i b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {i b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}+\frac {a (e x)^{2 n}}{2 e n}-\frac {2 b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2279
Rule 2391
Rule 4183
Rule 4205
Rule 4209
Rubi steps
\begin {align*} \int (e x)^{-1+2 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx &=\int \left (a (e x)^{-1+2 n}+b (e x)^{-1+2 n} \csc \left (c+d x^n\right )\right ) \, dx\\ &=\frac {a (e x)^{2 n}}{2 e n}+b \int (e x)^{-1+2 n} \csc \left (c+d x^n\right ) \, dx\\ &=\frac {a (e x)^{2 n}}{2 e n}+\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \int x^{-1+2 n} \csc \left (c+d x^n\right ) \, dx}{e}\\ &=\frac {a (e x)^{2 n}}{2 e n}+\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int x \csc (c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac {a (e x)^{2 n}}{2 e n}-\frac {2 b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d e n}+\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d e n}\\ &=\frac {a (e x)^{2 n}}{2 e n}-\frac {2 b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}\\ &=\frac {a (e x)^{2 n}}{2 e n}-\frac {2 b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {i b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {i b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (e^{i \left (c+d x^n\right )}\right )}{d^2 e n}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 185, normalized size = 1.31 \[ \frac {x^{-2 n} (e x)^{2 n} \left (a d^2 x^{2 n}+2 i b \text {Li}_2\left (-e^{i \left (d x^n+c\right )}\right )-2 i b \text {Li}_2\left (e^{i \left (d x^n+c\right )}\right )+2 b d x^n \log \left (1-e^{i \left (c+d x^n\right )}\right )-2 b d x^n \log \left (1+e^{i \left (c+d x^n\right )}\right )+2 b c \log \left (1-e^{i \left (c+d x^n\right )}\right )-2 b c \log \left (1+e^{i \left (c+d x^n\right )}\right )-2 b c \log \left (\tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )\right )}{2 d^2 e n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 384, normalized size = 2.72 \[ \frac {a d^{2} e^{2 \, n - 1} x^{2 \, n} - b d e^{2 \, n - 1} x^{n} \log \left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + 1\right ) - b d e^{2 \, n - 1} x^{n} \log \left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + 1\right ) - b c e^{2 \, n - 1} \log \left (-\frac {1}{2} \, \cos \left (d x^{n} + c\right ) + \frac {1}{2} i \, \sin \left (d x^{n} + c\right ) + \frac {1}{2}\right ) - b c e^{2 \, n - 1} \log \left (-\frac {1}{2} \, \cos \left (d x^{n} + c\right ) - \frac {1}{2} i \, \sin \left (d x^{n} + c\right ) + \frac {1}{2}\right ) - i \, b e^{2 \, n - 1} {\rm Li}_2\left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) + i \, b e^{2 \, n - 1} {\rm Li}_2\left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) - i \, b e^{2 \, n - 1} {\rm Li}_2\left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) + i \, b e^{2 \, n - 1} {\rm Li}_2\left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) + {\left (b d e^{2 \, n - 1} x^{n} + b c e^{2 \, n - 1}\right )} \log \left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + 1\right ) + {\left (b d e^{2 \, n - 1} x^{n} + b c e^{2 \, n - 1}\right )} \log \left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + 1\right )}{2 \, d^{2} n} \]
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 \[ \int {\left (b \csc \left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.88, size = 731, normalized size = 5.18 \[ \text {result too large to display} \]
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 \[ {\left (e^{2 \, n + 1} \int \frac {x^{2 \, n} \sin \left (d x^{n} + c\right )}{e^{2} x \cos \left (d x^{n} + c\right )^{2} + e^{2} x \sin \left (d x^{n} + c\right )^{2} + 2 \, e^{2} x \cos \left (d x^{n} + c\right ) + e^{2} x}\,{d x} + e^{2 \, n + 1} \int \frac {x^{2 \, n} \sin \left (d x^{n} + c\right )}{e^{2} x \cos \left (d x^{n} + c\right )^{2} + e^{2} x \sin \left (d x^{n} + c\right )^{2} - 2 \, e^{2} x \cos \left (d x^{n} + c\right ) + e^{2} x}\,{d x}\right )} b + \frac {\left (e x\right )^{2 \, n} a}{2 \, e n} \]
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 \[ \int \left (a+\frac {b}{\sin \left (c+d\,x^n\right )}\right )\,{\left (e\,x\right )}^{2\,n-1} \,d x \]
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 \[ \int \left (e x\right )^{2 n - 1} \left (a + b \csc {\left (c + d x^{n} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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