Optimal. Leaf size=221 \[ \frac {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \text {PolyLog}\left (2,-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \text {PolyLog}\left (2,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 b x^{-3 n} (e x)^{3 n} \text {PolyLog}\left (3,-e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac {2 b x^{-3 n} (e x)^{3 n} \text {PolyLog}\left (3,e^{i \left (c+d x^n\right )}\right )}{d^3 e n} \]
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Rubi [A]
time = 0.13, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {14, 4294,
4290, 4268, 2611, 2320, 6724} \begin {gather*} \frac {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-e^{i \left (d x^n+c\right )}\right )}{d^3 e n}+\frac {2 b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (e^{i \left (d x^n+c\right )}\right )}{d^3 e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {2 i b x^{-2 n} (e x)^{3 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)^{3 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2320
Rule 2611
Rule 4268
Rule 4290
Rule 4294
Rule 6724
Rubi steps
\begin {align*} \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx &=\int \left (a (e x)^{-1+3 n}+b (e x)^{-1+3 n} \csc \left (c+d x^n\right )\right ) \, dx\\ &=\frac {a (e x)^{3 n}}{3 e n}+b \int (e x)^{-1+3 n} \csc \left (c+d x^n\right ) \, dx\\ &=\frac {a (e x)^{3 n}}{3 e n}+\frac {\left (b x^{-3 n} (e x)^{3 n}\right ) \int x^{-1+3 n} \csc \left (c+d x^n\right ) \, dx}{e}\\ &=\frac {a (e x)^{3 n}}{3 e n}+\frac {\left (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 {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac {\left (2 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 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 {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {2 i 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 {2 i 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 (2 i b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d^2 e n}+\frac {\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d^2 e n}\\ &=\frac {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {2 i 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 {2 i 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 (2 b x^{-3 n} (e x)^{3 n}\right ) \text {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 x^{-3 n} (e x)^{3 n}\right ) \text {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 {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {2 i 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 {2 i 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 {2 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 {2 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*}
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Mathematica [F]
time = 3.70, size = 0, normalized size = 0.00 \begin {gather*} \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{-1+3 n} \left (a +b \csc \left (c +d \,x^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 540 vs. \(2 (211) = 422\).
time = 5.10, size = 540, normalized size = 2.44 \begin {gather*} \frac {2 \, a d^{3} x^{3 \, n} e^{\left (3 \, n - 1\right )} - 3 \, b d^{2} x^{2 \, n} e^{\left (3 \, n - 1\right )} \log \left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + 1\right ) - 3 \, b d^{2} x^{2 \, n} e^{\left (3 \, n - 1\right )} \log \left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + 1\right ) - 6 i \, b d x^{n} {\rm Li}_2\left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) e^{\left (3 \, n - 1\right )} + 6 i \, b d x^{n} {\rm Li}_2\left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) e^{\left (3 \, n - 1\right )} - 6 i \, b d x^{n} {\rm Li}_2\left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) e^{\left (3 \, n - 1\right )} + 6 i \, b d x^{n} {\rm Li}_2\left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) e^{\left (3 \, n - 1\right )} + 3 \, b c^{2} e^{\left (3 \, n - 1\right )} \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 ) + 3 \, b c^{2} e^{\left (3 \, n - 1\right )} \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 ) + 6 \, b e^{\left (3 \, n - 1\right )} {\rm polylog}\left (3, \cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) + 6 \, b e^{\left (3 \, n - 1\right )} {\rm polylog}\left (3, \cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) - 6 \, b e^{\left (3 \, n - 1\right )} {\rm polylog}\left (3, -\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) - 6 \, b e^{\left (3 \, n - 1\right )} {\rm polylog}\left (3, -\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) + 3 \, {\left (b d^{2} x^{2 \, n} e^{\left (3 \, n - 1\right )} - b c^{2} e^{\left (3 \, n - 1\right )}\right )} \log \left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + 1\right ) + 3 \, {\left (b d^{2} x^{2 \, n} e^{\left (3 \, n - 1\right )} - b c^{2} e^{\left (3 \, n - 1\right )}\right )} \log \left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + 1\right )}{6 \, d^{3} n} \end {gather*}
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 \begin {gather*} \int \left (e x\right )^{3 n - 1} \left (a + b \csc {\left (c + d x^{n} \right )}\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a+\frac {b}{\sin \left (c+d\,x^n\right )}\right )\,{\left (e\,x\right )}^{3\,n-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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