Optimal. Leaf size=235 \[ \frac {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-i e^{i \left (d x^n+c\right )}\right )}{d^3 e n}+\frac {2 b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (i e^{i \left (d x^n+c\right )}\right )}{d^3 e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-i e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (i e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {2 i b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n} \]
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Rubi [A] time = 0.19, antiderivative size = 235, 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, 4208, 4204, 4181, 2531, 2282, 6589} \[ -\frac {2 b x^{-3 n} (e x)^{3 n} \text {PolyLog}\left (3,-i e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac {2 b x^{-3 n} (e x)^{3 n} \text {PolyLog}\left (3,i e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \text {PolyLog}\left (2,-i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \text {PolyLog}\left (2,i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}+\frac {a (e x)^{3 n}}{3 e n}-\frac {2 i b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2282
Rule 2531
Rule 4181
Rule 4204
Rule 4208
Rule 6589
Rubi steps
\begin {align*} \int (e x)^{-1+3 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx &=\int \left (a (e x)^{-1+3 n}+b (e x)^{-1+3 n} \sec \left (c+d x^n\right )\right ) \, dx\\ &=\frac {a (e x)^{3 n}}{3 e n}+b \int (e x)^{-1+3 n} \sec \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} \sec \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 ) \operatorname {Subst}\left (\int x^2 \sec (c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac {a (e x)^{3 n}}{3 e n}-\frac {2 i b x^{-n} (e x)^{3 n} \tan ^{-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 ) \operatorname {Subst}\left (\int x \log \left (1-i 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 ) \operatorname {Subst}\left (\int x \log \left (1+i 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 i b x^{-n} (e x)^{3 n} \tan ^{-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 (-i 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 (i 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 ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i 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 ) \operatorname {Subst}\left (\int \text {Li}_2\left (i 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 i b x^{-n} (e x)^{3 n} \tan ^{-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 (-i 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 (i 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 ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i 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 ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i 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 i b x^{-n} (e x)^{3 n} \tan ^{-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 (-i 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 (i 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 (-i 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 (i e^{i \left (c+d x^n\right )}\right )}{d^3 e n}\\ \end {align*}
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Mathematica [F] time = 1.32, size = 0, normalized size = 0.00 \[ \int (e x)^{-1+3 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.83, size = 655, normalized size = 2.79 \[ \frac {2 \, a d^{3} e^{3 \, n - 1} x^{3 \, n} - 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right )\right ) - 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right )\right ) + 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (-i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right )\right ) + 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (-i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right )\right ) + 3 \, b c^{2} e^{3 \, n - 1} \log \left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + i\right ) - 3 \, b c^{2} e^{3 \, n - 1} \log \left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + i\right ) + 3 \, b c^{2} e^{3 \, n - 1} \log \left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + i\right ) - 3 \, b c^{2} e^{3 \, n - 1} \log \left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + i\right ) - 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right )\right ) + 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right )\right ) - 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, -i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right )\right ) + 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, -i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right )\right ) + 3 \, {\left (b d^{2} e^{3 \, n - 1} x^{2 \, n} - b c^{2} e^{3 \, n - 1}\right )} \log \left (i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right ) + 1\right ) - 3 \, {\left (b d^{2} e^{3 \, n - 1} x^{2 \, n} - b c^{2} e^{3 \, n - 1}\right )} \log \left (i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right ) + 1\right ) + 3 \, {\left (b d^{2} e^{3 \, n - 1} x^{2 \, n} - b c^{2} e^{3 \, n - 1}\right )} \log \left (-i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right ) + 1\right ) - 3 \, {\left (b d^{2} e^{3 \, n - 1} x^{2 \, n} - b c^{2} e^{3 \, n - 1}\right )} \log \left (-i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right ) + 1\right )}{6 \, d^{3} 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 \[ \int {\left (b \sec \left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{3 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.27, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{-1+3 n} \left (a +b \sec \left (c +d \,x^{n}\right )\right )\, dx \]
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 \[ 2 \, b e^{3 \, n} \int \frac {x^{3 \, n} \cos \left (2 \, d x^{n} + 2 \, c\right ) \cos \left (d x^{n} + c\right ) + x^{3 \, n} \sin \left (2 \, d x^{n} + 2 \, c\right ) \sin \left (d x^{n} + c\right ) + x^{3 \, n} \cos \left (d x^{n} + c\right )}{e x \cos \left (2 \, d x^{n} + 2 \, c\right )^{2} + e x \sin \left (2 \, d x^{n} + 2 \, c\right )^{2} + 2 \, e x \cos \left (2 \, d x^{n} + 2 \, c\right ) + e x}\,{d x} + \frac {\left (e x\right )^{3 \, n} a}{3 \, e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (a+\frac {b}{\cos \left (c+d\,x^n\right )}\right )\,{\left (e\,x\right )}^{3\,n-1} \,d x \]
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 \[ \int \left (e x\right )^{3 n - 1} \left (a + b \sec {\left (c + d x^{n} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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