Optimal. Leaf size=149 \[ \frac {a (e x)^{2 n}}{2 e n}+\frac {i b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-i e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {i b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (i e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {2 i b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n} \]
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Rubi [A] time = 0.11, antiderivative size = 149, 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, 4208, 4204, 4181, 2279, 2391} \[ \frac {i b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {i b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}+\frac {a (e x)^{2 n}}{2 e n}-\frac {2 i b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2279
Rule 2391
Rule 4181
Rule 4204
Rule 4208
Rubi steps
\begin {align*} \int (e x)^{-1+2 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx &=\int \left (a (e x)^{-1+2 n}+b (e x)^{-1+2 n} \sec \left (c+d x^n\right )\right ) \, dx\\ &=\frac {a (e x)^{2 n}}{2 e n}+b \int (e x)^{-1+2 n} \sec \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} \sec \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 \sec (c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac {a (e x)^{2 n}}{2 e n}-\frac {2 i b x^{-n} (e x)^{2 n} \tan ^{-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-i 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+i 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 i b x^{-n} (e x)^{2 n} \tan ^{-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-i 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+i 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 i b x^{-n} (e x)^{2 n} \tan ^{-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 (-i 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 (i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}\\ \end {align*}
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Mathematica [A] time = 0.65, size = 188, normalized size = 1.26 \[ \frac {(e x)^{2 n} \cos \left (c+d x^n\right ) \left (a+b \sec \left (c+d x^n\right )\right ) \left (a+\frac {b x^{-2 n} \left (2 i \left (\text {Li}_2\left (-i e^{-i \left (d x^n+c\right )}\right )-\text {Li}_2\left (i e^{-i \left (d x^n+c\right )}\right )\right )+\left (-2 c-2 d x^n+\pi \right ) \left (\log \left (1-i e^{-i \left (c+d x^n\right )}\right )-\log \left (1+i e^{-i \left (c+d x^n\right )}\right )\right )-(\pi -2 c) \log \left (\cot \left (\frac {1}{4} \left (2 c+2 d x^n+\pi \right )\right )\right )\right )}{d^2}\right )}{2 e n \left (a \cos \left (c+d x^n\right )+b\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.48, size = 470, normalized size = 3.15 \[ \frac {a d^{2} e^{2 \, n - 1} x^{2 \, n} - b c e^{2 \, n - 1} \log \left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + i\right ) + b c e^{2 \, n - 1} \log \left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + i\right ) - b c e^{2 \, n - 1} \log \left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + i\right ) + b c e^{2 \, n - 1} \log \left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + i\right ) - i \, b e^{2 \, n - 1} {\rm Li}_2\left (i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right )\right ) - i \, b e^{2 \, n - 1} {\rm Li}_2\left (i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right )\right ) + i \, b e^{2 \, n - 1} {\rm Li}_2\left (-i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right )\right ) + i \, b e^{2 \, n - 1} {\rm Li}_2\left (-i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right )\right ) + {\left (b d e^{2 \, n - 1} x^{n} + b c e^{2 \, n - 1}\right )} \log \left (i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right ) + 1\right ) - {\left (b d e^{2 \, n - 1} x^{n} + b c e^{2 \, n - 1}\right )} \log \left (i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right ) + 1\right ) + {\left (b d e^{2 \, n - 1} x^{n} + b c e^{2 \, n - 1}\right )} \log \left (-i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right ) + 1\right ) - {\left (b d e^{2 \, n - 1} x^{n} + b c e^{2 \, n - 1}\right )} \log \left (-i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right ) + 1\right )}{2 \, d^{2} n} \]
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 \[ \int {\left (b \sec \left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.87, size = 873, normalized size = 5.86 \[ \text {result too large to display} \]
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 \[ 2 \, b e^{2 \, n} \int \frac {x^{2 \, n} \cos \left (2 \, d x^{n} + 2 \, c\right ) \cos \left (d x^{n} + c\right ) + x^{2 \, n} \sin \left (2 \, d x^{n} + 2 \, c\right ) \sin \left (d x^{n} + c\right ) + x^{2 \, n} \cos \left (d x^{n} + c\right )}{e x \cos \left (2 \, d x^{n} + 2 \, c\right )^{2} + e x \sin \left (2 \, d x^{n} + 2 \, c\right )^{2} + 2 \, e x \cos \left (2 \, d x^{n} + 2 \, c\right ) + e x}\,{d x} + \frac {\left (e x\right )^{2 \, n} a}{2 \, e n} \]
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 \[ \int \left (a+\frac {b}{\cos \left (c+d\,x^n\right )}\right )\,{\left (e\,x\right )}^{2\,n-1} \,d x \]
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 \[ \int \left (e x\right )^{2 n - 1} \left (a + b \sec {\left (c + d x^{n} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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