Optimal. Leaf size=221 \[ \frac {a^2 (e x)^{2 n}}{2 e n}+\frac {2 i a b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-i e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {2 i a b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (i e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {4 i a b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (\cos \left (c+d x^n\right )\right )}{d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \tan \left (c+d x^n\right )}{d e n} \]
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Rubi [A] time = 0.20, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4208, 4204, 4190, 4181, 2279, 2391, 4184, 3475} \[ \frac {2 i a b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 i a b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}+\frac {a^2 (e x)^{2 n}}{2 e n}-\frac {4 i a b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (\cos \left (c+d x^n\right )\right )}{d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \tan \left (c+d x^n\right )}{d e n} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 3475
Rule 4181
Rule 4184
Rule 4190
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 )^2 \, dx &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \int x^{-1+2 n} \left (a+b \sec \left (c+d x^n\right )\right )^2 \, dx}{e}\\ &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int x (a+b \sec (c+d x))^2 \, dx,x,x^n\right )}{e n}\\ &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \left (a^2 x+2 a b x \sec (c+d x)+b^2 x \sec ^2(c+d x)\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac {a^2 (e x)^{2 n}}{2 e n}+\frac {\left (2 a 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 {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int x \sec ^2(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac {a^2 (e x)^{2 n}}{2 e n}-\frac {4 i a b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {b^2 x^{-n} (e x)^{2 n} \tan \left (c+d x^n\right )}{d e n}-\frac {\left (2 a 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 (2 a 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^2 x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \tan (c+d x) \, dx,x,x^n\right )}{d e n}\\ &=\frac {a^2 (e x)^{2 n}}{2 e n}-\frac {4 i a b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (\cos \left (c+d x^n\right )\right )}{d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \tan \left (c+d x^n\right )}{d e n}+\frac {\left (2 i a 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 (2 i a 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^2 (e x)^{2 n}}{2 e n}-\frac {4 i a b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (\cos \left (c+d x^n\right )\right )}{d^2 e n}+\frac {2 i a 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 {2 i a 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 {b^2 x^{-n} (e x)^{2 n} \tan \left (c+d x^n\right )}{d e n}\\ \end {align*}
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Mathematica [A] time = 5.16, size = 347, normalized size = 1.57 \[ \frac {x^{-2 n} (e x)^{2 n} \left (d x^n \left (a^2 d x^n+2 b^2 \tan (c)\right )-\frac {4 a b \csc (c) \left (i \text {Li}_2\left (-e^{i \left (d x^n-\tan ^{-1}(\cot (c))\right )}\right )-i \text {Li}_2\left (e^{i \left (d x^n-\tan ^{-1}(\cot (c))\right )}\right )+\left (d x^n-\tan ^{-1}(\cot (c))\right ) \left (\log \left (1-e^{i \left (d x^n-\tan ^{-1}(\cot (c))\right )}\right )-\log \left (1+e^{i \left (d x^n-\tan ^{-1}(\cot (c))\right )}\right )\right )\right )}{\sqrt {\csc ^2(c)}}+8 a b \tan ^{-1}(\cot (c)) \tanh ^{-1}\left (\cos (c) \tan \left (\frac {d x^n}{2}\right )+\sin (c)\right )-2 b^2 d \tan (c) x^n+\frac {2 b^2 d x^n \sin \left (\frac {d x^n}{2}\right )}{\left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} \left (c+d x^n\right )\right )-\sin \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}+\frac {2 b^2 d x^n \sin \left (\frac {d x^n}{2}\right )}{\left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} \left (c+d x^n\right )\right )+\cos \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}+2 b^2 \left (d \tan (c) x^n+\log \left (\cos \left (c+d x^n\right )\right )\right )\right )}{2 d^2 e n} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.94, size = 656, normalized size = 2.97 \[ \frac {a^{2} d^{2} e^{2 \, n - 1} x^{2 \, n} \cos \left (d x^{n} + c\right ) + 2 \, b^{2} d e^{2 \, n - 1} x^{n} \sin \left (d x^{n} + c\right ) - 2 i \, a b e^{2 \, n - 1} \cos \left (d x^{n} + c\right ) {\rm Li}_2\left (i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right )\right ) - 2 i \, a b e^{2 \, n - 1} \cos \left (d x^{n} + c\right ) {\rm Li}_2\left (i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right )\right ) + 2 i \, a b e^{2 \, n - 1} \cos \left (d x^{n} + c\right ) {\rm Li}_2\left (-i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right )\right ) + 2 i \, a b e^{2 \, n - 1} \cos \left (d x^{n} + c\right ) {\rm Li}_2\left (-i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right )\right ) - {\left (2 \, a b c - b^{2}\right )} e^{2 \, n - 1} \cos \left (d x^{n} + c\right ) \log \left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + i\right ) + {\left (2 \, a b c + b^{2}\right )} e^{2 \, n - 1} \cos \left (d x^{n} + c\right ) \log \left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + i\right ) - {\left (2 \, a b c - b^{2}\right )} e^{2 \, n - 1} \cos \left (d x^{n} + c\right ) \log \left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + i\right ) + {\left (2 \, a b c + b^{2}\right )} e^{2 \, n - 1} \cos \left (d x^{n} + c\right ) \log \left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + i\right ) + 2 \, {\left (a b d e^{2 \, n - 1} x^{n} + a b c e^{2 \, n - 1}\right )} \cos \left (d x^{n} + c\right ) \log \left (i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right ) + 1\right ) - 2 \, {\left (a b d e^{2 \, n - 1} x^{n} + a b c e^{2 \, n - 1}\right )} \cos \left (d x^{n} + c\right ) \log \left (i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right ) + 1\right ) + 2 \, {\left (a b d e^{2 \, n - 1} x^{n} + a b c e^{2 \, n - 1}\right )} \cos \left (d x^{n} + c\right ) \log \left (-i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right ) + 1\right ) - 2 \, {\left (a b d e^{2 \, n - 1} x^{n} + a b c e^{2 \, n - 1}\right )} \cos \left (d x^{n} + c\right ) \log \left (-i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right ) + 1\right )}{2 \, d^{2} n \cos \left (d x^{n} + c\right )} \]
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 \[ \int {\left (b \sec \left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.20, size = 1096, normalized size = 4.96 \[ \text {result too large to display} \]
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 \[ \frac {\left (e x\right )^{2 \, n} a^{2}}{2 \, e n} + \frac {2 \, b^{2} e^{2 \, n} x^{n} \sin \left (2 \, d x^{n} + 2 \, c\right ) + 2 \, {\left (d e n \cos \left (2 \, d x^{n} + 2 \, c\right )^{2} + d e n \sin \left (2 \, d x^{n} + 2 \, c\right )^{2} + 2 \, d e n \cos \left (2 \, d x^{n} + 2 \, c\right ) + d e n\right )} \int \frac {2 \, a b d e^{2 \, n} x^{2 \, n} \cos \left (2 \, d x^{n} + 2 \, c\right ) \cos \left (d x^{n} + c\right ) + 2 \, a b d e^{2 \, n} x^{2 \, n} \cos \left (d x^{n} + c\right ) + {\left (2 \, a b d e^{2 \, n} x^{2 \, n} \sin \left (d x^{n} + c\right ) - b^{2} e^{2 \, n} x^{n}\right )} \sin \left (2 \, d x^{n} + 2 \, c\right )}{d e x \cos \left (2 \, d x^{n} + 2 \, c\right )^{2} + d e x \sin \left (2 \, d x^{n} + 2 \, c\right )^{2} + 2 \, d e x \cos \left (2 \, d x^{n} + 2 \, c\right ) + d e x}\,{d x}}{d e n \cos \left (2 \, d x^{n} + 2 \, c\right )^{2} + d e n \sin \left (2 \, d x^{n} + 2 \, c\right )^{2} + 2 \, d e n \cos \left (2 \, d x^{n} + 2 \, c\right ) + d e n} \]
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 \[ \int {\left (a+\frac {b}{\cos \left (c+d\,x^n\right )}\right )}^2\,{\left (e\,x\right )}^{2\,n-1} \,d x \]
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 \[ \int \left (e x\right )^{2 n - 1} \left (a + b \sec {\left (c + d x^{n} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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