Optimal. Leaf size=44 \[ \frac {a (e x)^n}{e n}+\frac {b x^{-n} (e x)^n \tanh ^{-1}\left (\sin \left (c+d x^n\right )\right )}{d e n} \]
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Rubi [A] time = 0.05, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {14, 4208, 4204, 3770} \[ \frac {a (e x)^n}{e n}+\frac {b x^{-n} (e x)^n \tanh ^{-1}\left (\sin \left (c+d x^n\right )\right )}{d e n} \]
Antiderivative was successfully verified.
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Rule 14
Rule 3770
Rule 4204
Rule 4208
Rubi steps
\begin {align*} \int (e x)^{-1+n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx &=\int \left (a (e x)^{-1+n}+b (e x)^{-1+n} \sec \left (c+d x^n\right )\right ) \, dx\\ &=\frac {a (e x)^n}{e n}+b \int (e x)^{-1+n} \sec \left (c+d x^n\right ) \, dx\\ &=\frac {a (e x)^n}{e n}+\frac {\left (b x^{-n} (e x)^n\right ) \int x^{-1+n} \sec \left (c+d x^n\right ) \, dx}{e}\\ &=\frac {a (e x)^n}{e n}+\frac {\left (b x^{-n} (e x)^n\right ) \operatorname {Subst}\left (\int \sec (c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac {a (e x)^n}{e n}+\frac {b x^{-n} (e x)^n \tanh ^{-1}\left (\sin \left (c+d x^n\right )\right )}{d e n}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 38, normalized size = 0.86 \[ \frac {x^{-n} (e x)^n \left (a d x^n+b \tanh ^{-1}\left (\sin \left (c+d x^n\right )\right )\right )}{d e n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 60, normalized size = 1.36 \[ \frac {2 \, a d e^{n - 1} x^{n} + b e^{n - 1} \log \left (\sin \left (d x^{n} + c\right ) + 1\right ) - b e^{n - 1} \log \left (-\sin \left (d x^{n} + c\right ) + 1\right )}{2 \, d n} \]
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 \[ \int {\left (b \sec \left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.58, size = 159, normalized size = 3.61 \[ \frac {a x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \relax (x )+2 \ln \relax (e )\right )}{2}}}{n}-\frac {2 i \arctan \left ({\mathrm e}^{i \left (c +d \,x^{n}\right )}\right ) e^{n} b \,{\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-1+n \right ) \left (\mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i e x \right )+\mathrm {csgn}\left (i e \right )\right )}{2}}}{d e n} \]
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 \[ 2 \, b e^{n} \int \frac {x^{n} \cos \left (2 \, d x^{n} + 2 \, c\right ) \cos \left (d x^{n} + c\right ) + x^{n} \sin \left (2 \, d x^{n} + 2 \, c\right ) \sin \left (d x^{n} + c\right ) + x^{n} \cos \left (d x^{n} + c\right )}{e x \cos \left (2 \, d x^{n} + 2 \, c\right )^{2} + e x \sin \left (2 \, d x^{n} + 2 \, c\right )^{2} + 2 \, e x \cos \left (2 \, d x^{n} + 2 \, c\right ) + e x}\,{d x} + \frac {\left (e x\right )^{n} a}{e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.66, size = 104, normalized size = 2.36 \[ \frac {{\left (e\,x\right )}^n\,\left (b\,\ln \left (-b\,{\left (e\,x\right )}^{n-1}\,2{}\mathrm {i}-2\,b\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x^n\,1{}\mathrm {i}}\,{\left (e\,x\right )}^{n-1}\right )-b\,\ln \left (b\,{\left (e\,x\right )}^{n-1}\,2{}\mathrm {i}-2\,b\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x^n\,1{}\mathrm {i}}\,{\left (e\,x\right )}^{n-1}\right )+a\,d\,x^n\right )}{d\,e\,n\,x^n} \]
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 \[ \int \left (e x\right )^{n - 1} \left (a + b \sec {\left (c + d x^{n} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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