Optimal. Leaf size=132 \[ -\frac {\sqrt {2} x^{-n} (e x)^n F_1\left (\frac {1}{2};\frac {1}{2},-p;\frac {3}{2};\frac {1}{2} \left (1-\sin \left (c+d x^n\right )\right ),\frac {b \left (1-\sin \left (c+d x^n\right )\right )}{a+b}\right ) \cos \left (c+d x^n\right ) \left (a+b \sin \left (c+d x^n\right )\right )^p \left (\frac {a+b \sin \left (c+d x^n\right )}{a+b}\right )^{-p}}{d e n \sqrt {1+\sin \left (c+d x^n\right )}} \]
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Rubi [A]
time = 0.13, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {3462, 3460,
2744, 144, 143} \begin {gather*} -\frac {\sqrt {2} x^{-n} (e x)^n \cos \left (c+d x^n\right ) \left (a+b \sin \left (c+d x^n\right )\right )^p \left (\frac {a+b \sin \left (c+d x^n\right )}{a+b}\right )^{-p} F_1\left (\frac {1}{2};\frac {1}{2},-p;\frac {3}{2};\frac {1}{2} \left (1-\sin \left (d x^n+c\right )\right ),\frac {b \left (1-\sin \left (d x^n+c\right )\right )}{a+b}\right )}{d e n \sqrt {\sin \left (c+d x^n\right )+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 143
Rule 144
Rule 2744
Rule 3460
Rule 3462
Rubi steps
\begin {align*} \int (e x)^{-1+n} \left (a+b \sin \left (c+d x^n\right )\right )^p \, dx &=\frac {\left (x^{-n} (e x)^n\right ) \int x^{-1+n} \left (a+b \sin \left (c+d x^n\right )\right )^p \, dx}{e}\\ &=\frac {\left (x^{-n} (e x)^n\right ) \text {Subst}\left (\int (a+b \sin (c+d x))^p \, dx,x,x^n\right )}{e n}\\ &=\frac {\left (x^{-n} (e x)^n \cos \left (c+d x^n\right )\right ) \text {Subst}\left (\int \frac {(a+b x)^p}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin \left (c+d x^n\right )\right )}{d e n \sqrt {1-\sin \left (c+d x^n\right )} \sqrt {1+\sin \left (c+d x^n\right )}}\\ &=\frac {\left (x^{-n} (e x)^n \cos \left (c+d x^n\right ) \left (a+b \sin \left (c+d x^n\right )\right )^p \left (-\frac {a+b \sin \left (c+d x^n\right )}{-a-b}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^p}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin \left (c+d x^n\right )\right )}{d e n \sqrt {1-\sin \left (c+d x^n\right )} \sqrt {1+\sin \left (c+d x^n\right )}}\\ &=-\frac {\sqrt {2} x^{-n} (e x)^n F_1\left (\frac {1}{2};\frac {1}{2},-p;\frac {3}{2};\frac {1}{2} \left (1-\sin \left (c+d x^n\right )\right ),\frac {b \left (1-\sin \left (c+d x^n\right )\right )}{a+b}\right ) \cos \left (c+d x^n\right ) \left (a+b \sin \left (c+d x^n\right )\right )^p \left (\frac {a+b \sin \left (c+d x^n\right )}{a+b}\right )^{-p}}{d e n \sqrt {1+\sin \left (c+d x^n\right )}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 148, normalized size = 1.12 \begin {gather*} \frac {x^{-n} (e x)^n F_1\left (1+p;\frac {1}{2},\frac {1}{2};2+p;\frac {a+b \sin \left (c+d x^n\right )}{a-b},\frac {a+b \sin \left (c+d x^n\right )}{a+b}\right ) \sec \left (c+d x^n\right ) \sqrt {-\frac {b \left (-1+\sin \left (c+d x^n\right )\right )}{a+b}} \sqrt {\frac {b \left (1+\sin \left (c+d x^n\right )\right )}{-a+b}} \left (a+b \sin \left (c+d x^n\right )\right )^{1+p}}{b d e n (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{-1+n} \left (a +b \sin \left (c +d \,x^{n}\right )\right )^{p}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.01 \begin {gather*} \int {\left (e\,x\right )}^{n-1}\,{\left (a+b\,\sin \left (c+d\,x^n\right )\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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