Optimal. Leaf size=132 \[ -\frac {\sqrt {2} x^{-n} (e x)^n \cos \left (c+d x^n\right ) \left (a+b \sin \left (c+d x^n\right )\right )^p \left (\frac {a+b \sin \left (c+d x^n\right )}{a+b}\right )^{-p} F_1\left (\frac {1}{2};\frac {1}{2},-p;\frac {3}{2};\frac {1}{2} \left (1-\sin \left (d x^n+c\right )\right ),\frac {b \left (1-\sin \left (d x^n+c\right )\right )}{a+b}\right )}{d e n \sqrt {\sin \left (c+d x^n\right )+1}} \]
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Rubi [A] time = 0.19, 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 = {3381, 3379, 2665, 139, 138} \[ -\frac {\sqrt {2} x^{-n} (e x)^n \cos \left (c+d x^n\right ) \left (a+b \sin \left (c+d x^n\right )\right )^p \left (\frac {a+b \sin \left (c+d x^n\right )}{a+b}\right )^{-p} F_1\left (\frac {1}{2};\frac {1}{2},-p;\frac {3}{2};\frac {1}{2} \left (1-\sin \left (d x^n+c\right )\right ),\frac {b \left (1-\sin \left (d x^n+c\right )\right )}{a+b}\right )}{d e n \sqrt {\sin \left (c+d x^n\right )+1}} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 2665
Rule 3379
Rule 3381
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.46, size = 148, normalized size = 1.12 \[ \frac {x^{-n} (e x)^n \sec \left (c+d x^n\right ) \sqrt {-\frac {b \left (\sin \left (c+d x^n\right )-1\right )}{a+b}} \sqrt {\frac {b \left (\sin \left (c+d x^n\right )+1\right )}{b-a}} \left (a+b \sin \left (c+d x^n\right )\right )^{p+1} F_1\left (p+1;\frac {1}{2},\frac {1}{2};p+2;\frac {a+b \sin \left (d x^n+c\right )}{a-b},\frac {a+b \sin \left (d x^n+c\right )}{a+b}\right )}{b d e n (p+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{n - 1} {\left (b \sin \left (d x^{n} + c\right ) + a\right )}^{p}, x\right ) \]
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 \[ \int \left (e x\right )^{n - 1} {\left (b \sin \left (d x^{n} + c\right ) + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.38, 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 \[ \int \left (e x\right )^{n - 1} {\left (b \sin \left (d x^{n} + c\right ) + a\right )}^{p}\,{d x} \]
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 \[ \int {\left (e\,x\right )}^{n-1}\,{\left (a+b\,\sin \left (c+d\,x^n\right )\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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