Optimal. Leaf size=90 \[ \frac{2^{m+\frac{1}{2}} \cos (e+f x) (-\sin (e+f x))^{-n} (d \sin (e+f x))^n F_1\left (\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};\sin (e+f x)+1,\frac{1}{2} (\sin (e+f x)+1)\right )}{f \sqrt{1-\sin (e+f x)}} \]
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Rubi [A] time = 0.109557, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2786, 2785, 133} \[ \frac{2^{m+\frac{1}{2}} \cos (e+f x) (-\sin (e+f x))^{-n} (d \sin (e+f x))^n F_1\left (\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};\sin (e+f x)+1,\frac{1}{2} (\sin (e+f x)+1)\right )}{f \sqrt{1-\sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2786
Rule 2785
Rule 133
Rubi steps
\begin{align*} \int (1-\sin (e+f x))^m (d \sin (e+f x))^n \, dx &=\left ((-\sin (e+f x))^{-n} (d \sin (e+f x))^n\right ) \int (1-\sin (e+f x))^m (-\sin (e+f x))^n \, dx\\ &=\frac{\left (\cos (e+f x) (-\sin (e+f x))^{-n} (d \sin (e+f x))^n\right ) \operatorname{Subst}\left (\int \frac{(1-x)^n (2-x)^{-\frac{1}{2}+m}}{\sqrt{x}} \, dx,x,1+\sin (e+f x)\right )}{f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}\\ &=\frac{2^{\frac{1}{2}+m} F_1\left (\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1+\sin (e+f x),\frac{1}{2} (1+\sin (e+f x))\right ) \cos (e+f x) (-\sin (e+f x))^{-n} (d \sin (e+f x))^n}{f \sqrt{1-\sin (e+f x)}}\\ \end{align*}
Mathematica [B] time = 0.712451, size = 300, normalized size = 3.33 \[ -\frac{(2 m+3) \cos (e+f x) (1-\sin (e+f x))^m (d \sin (e+f x))^n F_1\left (m+\frac{1}{2};-n,m+n+1;m+\frac{3}{2};\cot ^2\left (\frac{1}{4} (2 e+2 f x+\pi )\right ),-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )}{f (2 m+1) \left ((2 m+3) F_1\left (m+\frac{1}{2};-n,m+n+1;m+\frac{3}{2};\cot ^2\left (\frac{1}{4} (2 e+2 f x+\pi )\right ),-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )-2 \tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right ) \left (n F_1\left (m+\frac{3}{2};1-n,m+n+1;m+\frac{5}{2};\cot ^2\left (\frac{1}{4} (2 e+2 f x+\pi )\right ),-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )+(m+n+1) F_1\left (m+\frac{3}{2};-n,m+n+2;m+\frac{5}{2};\cot ^2\left (\frac{1}{4} (2 e+2 f x+\pi )\right ),-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.597, size = 0, normalized size = 0. \begin{align*} \int \left ( 1-\sin \left ( fx+e \right ) \right ) ^{m} \left ( d\sin \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sin \left (f x + e\right )\right )^{n}{\left (-\sin \left (f x + e\right ) + 1\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d \sin \left (f x + e\right )\right )^{n}{\left (-\sin \left (f x + e\right ) + 1\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sin{\left (e + f x \right )}\right )^{n} \left (1 - \sin{\left (e + f x \right )}\right )^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sin \left (f x + e\right )\right )^{n}{\left (-\sin \left (f x + e\right ) + 1\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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