Optimal. Leaf size=90 \[ \frac {2^{\frac {1}{2}+m} F_1\left (\frac {1}{2};-n,\frac {1}{2}-m;\frac {3}{2};1+\sin (e+f x),\frac {1}{2} (1+\sin (e+f x))\right ) \cos (e+f x) (-\sin (e+f x))^{-n} (d \sin (e+f x))^n}{f \sqrt {1-\sin (e+f x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2865, 2864,
138} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 2864
Rule 2865
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 ) \text {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*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(300\) vs. \(2(90)=180\).
time = 0.44, size = 300, normalized size = 3.33 \begin {gather*} -\frac {(3+2 m) F_1\left (\frac {1}{2}+m;-n,1+m+n;\frac {3}{2}+m;\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right ) \cos (e+f x) (1-\sin (e+f x))^m (d \sin (e+f x))^n}{f (1+2 m) \left ((3+2 m) F_1\left (\frac {1}{2}+m;-n,1+m+n;\frac {3}{2}+m;\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )-2 \left (n F_1\left (\frac {3}{2}+m;1-n,1+m+n;\frac {5}{2}+m;\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )+(1+m+n) F_1\left (\frac {3}{2}+m;-n,2+m+n;\frac {5}{2}+m;\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )\right ) \tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \left (1-\sin \left (f x +e \right )\right )^{m} \left (d \sin \left (f x +e \right )\right )^{n}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d \sin {\left (e + f x \right )}\right )^{n} \left (1 - \sin {\left (e + f x \right )}\right )^{m}\, dx \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 (d\,\sin \left (e+f\,x\right )\right )}^n\,{\left (1-\sin \left (e+f\,x\right )\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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