Optimal. Leaf size=91 \[ -\frac {2^{\frac {1}{2}+m} F_1\left (\frac {1}{2};-n,\frac {1}{2}-m;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt {1+\sin (e+f x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2865, 2864,
138} \begin {gather*} -\frac {2^{m+\frac {1}{2}} \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n 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 )}{f \sqrt {\sin (e+f x)+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 2864
Rule 2865
Rubi steps
\begin {align*} \int (d \sin (e+f x))^n (1+\sin (e+f x))^m \, dx &=\left (\sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \int \sin ^n(e+f x) (1+\sin (e+f x))^m \, dx\\ &=-\frac {\left (\cos (e+f x) \sin ^{-n}(e+f x) (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 ^{-n}(e+f x) (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. \(993\) vs. \(2(91)=182\).
time = 2.68, size = 993, normalized size = 10.91 \begin {gather*} \frac {15 F_1\left (\frac {1}{2};-n,1+m+n;\frac {3}{2};\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) (d \sin (e+f x))^n (1+\sin (e+f x))^m}{f \left (15 n F_1\left (\frac {1}{2};-n,1+m+n;\frac {3}{2};\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) \cot (e+f x)+10 \left (n F_1\left (\frac {3}{2};1-n,1+m+n;\frac {5}{2};\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};-n,2+m+n;\frac {5}{2};\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 ) \cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )+\frac {9 F_1\left (\frac {1}{2};-n,1+m+n;\frac {3}{2};\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) \cot \left (\frac {1}{4} (2 e+\pi +2 f x)\right ) \left (-5 n F_1\left (\frac {3}{2};1-n,1+m+n;\frac {5}{2};\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 )-5 (1+m+n) F_1\left (\frac {3}{2};-n,2+m+n;\frac {5}{2};\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 (2 n (1+m+n) F_1\left (\frac {5}{2};1-n,2+m+n;\frac {7}{2};\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+n) n F_1\left (\frac {5}{2};2-n,1+m+n;\frac {7}{2};\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 )+\left (2+m^2+3 n+n^2+m (3+2 n)\right ) F_1\left (\frac {5}{2};-n,3+m+n;\frac {7}{2};\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 ) \cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )\right ) \csc ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )}{3 F_1\left (\frac {1}{2};-n,1+m+n;\frac {3}{2};\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};1-n,1+m+n;\frac {5}{2};\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};-n,2+m+n;\frac {5}{2};\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 ) \cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )}-15 F_1\left (\frac {1}{2};-n,1+m+n;\frac {3}{2};\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 ) \sin (e+f x)+30 m F_1\left (\frac {1}{2};-n,1+m+n;\frac {3}{2};\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 ) \sin ^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 = 0.10, size = 0, normalized size = 0.00 \[\int \left (d \sin \left (f x +e \right )\right )^{n} \left (1+\sin \left (f x +e \right )\right )^{m}\, dx\]
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 (\sin {\left (e + f x \right )} + 1\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 (\sin \left (e+f\,x\right )+1\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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