Optimal. Leaf size=89 \[ \frac {2^{\frac {3}{2}-m} (e \cos (c+d x))^{3-2 m} \, _2F_1\left (\frac {3}{2},\frac {1}{2} (-1+2 m);\frac {5}{2};\frac {1}{2} (1+\sin (c+d x))\right ) (1-\sin (c+d x))^{-\frac {3}{2}+m} (a+a \sin (c+d x))^m}{3 d e} \]
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Rubi [A]
time = 0.07, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2768, 7, 72, 71}
\begin {gather*} \frac {2^{\frac {3}{2}-m} (1-\sin (c+d x))^{m-\frac {3}{2}} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{3-2 m} \, _2F_1\left (\frac {3}{2},\frac {1}{2} (2 m-1);\frac {5}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{3 d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 7
Rule 71
Rule 72
Rule 2768
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{2-2 m} (a+a \sin (c+d x))^m \, dx &=\frac {\left (a^2 (e \cos (c+d x))^{3-2 m} (a-a \sin (c+d x))^{\frac {1}{2} (-3+2 m)} (a+a \sin (c+d x))^{\frac {1}{2} (-3+2 m)}\right ) \text {Subst}\left (\int (a-a x)^{\frac {1}{2} (1-2 m)} (a+a x)^{\frac {1}{2} (1-2 m)+m} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=\frac {\left (a^2 (e \cos (c+d x))^{3-2 m} (a-a \sin (c+d x))^{\frac {1}{2} (-3+2 m)} (a+a \sin (c+d x))^{\frac {1}{2} (-3+2 m)}\right ) \text {Subst}\left (\int (a-a x)^{\frac {1}{2} (1-2 m)} \sqrt {a+a x} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=\frac {\left (2^{\frac {1}{2}-m} a^2 (e \cos (c+d x))^{3-2 m} (a-a \sin (c+d x))^{\frac {1}{2}-m+\frac {1}{2} (-3+2 m)} \left (\frac {a-a \sin (c+d x)}{a}\right )^{-\frac {1}{2}+m} (a+a \sin (c+d x))^{\frac {1}{2} (-3+2 m)}\right ) \text {Subst}\left (\int \left (\frac {1}{2}-\frac {x}{2}\right )^{\frac {1}{2} (1-2 m)} \sqrt {a+a x} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=\frac {2^{\frac {3}{2}-m} (e \cos (c+d x))^{3-2 m} \, _2F_1\left (\frac {3}{2},\frac {1}{2} (-1+2 m);\frac {5}{2};\frac {1}{2} (1+\sin (c+d x))\right ) (1-\sin (c+d x))^{-\frac {3}{2}+m} (a+a \sin (c+d x))^m}{3 d e}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 96, normalized size = 1.08 \begin {gather*} \frac {2 \sqrt {2} e^2 \cos ^3(c+d x) (e \cos (c+d x))^{-2 m} \, _2F_1\left (-\frac {1}{2},\frac {3}{2}-m;\frac {5}{2}-m;\frac {1}{2} (1-\sin (c+d x))\right ) (a (1+\sin (c+d x)))^m}{d (-3+2 m) (1+\sin (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \left (e \cos \left (d x +c \right )\right )^{2-2 m} \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx\]
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 (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \left (e \cos {\left (c + d x \right )}\right )^{2 - 2 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 (e\,\cos \left (c+d\,x\right )\right )}^{2-2\,m}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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