Optimal. Leaf size=86 \[ \frac {2^{\frac {1}{2}-m} (e \cos (c+d x))^{1-2 m} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+2 m);\frac {3}{2};\frac {1}{2} (1+\sin (c+d x))\right ) (1-\sin (c+d x))^{-\frac {1}{2}+m} (a+a \sin (c+d x))^m}{d e} \]
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Rubi [A]
time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2768, 7, 72, 71}
\begin {gather*} \frac {2^{\frac {1}{2}-m} (1-\sin (c+d x))^{m-\frac {1}{2}} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{1-2 m} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (2 m+1);\frac {3}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{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 m} (a+a \sin (c+d x))^m \, dx &=\frac {\left (a^2 (e \cos (c+d x))^{1-2 m} (a-a \sin (c+d x))^{\frac {1}{2} (-1+2 m)} (a+a \sin (c+d x))^{\frac {1}{2} (-1+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))^{1-2 m} (a-a \sin (c+d x))^{\frac {1}{2} (-1+2 m)} (a+a \sin (c+d x))^{\frac {1}{2} (-1+2 m)}\right ) \text {Subst}\left (\int \frac {(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))^{1-2 m} (a-a \sin (c+d x))^{-\frac {1}{2}-m+\frac {1}{2} (-1+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} (-1+2 m)}\right ) \text {Subst}\left (\int \frac {\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 {1}{2}-m} (e \cos (c+d x))^{1-2 m} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+2 m);\frac {3}{2};\frac {1}{2} (1+\sin (c+d x))\right ) (1-\sin (c+d x))^{-\frac {1}{2}+m} (a+a \sin (c+d x))^m}{d e}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 90, normalized size = 1.05 \begin {gather*} \frac {\sqrt {2} \cos (c+d x) (e \cos (c+d x))^{-2 m} \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2}-m;\frac {1}{2} (1-\sin (c+d x))\right ) (a (1+\sin (c+d x)))^m}{d (-1+2 m) \sqrt {1+\sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (d x +c \right )\right )^{m} \left (e \cos \left (d x +c \right )\right )^{-2 m}\, dx\]
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 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 \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{2\,m}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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