Optimal. Leaf size=81 \[ -\frac {2^{\frac {3}{2}+m} a \cos ^3(c+d x) \, _2F_1\left (\frac {3}{2},-\frac {1}{2}-m;\frac {5}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-\frac {1}{2}-m} (a+a \sin (c+d x))^{-1+m}}{3 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 81, 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 = {2768, 72, 71}
\begin {gather*} -\frac {a 2^{m+\frac {3}{2}} \cos ^3(c+d x) (\sin (c+d x)+1)^{-m-\frac {1}{2}} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left (\frac {3}{2},-m-\frac {1}{2};\frac {5}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 2768
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac {\left (a^2 \cos ^3(c+d x)\right ) \text {Subst}\left (\int \sqrt {a-a x} (a+a x)^{\frac {1}{2}+m} \, dx,x,\sin (c+d x)\right )}{d (a-a \sin (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}\\ &=\frac {\left (2^{\frac {1}{2}+m} a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^{-1+m} \left (\frac {a+a \sin (c+d x)}{a}\right )^{-\frac {1}{2}-m}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {1}{2}+m} \sqrt {a-a x} \, dx,x,\sin (c+d x)\right )}{d (a-a \sin (c+d x))^{3/2}}\\ &=-\frac {2^{\frac {3}{2}+m} a \cos ^3(c+d x) \, _2F_1\left (\frac {3}{2},-\frac {1}{2}-m;\frac {5}{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))^{-1+m}}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 78, normalized size = 0.96 \begin {gather*} -\frac {2^{\frac {3}{2}+m} \cos ^3(c+d x) \, _2F_1\left (\frac {3}{2},-\frac {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 (1+\sin (c+d x)))^m}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \left (\cos ^{2}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx\]
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} \cos ^{2}{\left (c + d x \right )}\, 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 {\cos \left (c+d\,x\right )}^2\,{\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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