Optimal. Leaf size=127 \[ \frac{\cos (c+d x) (a+b \sin (c+d x))^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right )}{b d (m+1) \sqrt{1-\frac{a+b \sin (c+d x)}{a-b}} \sqrt{1-\frac{a+b \sin (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.0889027, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2704, 138} \[ \frac{\cos (c+d x) (a+b \sin (c+d x))^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right )}{b d (m+1) \sqrt{1-\frac{a+b \sin (c+d x)}{a-b}} \sqrt{1-\frac{a+b \sin (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2704
Rule 138
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+b \sin (c+d x))^m \, dx &=\frac{\cos (c+d x) \operatorname{Subst}\left (\int (a+b x)^m \sqrt{-\frac{b}{a-b}-\frac{b x}{a-b}} \sqrt{\frac{b}{a+b}-\frac{b x}{a+b}} \, dx,x,\sin (c+d x)\right )}{d \sqrt{1-\frac{a+b \sin (c+d x)}{a-b}} \sqrt{1-\frac{a+b \sin (c+d x)}{a+b}}}\\ &=\frac{F_1\left (1+m;-\frac{1}{2},-\frac{1}{2};2+m;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right ) \cos (c+d x) (a+b \sin (c+d x))^{1+m}}{b d (1+m) \sqrt{1-\frac{a+b \sin (c+d x)}{a-b}} \sqrt{1-\frac{a+b \sin (c+d x)}{a+b}}}\\ \end{align*}
Mathematica [F] time = 5.86056, size = 0, normalized size = 0. \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.142, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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