Optimal. Leaf size=131 \[ -\frac{a \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )}{d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{a \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right )}{d (m+2) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0637415, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2748, 2643} \[ -\frac{a \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )}{d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{a \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right )}{d (m+2) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \cos ^m(c+d x) (a+a \cos (c+d x)) \, dx &=a \int \cos ^m(c+d x) \, dx+a \int \cos ^{1+m}(c+d x) \, dx\\ &=-\frac{a \cos ^{1+m}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+m) \sqrt{\sin ^2(c+d x)}}-\frac{a \cos ^{2+m}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2+m) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.997566, size = 208, normalized size = 1.59 \[ \frac{i a 2^{-m-2} \left (e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^{m+1} (\cos (c+d x)+1) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left ((m-1) m \, _2F_1\left (1,\frac{m+1}{2};\frac{1-m}{2};-e^{2 i (c+d x)}\right )+(m+1) e^{i (c+d x)} \left (2 (m-1) \, _2F_1\left (1,\frac{m+2}{2};1-\frac{m}{2};-e^{2 i (c+d x)}\right )+m e^{i (c+d x)} \, _2F_1\left (1,\frac{m+3}{2};\frac{3-m}{2};-e^{2 i (c+d x)}\right )\right )\right )}{d (m-1) m (m+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.062, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{m} \left ( a+\cos \left ( dx+c \right ) a \right ) \, 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 (a \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{m}\,{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 (a \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \cos{\left (c + d x \right )} \cos ^{m}{\left (c + d x \right )}\, dx + \int \cos ^{m}{\left (c + d x \right )}\, dx\right ) \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 (a \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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