Optimal. Leaf size=229 \[ \frac{(1-2 m) m \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )}{3 a^2 d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{2 (1-m) (m+1) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right )}{3 a^2 d (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{2 (1-m) \sin (c+d x) \cos ^{m+1}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x) \cos ^{m+1}(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.304709, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2766, 2978, 2748, 2643} \[ \frac{(1-2 m) m \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )}{3 a^2 d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{2 (1-m) (m+1) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right )}{3 a^2 d (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{2 (1-m) \sin (c+d x) \cos ^{m+1}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x) \cos ^{m+1}(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2766
Rule 2978
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \frac{\cos ^m(c+d x)}{(a+a \cos (c+d x))^2} \, dx &=-\frac{\cos ^{1+m}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{\cos ^m(c+d x) (a (2-m)+a m \cos (c+d x))}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac{2 (1-m) \cos ^{1+m}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac{\cos ^{1+m}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \cos ^m(c+d x) \left (-a^2 (1-2 m) m+2 a^2 (1-m) (1+m) \cos (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{2 (1-m) \cos ^{1+m}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac{\cos ^{1+m}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{((1-2 m) m) \int \cos ^m(c+d x) \, dx}{3 a^2}+\frac{(2 (1-m) (1+m)) \int \cos ^{1+m}(c+d x) \, dx}{3 a^2}\\ &=-\frac{2 (1-m) \cos ^{1+m}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac{\cos ^{1+m}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{(1-2 m) m \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)}{3 a^2 d (1+m) \sqrt{\sin ^2(c+d x)}}-\frac{2 (1-m) (1+m) \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)}{3 a^2 d (2+m) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [F] time = 1.05838, size = 0, normalized size = 0. \[ \int \frac{\cos ^m(c+d x)}{(a+a \cos (c+d x))^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.492, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{m}}{ \left ( a+\cos \left ( dx+c \right ) a \right ) ^{2}}}\, 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 \frac{\cos \left (d x + c\right )^{m}}{{\left (a \cos \left (d x + c\right ) + a\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 (\frac{\cos \left (d x + c\right )^{m}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}}, 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*} \frac{\int \frac{\cos ^{m}{\left (c + d x \right )}}{\cos ^{2}{\left (c + d x \right )} + 2 \cos{\left (c + d x \right )} + 1}\, dx}{a^{2}} \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 \frac{\cos \left (d x + c\right )^{m}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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