Optimal. Leaf size=217 \[ -\frac{a (A (m+2)+C (m+1)) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{a C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}-\frac{b (A (m+3)+C (m+2)) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right )}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3)} \]
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Rubi [A] time = 0.371077, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {3034, 3023, 2748, 2643} \[ -\frac{a (A (m+2)+C (m+1)) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{a C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}-\frac{b (A (m+3)+C (m+2)) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right )}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3)} \]
Antiderivative was successfully verified.
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Rule 3034
Rule 3023
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \cos ^m(c+d x) (a+b \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{b C \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m)}+\frac{\int \cos ^m(c+d x) \left (a A (3+m)+b (C (2+m)+A (3+m)) \cos (c+d x)+a C (3+m) \cos ^2(c+d x)\right ) \, dx}{3+m}\\ &=\frac{a C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m)}+\frac{b C \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m)}+\frac{\int \cos ^m(c+d x) (a (3+m) (C (1+m)+A (2+m))+b (2+m) (C (2+m)+A (3+m)) \cos (c+d x)) \, dx}{6+5 m+m^2}\\ &=\frac{a C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m)}+\frac{b C \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m)}+\frac{(a (3+m) (C (1+m)+A (2+m))) \int \cos ^m(c+d x) \, dx}{6+5 m+m^2}+\frac{(b (2+m) (C (2+m)+A (3+m))) \int \cos ^{1+m}(c+d x) \, dx}{6+5 m+m^2}\\ &=\frac{a C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m)}+\frac{b C \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m)}-\frac{a (C (1+m)+A (2+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)}{d (1+m) (2+m) \sqrt{\sin ^2(c+d x)}}-\frac{b (C (2+m)+A (3+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)}{d \left (6+5 m+m^2\right ) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.00144, size = 194, normalized size = 0.89 \[ \frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left (\cos (c+d x) \left (C \cos (c+d x) \left (-\frac{a \, _2F_1\left (\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right )}{m+3}-\frac{b \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\cos ^2(c+d x)\right )}{m+4}\right )-\frac{A b \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right )}{m+2}\right )-\frac{a A \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )}{m+1}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.85, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{m} \left ( a+b\cos \left ( dx+c \right ) \right ) \left ( A+C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \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 (C \cos \left (d x + c\right )^{2} + A\right )}{\left (b \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 (C b \cos \left (d x + c\right )^{3} + C a \cos \left (d x + c\right )^{2} + A b \cos \left (d x + c\right ) + A 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(-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 (C \cos \left (d x + c\right )^{2} + A\right )}{\left (b \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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