Optimal. Leaf size=318 \[ -\frac{\sin (c+d x) \left (a^2 (m+4) (A (m+2)+C (m+1))+b^2 (m+1) (A (m+4)+C (m+3))\right ) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )}{d (m+1) (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \left (2 a^2 C+b^2 (A (m+4)+C (m+3))\right ) \cos ^{m+1}(c+d x)}{d (m+2) (m+4)}-\frac{2 a b (A (m+3)+C (m+2)) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right )}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{C \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))^2}{d (m+4)}+\frac{2 a b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3) (m+4)} \]
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Rubi [A] time = 0.858477, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {3050, 3033, 3023, 2748, 2643} \[ -\frac{\sin (c+d x) \left (a^2 (m+4) (A (m+2)+C (m+1))+b^2 (m+1) (A (m+4)+C (m+3))\right ) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )}{d (m+1) (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \left (2 a^2 C+b^2 (A (m+4)+C (m+3))\right ) \cos ^{m+1}(c+d x)}{d (m+2) (m+4)}-\frac{2 a b (A (m+3)+C (m+2)) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right )}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{C \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))^2}{d (m+4)}+\frac{2 a b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3) (m+4)} \]
Antiderivative was successfully verified.
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Rule 3050
Rule 3033
Rule 3023
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \cos ^m(c+d x) (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{C \cos ^{1+m}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{d (4+m)}+\frac{\int \cos ^m(c+d x) (a+b \cos (c+d x)) \left (a (C (1+m)+A (4+m))+b (C (3+m)+A (4+m)) \cos (c+d x)+2 a C \cos ^2(c+d x)\right ) \, dx}{4+m}\\ &=\frac{2 a b C \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m) (4+m)}+\frac{C \cos ^{1+m}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{d (4+m)}+\frac{\int \cos ^m(c+d x) \left (a^2 (3+m) (C (1+m)+A (4+m))+2 a b (4+m) (C (2+m)+A (3+m)) \cos (c+d x)+(3+m) \left (2 a^2 C+b^2 (C (3+m)+A (4+m))\right ) \cos ^2(c+d x)\right ) \, dx}{12+7 m+m^2}\\ &=\frac{\left (2 a^2 C+b^2 C (3+m)+A b^2 (4+m)\right ) \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m) (4+m)}+\frac{2 a b C \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m) (4+m)}+\frac{C \cos ^{1+m}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{d (4+m)}+\frac{\int \cos ^m(c+d x) \left ((3+m) \left (a^2 (4+m) (C (1+m)+A (2+m))+b^2 (1+m) (C (3+m)+A (4+m))\right )+2 a b (2+m) (4+m) (C (2+m)+A (3+m)) \cos (c+d x)\right ) \, dx}{24+26 m+9 m^2+m^3}\\ &=\frac{\left (2 a^2 C+b^2 C (3+m)+A b^2 (4+m)\right ) \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m) (4+m)}+\frac{2 a b C \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m) (4+m)}+\frac{C \cos ^{1+m}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{d (4+m)}+\frac{(2 a b (2+m) (4+m) (C (2+m)+A (3+m))) \int \cos ^{1+m}(c+d x) \, dx}{24+26 m+9 m^2+m^3}+\frac{\left (a^2 (4+m) (C (1+m)+A (2+m))+b^2 (1+m) (C (3+m)+A (4+m))\right ) \int \cos ^m(c+d x) \, dx}{8+6 m+m^2}\\ &=\frac{\left (2 a^2 C+b^2 C (3+m)+A b^2 (4+m)\right ) \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m) (4+m)}+\frac{2 a b C \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m) (4+m)}+\frac{C \cos ^{1+m}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{d (4+m)}-\frac{\left (a^2 (4+m) (C (1+m)+A (2+m))+b^2 (1+m) (C (3+m)+A (4+m))\right ) \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) \left (8+6 m+m^2\right ) \sqrt{\sin ^2(c+d x)}}-\frac{2 a 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 = 2.3779, size = 250, normalized size = 0.79 \[ \frac{\sin (c+d x) \cos ^{m+1}(c+d x) \left (\cos (c+d x) \left (\cos (c+d x) \left (b C \cos (c+d x) \left (-\frac{2 a \, _2F_1\left (\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\cos ^2(c+d x)\right )}{m+4}-\frac{b \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+5}{2};\frac{m+7}{2};\cos ^2(c+d x)\right )}{m+5}\right )-\frac{\left (a^2 C+A b^2\right ) \, _2F_1\left (\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right )}{m+3}\right )-\frac{2 a A b \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right )}{m+2}\right )-\frac{a^2 A \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )}{m+1}\right )}{d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.61, 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 ) ^{2} \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 )}^{2} \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^{2} \cos \left (d x + c\right )^{4} + 2 \, C a b \cos \left (d x + c\right )^{3} + 2 \, A a b \cos \left (d x + c\right ) + A a^{2} +{\left (C a^{2} + A b^{2}\right )} \cos \left (d x + c\right )^{2}\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 )}^{2} \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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