Optimal. Leaf size=229 \[ -\frac{3 (A (3 m+8)+C (3 m+5)) \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right )}{d (3 m+5) (3 m+8) \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+8);\frac{1}{6} (3 m+14);\cos ^2(c+d x)\right )}{d (3 m+8) \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x)}{d (3 m+8)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.212513, antiderivative size = 219, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {20, 3023, 2748, 2643} \[ -\frac{3 \left (\frac{A}{3 m+5}+\frac{C}{3 m+8}\right ) \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right )}{d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+8);\frac{1}{6} (3 m+14);\cos ^2(c+d x)\right )}{d (3 m+8) \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x)}{d (3 m+8)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 20
Rule 3023
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{(b \cos (c+d x))^{2/3} \int \cos ^{\frac{2}{3}+m}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{\cos ^{\frac{2}{3}}(c+d x)}\\ &=\frac{3 C \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \sin (c+d x)}{d (8+3 m)}+\frac{\left (3 (b \cos (c+d x))^{2/3}\right ) \int \cos ^{\frac{2}{3}+m}(c+d x) \left (\frac{1}{3} \left (3 C \left (\frac{5}{3}+m\right )+3 A \left (\frac{8}{3}+m\right )\right )+\frac{1}{3} B (8+3 m) \cos (c+d x)\right ) \, dx}{(8+3 m) \cos ^{\frac{2}{3}}(c+d x)}\\ &=\frac{3 C \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \sin (c+d x)}{d (8+3 m)}+\frac{\left (B (b \cos (c+d x))^{2/3}\right ) \int \cos ^{\frac{5}{3}+m}(c+d x) \, dx}{\cos ^{\frac{2}{3}}(c+d x)}+\frac{\left ((C (5+3 m)+A (8+3 m)) (b \cos (c+d x))^{2/3}\right ) \int \cos ^{\frac{2}{3}+m}(c+d x) \, dx}{(8+3 m) \cos ^{\frac{2}{3}}(c+d x)}\\ &=\frac{3 C \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \sin (c+d x)}{d (8+3 m)}-\frac{3 \left (\frac{A}{5+3 m}+\frac{C}{8+3 m}\right ) \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{2},\frac{1}{6} (5+3 m);\frac{1}{6} (11+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \cos ^{2+m}(c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{2},\frac{1}{6} (8+3 m);\frac{1}{6} (14+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (8+3 m) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.425282, size = 166, normalized size = 0.72 \[ -\frac{3 \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \left ((A (3 m+8)+C (3 m+5)) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right )+(3 m+5) \left (B \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+8);\frac{m}{2}+\frac{7}{3};\cos ^2(c+d x)\right )-C \sqrt{\sin ^2(c+d x)}\right )\right )}{d (3 m+5) (3 m+8) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.313, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{m} \left ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}} \left ( A+B\cos \left ( dx+c \right ) +C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \cos \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \cos \left (d x + c\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \cos \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]