Optimal. Leaf size=290 \[ \frac{\sin (c+d x) \left (3 a^2 C-8 a b B+8 A b^2+5 b^2 C\right ) (a+b \cos (c+d x))^{2/3} F_1\left (\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right )}{4 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3}}+\frac{(a+b) (8 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left (\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right )}{4 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3}}+\frac{3 C \sin (c+d x) (a+b \cos (c+d x))^{5/3}}{8 b d} \]
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Rubi [A] time = 0.368263, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3023, 2756, 2665, 139, 138} \[ \frac{\sin (c+d x) \left (3 a^2 C-8 a b B+8 A b^2+5 b^2 C\right ) (a+b \cos (c+d x))^{2/3} F_1\left (\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right )}{4 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3}}+\frac{(a+b) (8 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left (\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right )}{4 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3}}+\frac{3 C \sin (c+d x) (a+b \cos (c+d x))^{5/3}}{8 b d} \]
Antiderivative was successfully verified.
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Rule 3023
Rule 2756
Rule 2665
Rule 139
Rule 138
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^{2/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{3 C (a+b \cos (c+d x))^{5/3} \sin (c+d x)}{8 b d}+\frac{3 \int (a+b \cos (c+d x))^{2/3} \left (\frac{1}{3} b (8 A+5 C)+\frac{1}{3} (8 b B-3 a C) \cos (c+d x)\right ) \, dx}{8 b}\\ &=\frac{3 C (a+b \cos (c+d x))^{5/3} \sin (c+d x)}{8 b d}+\frac{(8 b B-3 a C) \int (a+b \cos (c+d x))^{5/3} \, dx}{8 b^2}+\frac{\left (3 \left (\frac{1}{3} b^2 (8 A+5 C)-\frac{1}{3} a (8 b B-3 a C)\right )\right ) \int (a+b \cos (c+d x))^{2/3} \, dx}{8 b^2}\\ &=\frac{3 C (a+b \cos (c+d x))^{5/3} \sin (c+d x)}{8 b d}-\frac{((8 b B-3 a C) \sin (c+d x)) \operatorname{Subst}\left (\int \frac{(a+b x)^{5/3}}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\cos (c+d x)\right )}{8 b^2 d \sqrt{1-\cos (c+d x)} \sqrt{1+\cos (c+d x)}}-\frac{\left (3 \left (\frac{1}{3} b^2 (8 A+5 C)-\frac{1}{3} a (8 b B-3 a C)\right ) \sin (c+d x)\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{2/3}}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\cos (c+d x)\right )}{8 b^2 d \sqrt{1-\cos (c+d x)} \sqrt{1+\cos (c+d x)}}\\ &=\frac{3 C (a+b \cos (c+d x))^{5/3} \sin (c+d x)}{8 b d}+\frac{\left ((-a-b) (8 b B-3 a C) (a+b \cos (c+d x))^{2/3} \sin (c+d x)\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{-a-b}-\frac{b x}{-a-b}\right )^{5/3}}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\cos (c+d x)\right )}{8 b^2 d \sqrt{1-\cos (c+d x)} \sqrt{1+\cos (c+d x)} \left (-\frac{a+b \cos (c+d x)}{-a-b}\right )^{2/3}}-\frac{\left (3 \left (\frac{1}{3} b^2 (8 A+5 C)-\frac{1}{3} a (8 b B-3 a C)\right ) (a+b \cos (c+d x))^{2/3} \sin (c+d x)\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{-a-b}-\frac{b x}{-a-b}\right )^{2/3}}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\cos (c+d x)\right )}{8 b^2 d \sqrt{1-\cos (c+d x)} \sqrt{1+\cos (c+d x)} \left (-\frac{a+b \cos (c+d x)}{-a-b}\right )^{2/3}}\\ &=\frac{3 C (a+b \cos (c+d x))^{5/3} \sin (c+d x)}{8 b d}+\frac{(a+b) (8 b B-3 a C) F_1\left (\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right ) (a+b \cos (c+d x))^{2/3} \sin (c+d x)}{4 \sqrt{2} b^2 d \sqrt{1+\cos (c+d x)} \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3}}+\frac{\left (8 A b^2-8 a b B+3 a^2 C+5 b^2 C\right ) F_1\left (\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right ) (a+b \cos (c+d x))^{2/3} \sin (c+d x)}{4 \sqrt{2} b^2 d \sqrt{1+\cos (c+d x)} \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 3.39797, size = 296, normalized size = 1.02 \[ -\frac{3 \csc (c+d x) (a+b \cos (c+d x))^{2/3} \left (4 \left (-6 a^2 C+16 a b B+40 A b^2+25 b^2 C\right ) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x)) F_1\left (\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right )+20 \left (b^2-a^2\right ) (8 b B-3 a C) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right )-20 b^2 \sin ^2(c+d x) (2 a C+8 b B+5 b C \cos (c+d x))\right )}{800 b^3 d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.303, size = 0, normalized size = 0. \begin{align*} \int \left ( a+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.
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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} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{2}{3}}\,{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 \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{2}{3}}, 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} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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