Optimal. Leaf size=290 \[ \frac {3 C (a+b \cos (c+d x))^{5/3} \sin (c+d x)}{8 b d}+\frac {(a+b) (8 b B-3 a C) F_1\left (\frac {1}{2};\frac {1}{2},-\frac {5}{3};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right ) (a+b \cos (c+d x))^{2/3} \sin (c+d x)}{4 \sqrt {2} b^2 d \sqrt {1+\cos (c+d x)} \left (\frac {a+b \cos (c+d x)}{a+b}\right )^{2/3}}+\frac {\left (8 A b^2-8 a b B+3 a^2 C+5 b^2 C\right ) F_1\left (\frac {1}{2};\frac {1}{2},-\frac {2}{3};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right ) (a+b \cos (c+d x))^{2/3} \sin (c+d x)}{4 \sqrt {2} b^2 d \sqrt {1+\cos (c+d x)} \left (\frac {a+b \cos (c+d x)}{a+b}\right )^{2/3}} \]
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Rubi [A]
time = 0.25, antiderivative size = 290, normalized size of antiderivative = 1.00, 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 = {3102, 2835,
2744, 144, 143} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 143
Rule 144
Rule 2744
Rule 2835
Rule 3102
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)) \text {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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 3.63, size = 296, normalized size = 1.02 \begin {gather*} -\frac {3 (a+b \cos (c+d x))^{2/3} \csc (c+d x) \left (20 \left (-a^2+b^2\right ) (8 b B-3 a C) 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 ) \sqrt {-\frac {b (-1+\cos (c+d x))}{a+b}} \sqrt {-\frac {b (1+\cos (c+d x))}{a-b}}+4 \left (40 A b^2+16 a b B-6 a^2 C+25 b^2 C\right ) 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 ) \sqrt {-\frac {b (-1+\cos (c+d x))}{a+b}} \sqrt {\frac {b (1+\cos (c+d x))}{-a+b}} (a+b \cos (c+d x))-20 b^2 (8 b B+2 a C+5 b C \cos (c+d x)) \sin ^2(c+d x)\right )}{800 b^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \left (a +b \cos \left (d x +c \right )\right )^{\frac {2}{3}} \left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\cos \left (c+d\,x\right )\right )}^{2/3}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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