Optimal. Leaf size=154 \[ \frac {3 C (b \cos (c+d x))^{8/3} \sin (c+d x)}{11 b^2 d}-\frac {3 (11 A+8 C) (b \cos (c+d x))^{8/3} \, _2F_1\left (\frac {1}{2},\frac {4}{3};\frac {7}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{88 b^2 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B (b \cos (c+d x))^{11/3} \, _2F_1\left (\frac {1}{2},\frac {11}{6};\frac {17}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{11 b^3 d \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.10, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {16, 3102, 2827,
2722} \begin {gather*} -\frac {3 (11 A+8 C) \sin (c+d x) (b \cos (c+d x))^{8/3} \, _2F_1\left (\frac {1}{2},\frac {4}{3};\frac {7}{3};\cos ^2(c+d x)\right )}{88 b^2 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B \sin (c+d x) (b \cos (c+d x))^{11/3} \, _2F_1\left (\frac {1}{2},\frac {11}{6};\frac {17}{6};\cos ^2(c+d x)\right )}{11 b^3 d \sqrt {\sin ^2(c+d x)}}+\frac {3 C \sin (c+d x) (b \cos (c+d x))^{8/3}}{11 b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2722
Rule 2827
Rule 3102
Rubi steps
\begin {align*} \int \cos (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 {\int (b \cos (c+d x))^{5/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{b}\\ &=\frac {3 C (b \cos (c+d x))^{8/3} \sin (c+d x)}{11 b^2 d}+\frac {3 \int (b \cos (c+d x))^{5/3} \left (\frac {1}{3} b (11 A+8 C)+\frac {11}{3} b B \cos (c+d x)\right ) \, dx}{11 b^2}\\ &=\frac {3 C (b \cos (c+d x))^{8/3} \sin (c+d x)}{11 b^2 d}+\frac {B \int (b \cos (c+d x))^{8/3} \, dx}{b^2}+\frac {(11 A+8 C) \int (b \cos (c+d x))^{5/3} \, dx}{11 b}\\ &=\frac {3 C (b \cos (c+d x))^{8/3} \sin (c+d x)}{11 b^2 d}-\frac {3 (11 A+8 C) (b \cos (c+d x))^{8/3} \, _2F_1\left (\frac {1}{2},\frac {4}{3};\frac {7}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{88 b^2 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B (b \cos (c+d x))^{11/3} \, _2F_1\left (\frac {1}{2},\frac {11}{6};\frac {17}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{11 b^3 d \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 109, normalized size = 0.71 \begin {gather*} -\frac {3 (b \cos (c+d x))^{8/3} \sin (c+d x) \left ((11 A+8 C) \, _2F_1\left (\frac {1}{2},\frac {4}{3};\frac {7}{3};\cos ^2(c+d x)\right )+8 B \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {11}{6};\frac {17}{6};\cos ^2(c+d x)\right )-8 C \sqrt {\sin ^2(c+d x)}\right )}{88 b^2 d \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \cos \left (d x +c \right ) \left (b \cos \left (d x +c \right )\right )^{\frac {2}{3}} \left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.01 \begin {gather*} \int \cos \left (c+d\,x\right )\,{\left (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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