Optimal. Leaf size=229 \[ \frac {3 C \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \sin (c+d x)}{d (8+3 m)}-\frac {3 (C (5+3 m)+A (8+3 m)) \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac {1}{2},\frac {1}{6} (5+3 m);\frac {1}{6} (11+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+3 m) (8+3 m) \sqrt {\sin ^2(c+d x)}}-\frac {3 B \cos ^{2+m}(c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac {1}{2},\frac {1}{6} (8+3 m);\frac {1}{6} (14+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (8+3 m) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.15, 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, 3102, 2827,
2722} \begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 2722
Rule 2827
Rule 3102
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*}
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Mathematica [A]
time = 0.46, size = 166, normalized size = 0.72 \begin {gather*} -\frac {3 \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \sin (c+d x) \left ((C (5+3 m)+A (8+3 m)) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (5+3 m);\frac {1}{6} (11+3 m);\cos ^2(c+d x)\right )+(5+3 m) \left (B \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (8+3 m);\frac {7}{3}+\frac {m}{2};\cos ^2(c+d x)\right )-C \sqrt {\sin ^2(c+d x)}\right )\right )}{d (5+3 m) (8+3 m) \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \left (\cos ^{m}\left (d x +c \right )\right ) \left (b \cos \left (d x +c \right )\right )^{\frac {2}{3}} \left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx\]
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(-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 {\cos \left (c+d\,x\right )}^m\,{\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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