Optimal. Leaf size=154 \[ \frac {3 C (b \cos (c+d x))^{2/3} \sin (c+d x)}{5 b^2 d}-\frac {3 (5 A+2 C) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{10 b^2 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{5 b^3 d \sqrt {\sin ^2(c+d x)}} \]
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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 (5 A+2 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\cos ^2(c+d x)\right )}{10 b^2 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(c+d x)\right )}{5 b^3 d \sqrt {\sin ^2(c+d x)}}+\frac {3 C \sin (c+d x) (b \cos (c+d x))^{2/3}}{5 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 \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx &=\frac {\int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx}{b}\\ &=\frac {3 C (b \cos (c+d x))^{2/3} \sin (c+d x)}{5 b^2 d}+\frac {3 \int \frac {\frac {1}{3} b (5 A+2 C)+\frac {5}{3} b B \cos (c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx}{5 b^2}\\ &=\frac {3 C (b \cos (c+d x))^{2/3} \sin (c+d x)}{5 b^2 d}+\frac {B \int (b \cos (c+d x))^{2/3} \, dx}{b^2}+\frac {(5 A+2 C) \int \frac {1}{\sqrt [3]{b \cos (c+d x)}} \, dx}{5 b}\\ &=\frac {3 C (b \cos (c+d x))^{2/3} \sin (c+d x)}{5 b^2 d}-\frac {3 (5 A+2 C) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{10 b^2 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{5 b^3 d \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 111, normalized size = 0.72 \begin {gather*} -\frac {3 \left ((5 A+2 C) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\cos ^2(c+d x)\right )+2 B \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(c+d x)\right )-2 C \sqrt {\sin ^2(c+d x)}\right ) \sin (2 (c+d x))}{20 b d \sqrt [3]{b \cos (c+d x)} \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {\cos \left (d x +c \right ) \left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx\]
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 \frac {\cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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