Optimal. Leaf size=119 \[ -\frac {3 A (b \cos (c+d x))^{10/3} \, _2F_1\left (\frac {1}{2},\frac {5}{3};\frac {8}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{10 b^2 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B (b \cos (c+d x))^{13/3} \, _2F_1\left (\frac {1}{2},\frac {13}{6};\frac {19}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{13 b^3 d \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.06, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {16, 2827, 2722}
\begin {gather*} -\frac {3 A \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left (\frac {1}{2},\frac {5}{3};\frac {8}{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))^{13/3} \, _2F_1\left (\frac {1}{2},\frac {13}{6};\frac {19}{6};\cos ^2(c+d x)\right )}{13 b^3 d \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2722
Rule 2827
Rubi steps
\begin {align*} \int \cos (c+d x) (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \, dx &=\frac {\int (b \cos (c+d x))^{7/3} (A+B \cos (c+d x)) \, dx}{b}\\ &=\frac {A \int (b \cos (c+d x))^{7/3} \, dx}{b}+\frac {B \int (b \cos (c+d x))^{10/3} \, dx}{b^2}\\ &=-\frac {3 A (b \cos (c+d x))^{10/3} \, _2F_1\left (\frac {1}{2},\frac {5}{3};\frac {8}{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))^{13/3} \, _2F_1\left (\frac {1}{2},\frac {13}{6};\frac {19}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{13 b^3 d \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 89, normalized size = 0.75 \begin {gather*} -\frac {3 (b \cos (c+d x))^{7/3} \cot (c+d x) \left (13 A \, _2F_1\left (\frac {1}{2},\frac {5}{3};\frac {8}{3};\cos ^2(c+d x)\right )+10 B \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {13}{6};\frac {19}{6};\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{130 b d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \cos \left (d x +c \right ) \left (b \cos \left (d x +c \right )\right )^{\frac {4}{3}} \left (A +B \cos \left (d x +c \right )\right )\, dx\]
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 )}^{4/3}\,\left (A+B\,\cos \left (c+d\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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