Optimal. Leaf size=95 \[ \frac {3 C \sin (c+d x) (b \cos (c+d x))^{5/3}}{8 b^3 d}-\frac {3 (8 A+5 C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(c+d x)\right )}{40 b^3 d \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {16, 3014, 2643} \[ \frac {3 C \sin (c+d x) (b \cos (c+d x))^{5/3}}{8 b^3 d}-\frac {3 (8 A+5 C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(c+d x)\right )}{40 b^3 d \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rule 3014
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx &=\frac {\int (b \cos (c+d x))^{2/3} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b^2}\\ &=\frac {3 C (b \cos (c+d x))^{5/3} \sin (c+d x)}{8 b^3 d}+\frac {(8 A+5 C) \int (b \cos (c+d x))^{2/3} \, dx}{8 b^2}\\ &=\frac {3 C (b \cos (c+d x))^{5/3} \sin (c+d x)}{8 b^3 d}-\frac {3 (8 A+5 C) (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)}{40 b^3 d \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 96, normalized size = 1.01 \[ -\frac {3 \sqrt {\sin ^2(c+d x)} \cos ^2(c+d x) \cot (c+d x) \left (11 A \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(c+d x)\right )+5 C \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {11}{6};\frac {17}{6};\cos ^2(c+d x)\right )\right )}{55 d (b \cos (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}}{b^{2}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{2}\left (d x +c \right )\right ) \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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