Optimal. Leaf size=95 \[ \frac {3 C (b \cos (c+d x))^{8/3} \sin (c+d x)}{11 b^3 d}-\frac {3 (11 A+8 C) (b \cos (c+d x))^{8/3} \, _2F_1\left (\frac {1}{2},\frac {4}{3};\frac {7}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{88 b^3 d \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.05, 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, 3093, 2722}
\begin {gather*} \frac {3 C \sin (c+d x) (b \cos (c+d x))^{8/3}}{11 b^3 d}-\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^3 d \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2722
Rule 3093
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt [3]{b \cos (c+d x)}} \, dx &=\frac {\int (b \cos (c+d x))^{5/3} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b^2}\\ &=\frac {3 C (b \cos (c+d x))^{8/3} \sin (c+d x)}{11 b^3 d}+\frac {(11 A+8 C) \int (b \cos (c+d x))^{5/3} \, dx}{11 b^2}\\ &=\frac {3 C (b \cos (c+d x))^{8/3} \sin (c+d x)}{11 b^3 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^3 d \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 96, normalized size = 1.01 \begin {gather*} -\frac {3 \cot (c+d x) \left (7 A \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {4}{3};\frac {7}{3};\cos ^2(c+d x)\right )+4 C \cos ^4(c+d x) \, _2F_1\left (\frac {1}{2},\frac {7}{3};\frac {10}{3};\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{56 d \sqrt [3]{b \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \frac {\left (\cos ^{2}\left (d x +c \right )\right ) \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {1}{3}}}\, dx\]
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 )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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