Optimal. Leaf size=95 \[ \frac {3 C (b \cos (c+d x))^{4/3} \sin (c+d x)}{7 b d}-\frac {3 (7 A+4 C) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{28 b d \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {3093, 2722}
\begin {gather*} \frac {3 C \sin (c+d x) (b \cos (c+d x))^{4/3}}{7 b d}-\frac {3 (7 A+4 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right )}{28 b d \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 3093
Rubi steps
\begin {align*} \int \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {3 C (b \cos (c+d x))^{4/3} \sin (c+d x)}{7 b d}+\frac {1}{7} (7 A+4 C) \int \sqrt [3]{b \cos (c+d x)} \, dx\\ &=\frac {3 C (b \cos (c+d x))^{4/3} \sin (c+d x)}{7 b d}-\frac {3 (7 A+4 C) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{28 b d \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 88, normalized size = 0.93 \begin {gather*} -\frac {3 \sqrt [3]{b \cos (c+d x)} \cot (c+d x) \left (5 A \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right )+2 C \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{3};\frac {8}{3};\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{20 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \left (b \cos \left (d x +c \right )\right )^{\frac {1}{3}} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx\]
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 = -0.01 \begin {gather*} \int \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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