Optimal. Leaf size=146 \[ \frac{3 C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (3 m+5) \sqrt [3]{b \cos (c+d x)}}-\frac{3 (A (3 m+5)+C (3 m+2)) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right )}{d (3 m+2) (3 m+5) \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
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Rubi [A] time = 0.105282, antiderivative size = 136, normalized size of antiderivative = 0.93, 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 = {20, 3014, 2643} \[ \frac{3 C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (3 m+5) \sqrt [3]{b \cos (c+d x)}}-\frac{3 \left (\frac{A}{3 m+2}+\frac{C}{3 m+5}\right ) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right )}{d \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 3014
Rule 2643
Rubi steps
\begin{align*} \int \frac{\cos ^m(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt [3]{b \cos (c+d x)}} \, dx &=\frac{\sqrt [3]{\cos (c+d x)} \int \cos ^{-\frac{1}{3}+m}(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx}{\sqrt [3]{b \cos (c+d x)}}\\ &=\frac{3 C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (5+3 m) \sqrt [3]{b \cos (c+d x)}}+\frac{\left (\left (C \left (\frac{2}{3}+m\right )+A \left (\frac{5}{3}+m\right )\right ) \sqrt [3]{\cos (c+d x)}\right ) \int \cos ^{-\frac{1}{3}+m}(c+d x) \, dx}{\left (\frac{5}{3}+m\right ) \sqrt [3]{b \cos (c+d x)}}\\ &=\frac{3 C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (5+3 m) \sqrt [3]{b \cos (c+d x)}}-\frac{3 (C (2+3 m)+A (5+3 m)) \cos ^{1+m}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (2+3 m);\frac{1}{6} (8+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2+3 m) (5+3 m) \sqrt [3]{b \cos (c+d x)} \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.270992, size = 142, normalized size = 0.97 \[ -\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left (A (3 m+8) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right )+C (3 m+2) \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+8);\frac{m}{2}+\frac{7}{3};\cos ^2(c+d x)\right )\right )}{d (3 m+2) (3 m+8) \sqrt [3]{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.302, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cos \left ( dx+c \right ) \right ) ^{m} \left ( A+C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ){\frac{1}{\sqrt [3]{b\cos \left ( dx+c \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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}} \cos \left (d x + c\right )^{m}}{b \cos \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \cos ^{m}{\left (c + d x \right )}}{\sqrt [3]{b \cos{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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