Optimal. Leaf size=144 \[ \frac {3 C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (3 m+4) (b \cos (c+d x))^{2/3}}-\frac {3 (A (3 m+4)+3 C m+C) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+1);\frac {1}{6} (3 m+7);\cos ^2(c+d x)\right )}{d (3 m+1) (3 m+4) \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}} \]
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Rubi [A] time = 0.10, antiderivative size = 144, 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 = {20, 3014, 2643} \[ \frac {3 C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (3 m+4) (b \cos (c+d x))^{2/3}}-\frac {3 (A (3 m+4)+3 C m+C) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+1);\frac {1}{6} (3 m+7);\cos ^2(c+d x)\right )}{d (3 m+1) (3 m+4) \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rule 3014
Rubi steps
\begin {align*} \int \frac {\cos ^m(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx &=\frac {\cos ^{\frac {2}{3}}(c+d x) \int \cos ^{-\frac {2}{3}+m}(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx}{(b \cos (c+d x))^{2/3}}\\ &=\frac {3 C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3}}+\frac {\left (\left (C \left (\frac {1}{3}+m\right )+A \left (\frac {4}{3}+m\right )\right ) \cos ^{\frac {2}{3}}(c+d x)\right ) \int \cos ^{-\frac {2}{3}+m}(c+d x) \, dx}{\left (\frac {4}{3}+m\right ) (b \cos (c+d x))^{2/3}}\\ &=\frac {3 C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3}}-\frac {3 (C+3 C m+A (4+3 m)) \cos ^{1+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (1+3 m);\frac {1}{6} (7+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+3 m) (4+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 142, normalized size = 0.99 \[ -\frac {3 \sqrt {\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left (A (3 m+7) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+1);\frac {1}{6} (3 m+7);\cos ^2(c+d x)\right )+C (3 m+1) \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+7);\frac {1}{6} (3 m+13);\cos ^2(c+d x)\right )\right )}{d (3 m+1) (3 m+7) (b \cos (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \cos \left (d x + c\right )^{m}}{b \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{m}\left (d x +c \right )\right ) \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^m\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \cos ^{m}{\left (c + d x \right )}}{\left (b \cos {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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