Optimal. Leaf size=167 \[ -\frac {3 A \cos ^{1+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (1+3 m);\frac {1}{6} (7+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}-\frac {3 B \cos ^{2+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (4+3 m);\frac {1}{6} (10+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {20, 2827, 2722}
\begin {gather*} -\frac {3 A \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) \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac {3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+4);\frac {1}{6} (3 m+10);\cos ^2(c+d x)\right )}{d (3 m+4) \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 2722
Rule 2827
Rubi steps
\begin {align*} \int \frac {\cos ^m(c+d x) (A+B \cos (c+d x))}{(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) (A+B \cos (c+d x)) \, dx}{(b \cos (c+d x))^{2/3}}\\ &=\frac {\left (A \cos ^{\frac {2}{3}}(c+d x)\right ) \int \cos ^{-\frac {2}{3}+m}(c+d x) \, dx}{(b \cos (c+d x))^{2/3}}+\frac {\left (B \cos ^{\frac {2}{3}}(c+d x)\right ) \int \cos ^{\frac {1}{3}+m}(c+d x) \, dx}{(b \cos (c+d x))^{2/3}}\\ &=-\frac {3 A \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) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}-\frac {3 B \cos ^{2+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (4+3 m);\frac {1}{6} (10+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (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 = 0.31, size = 140, normalized size = 0.84 \begin {gather*} -\frac {3 \cos ^{1+m}(c+d x) \csc (c+d x) \left (A (4+3 m) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (1+3 m);\frac {1}{6} (7+3 m);\cos ^2(c+d x)\right )+B (1+3 m) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (4+3 m);\frac {5}{3}+\frac {m}{2};\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (1+3 m) (4+3 m) (b \cos (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (\cos ^{m}\left (d x +c \right )\right ) \left (A +B \cos \left (d x +c \right )\right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx\]
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 \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B \cos {\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 \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 )}^m\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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