Optimal. Leaf size=232 \[ -\frac {3 b (A (3 m+10)+C (3 m+7)) \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \cos ^{m+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+7);\frac {1}{6} (3 m+13);\cos ^2(c+d x)\right )}{d (3 m+7) (3 m+10) \sqrt {\sin ^2(c+d x)}}-\frac {3 b B \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \cos ^{m+3}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+10);\frac {1}{6} (3 m+16);\cos ^2(c+d x)\right )}{d (3 m+10) \sqrt {\sin ^2(c+d x)}}+\frac {3 b C \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \cos ^{m+2}(c+d x)}{d (3 m+10)} \]
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Rubi [A] time = 0.21, antiderivative size = 222, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {20, 3023, 2748, 2643} \[ -\frac {3 b \left (\frac {A}{3 m+7}+\frac {C}{3 m+10}\right ) \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \cos ^{m+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+7);\frac {1}{6} (3 m+13);\cos ^2(c+d x)\right )}{d \sqrt {\sin ^2(c+d x)}}-\frac {3 b B \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \cos ^{m+3}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+10);\frac {1}{6} (3 m+16);\cos ^2(c+d x)\right )}{d (3 m+10) \sqrt {\sin ^2(c+d x)}}+\frac {3 b C \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \cos ^{m+2}(c+d x)}{d (3 m+10)} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rule 2748
Rule 3023
Rubi steps
\begin {align*} \int \cos ^m(c+d x) (b \cos (c+d x))^{4/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {\left (b \sqrt [3]{b \cos (c+d x)}\right ) \int \cos ^{\frac {4}{3}+m}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{\sqrt [3]{\cos (c+d x)}}\\ &=\frac {3 b C \cos ^{2+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \sin (c+d x)}{d (10+3 m)}+\frac {\left (3 b \sqrt [3]{b \cos (c+d x)}\right ) \int \cos ^{\frac {4}{3}+m}(c+d x) \left (\frac {1}{3} \left (3 C \left (\frac {7}{3}+m\right )+3 A \left (\frac {10}{3}+m\right )\right )+\frac {1}{3} B (10+3 m) \cos (c+d x)\right ) \, dx}{(10+3 m) \sqrt [3]{\cos (c+d x)}}\\ &=\frac {3 b C \cos ^{2+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \sin (c+d x)}{d (10+3 m)}+\frac {\left (b B \sqrt [3]{b \cos (c+d x)}\right ) \int \cos ^{\frac {7}{3}+m}(c+d x) \, dx}{\sqrt [3]{\cos (c+d x)}}+\frac {\left (b (C (7+3 m)+A (10+3 m)) \sqrt [3]{b \cos (c+d x)}\right ) \int \cos ^{\frac {4}{3}+m}(c+d x) \, dx}{(10+3 m) \sqrt [3]{\cos (c+d x)}}\\ &=\frac {3 b C \cos ^{2+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \sin (c+d x)}{d (10+3 m)}-\frac {3 b \left (\frac {A}{7+3 m}+\frac {C}{10+3 m}\right ) \cos ^{2+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac {1}{2},\frac {1}{6} (7+3 m);\frac {1}{6} (13+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d \sqrt {\sin ^2(c+d x)}}-\frac {3 b B \cos ^{3+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac {1}{2},\frac {1}{6} (10+3 m);\frac {1}{6} (16+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (10+3 m) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 169, normalized size = 0.73 \[ -\frac {3 \sin (c+d x) (b \cos (c+d x))^{4/3} \cos ^{m+1}(c+d x) \left ((A (3 m+10)+C (3 m+7)) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+7);\frac {1}{6} (3 m+13);\cos ^2(c+d x)\right )+B (3 m+7) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {m}{2}+\frac {5}{3};\frac {m}{2}+\frac {8}{3};\cos ^2(c+d x)\right )-C (3 m+7) \sqrt {\sin ^2(c+d x)}\right )}{d (3 m+7) (3 m+10) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{3} + B b \cos \left (d x + c\right )^{2} + A b \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \cos \left (d x + c\right )^{m}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}} \cos \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.55, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{m}\left (d x +c \right )\right ) \left (b \cos \left (d x +c \right )\right )^{\frac {4}{3}} \left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}} \cos \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^m\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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