Optimal. Leaf size=167 \[ -\frac {C \sin (c+d x) (a \cos (c+d x))^{m+3} (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m+n+3);\frac {1}{2} (m+n+5);\cos ^2(c+d x)\right )}{a^3 d (m+n+3) \sqrt {\sin ^2(c+d x)}}-\frac {B \sin (c+d x) (a \cos (c+d x))^{m+2} (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m+n+2);\frac {1}{2} (m+n+4);\cos ^2(c+d x)\right )}{a^2 d (m+n+2) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.16, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {20, 3010, 2748, 2643} \[ -\frac {B \sin (c+d x) (a \cos (c+d x))^{m+2} (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m+n+2);\frac {1}{2} (m+n+4);\cos ^2(c+d x)\right )}{a^2 d (m+n+2) \sqrt {\sin ^2(c+d x)}}-\frac {C \sin (c+d x) (a \cos (c+d x))^{m+3} (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m+n+3);\frac {1}{2} (m+n+5);\cos ^2(c+d x)\right )}{a^3 d (m+n+3) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rule 2748
Rule 3010
Rubi steps
\begin {align*} \int (a \cos (c+d x))^m (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\left ((a \cos (c+d x))^{-n} (b \cos (c+d x))^n\right ) \int (a \cos (c+d x))^{m+n} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {\left ((a \cos (c+d x))^{-n} (b \cos (c+d x))^n\right ) \int (a \cos (c+d x))^{1+m+n} (B+C \cos (c+d x)) \, dx}{a}\\ &=\frac {\left (B (a \cos (c+d x))^{-n} (b \cos (c+d x))^n\right ) \int (a \cos (c+d x))^{1+m+n} \, dx}{a}+\frac {\left (C (a \cos (c+d x))^{-n} (b \cos (c+d x))^n\right ) \int (a \cos (c+d x))^{2+m+n} \, dx}{a^2}\\ &=-\frac {B (a \cos (c+d x))^{2+m} (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (2+m+n);\frac {1}{2} (4+m+n);\cos ^2(c+d x)\right ) \sin (c+d x)}{a^2 d (2+m+n) \sqrt {\sin ^2(c+d x)}}-\frac {C (a \cos (c+d x))^{3+m} (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (3+m+n);\frac {1}{2} (5+m+n);\cos ^2(c+d x)\right ) \sin (c+d x)}{a^3 d (3+m+n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 136, normalized size = 0.81 \[ -\frac {\sqrt {\sin ^2(c+d x)} \cos (c+d x) \cot (c+d x) (a \cos (c+d x))^m (b \cos (c+d x))^n \left (B (m+n+3) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m+n+2);\frac {1}{2} (m+n+4);\cos ^2(c+d x)\right )+C (m+n+2) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m+n+3);\frac {1}{2} (m+n+5);\cos ^2(c+d x)\right )\right )}{d (m+n+2) (m+n+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \left (a \cos \left (d x + c\right )\right )^{m} \left (b \cos \left (d x + c\right )\right )^{n}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \left (a \cos \left (d x + c\right )\right )^{m} \left (b \cos \left (d x + c\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.14, size = 0, normalized size = 0.00 \[ \int \left (a \cos \left (d x +c \right )\right )^{m} \left (b \cos \left (d x +c \right )\right )^{n} \left (B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \left (a \cos \left (d x + c\right )\right )^{m} \left (b \cos \left (d x + c\right )\right )^{n}\,{d x} \]
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 \[ \int {\left (a\,\cos \left (c+d\,x\right )\right )}^m\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right ) \,d x \]
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 \[ \int \left (a \cos {\left (c + d x \right )}\right )^{m} \left (b \cos {\left (c + d x \right )}\right )^{n} \left (B + C \cos {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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