Optimal. Leaf size=217 \[ -\frac {a (A (m+2)+C (m+1)) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(c+d x)\right )}{d (m+1) (m+2) \sqrt {\sin ^2(c+d x)}}+\frac {a C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}-\frac {b (A (m+3)+C (m+2)) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(c+d x)\right )}{d (m+2) (m+3) \sqrt {\sin ^2(c+d x)}}+\frac {b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3)} \]
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Rubi [A] time = 0.37, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3034, 3023, 2748, 2643} \[ -\frac {a (A (m+2)+C (m+1)) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(c+d x)\right )}{d (m+1) (m+2) \sqrt {\sin ^2(c+d x)}}+\frac {a C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}-\frac {b (A (m+3)+C (m+2)) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(c+d x)\right )}{d (m+2) (m+3) \sqrt {\sin ^2(c+d x)}}+\frac {b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2748
Rule 3023
Rule 3034
Rubi steps
\begin {align*} \int \cos ^m(c+d x) (a+b \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {b C \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m)}+\frac {\int \cos ^m(c+d x) \left (a A (3+m)+b (C (2+m)+A (3+m)) \cos (c+d x)+a C (3+m) \cos ^2(c+d x)\right ) \, dx}{3+m}\\ &=\frac {a C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m)}+\frac {b C \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m)}+\frac {\int \cos ^m(c+d x) (a (3+m) (C (1+m)+A (2+m))+b (2+m) (C (2+m)+A (3+m)) \cos (c+d x)) \, dx}{6+5 m+m^2}\\ &=\frac {a C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m)}+\frac {b C \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m)}+\frac {(a (3+m) (C (1+m)+A (2+m))) \int \cos ^m(c+d x) \, dx}{6+5 m+m^2}+\frac {(b (2+m) (C (2+m)+A (3+m))) \int \cos ^{1+m}(c+d x) \, dx}{6+5 m+m^2}\\ &=\frac {a C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m)}+\frac {b C \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m)}-\frac {a (C (1+m)+A (2+m)) \cos ^{1+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+m) (2+m) \sqrt {\sin ^2(c+d x)}}-\frac {b (C (2+m)+A (3+m)) \cos ^{2+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d \left (6+5 m+m^2\right ) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.00, size = 194, normalized size = 0.89 \[ \frac {\sqrt {\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left (\cos (c+d x) \left (C \cos (c+d x) \left (-\frac {a \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};\cos ^2(c+d x)\right )}{m+3}-\frac {b \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+4}{2};\frac {m+6}{2};\cos ^2(c+d x)\right )}{m+4}\right )-\frac {A b \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(c+d x)\right )}{m+2}\right )-\frac {a A \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(c+d x)\right )}{m+1}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{3} + C a \cos \left (d x + c\right )^{2} + A b \cos \left (d x + c\right ) + A a\right )} \cos \left (d x + c\right )^{m}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 8.74, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{m}\left (d x +c \right )\right ) \left (a +b \cos \left (d x +c \right )\right ) \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{m}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^m\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,\left (a+b\,\cos \left (c+d\,x\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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