Optimal. Leaf size=117 \[ \frac {C \sin (c+d x) (b \cos (c+d x))^{m+1}}{b d (m+2)}-\frac {(A (m+2)+C (m+1)) \sin (c+d x) (b \cos (c+d x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(c+d x)\right )}{b d (m+1) (m+2) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3014, 2643} \[ \frac {C \sin (c+d x) (b \cos (c+d x))^{m+1}}{b d (m+2)}-\frac {(A (m+2)+C (m+1)) \sin (c+d x) (b \cos (c+d x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(c+d x)\right )}{b d (m+1) (m+2) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3014
Rubi steps
\begin {align*} \int (b \cos (c+d x))^m \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C (b \cos (c+d x))^{1+m} \sin (c+d x)}{b d (2+m)}+\left (A+\frac {C (1+m)}{2+m}\right ) \int (b \cos (c+d x))^m \, dx\\ &=\frac {C (b \cos (c+d x))^{1+m} \sin (c+d x)}{b d (2+m)}-\frac {\left (A+\frac {C (1+m)}{2+m}\right ) (b \cos (c+d x))^{1+m} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (1+m) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 114, normalized size = 0.97 \[ -\frac {\sqrt {\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^m \left (A (m+3) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(c+d x)\right )+C (m+1) \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};\cos ^2(c+d x)\right )\right )}{d (m+1) (m+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{m}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.23, size = 0, normalized size = 0.00 \[ \int \left (b \cos \left (d x +c \right )\right )^{m} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^m \,d x \]
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 \[ \int \left (b \cos {\left (c + d x \right )}\right )^{m} \left (A + C \cos ^{2}{\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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