Optimal. Leaf size=141 \[ -\frac {A (b \cos (c+d x))^{2+n} \, _2F_1\left (\frac {1}{2},\frac {2+n}{2};\frac {4+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^2 d (2+n) \sqrt {\sin ^2(c+d x)}}-\frac {B (b \cos (c+d x))^{3+n} \, _2F_1\left (\frac {1}{2},\frac {3+n}{2};\frac {5+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^3 d (3+n) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {16, 2827, 2722}
\begin {gather*} -\frac {A \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left (\frac {1}{2},\frac {n+2}{2};\frac {n+4}{2};\cos ^2(c+d x)\right )}{b^2 d (n+2) \sqrt {\sin ^2(c+d x)}}-\frac {B \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left (\frac {1}{2},\frac {n+3}{2};\frac {n+5}{2};\cos ^2(c+d x)\right )}{b^3 d (n+3) \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2722
Rule 2827
Rubi steps
\begin {align*} \int \cos (c+d x) (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx &=\frac {\int (b \cos (c+d x))^{1+n} (A+B \cos (c+d x)) \, dx}{b}\\ &=\frac {A \int (b \cos (c+d x))^{1+n} \, dx}{b}+\frac {B \int (b \cos (c+d x))^{2+n} \, dx}{b^2}\\ &=-\frac {A (b \cos (c+d x))^{2+n} \, _2F_1\left (\frac {1}{2},\frac {2+n}{2};\frac {4+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^2 d (2+n) \sqrt {\sin ^2(c+d x)}}-\frac {B (b \cos (c+d x))^{3+n} \, _2F_1\left (\frac {1}{2},\frac {3+n}{2};\frac {5+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^3 d (3+n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 118, normalized size = 0.84 \begin {gather*} -\frac {\cos (c+d x) (b \cos (c+d x))^n \cot (c+d x) \left (A (3+n) \, _2F_1\left (\frac {1}{2},\frac {2+n}{2};\frac {4+n}{2};\cos ^2(c+d x)\right )+B (2+n) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {3+n}{2};\frac {5+n}{2};\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (2+n) (3+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.27, size = 0, normalized size = 0.00 \[\int \cos \left (d x +c \right ) \left (b \cos \left (d x +c \right )\right )^{n} \left (A +B \cos \left (d x +c \right )\right )\, dx\]
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 \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 \left (b \cos {\left (c + d x \right )}\right )^{n} \left (A + B \cos {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}\, 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 \cos \left (c+d\,x\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n\,\left (A+B\,\cos \left (c+d\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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