Optimal. Leaf size=141 \[ -\frac {A (b \cos (c+d x))^{3+n} \, _2F_1\left (\frac {1}{2},\frac {3+n}{2};\frac {5+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^3 d (3+n) \sqrt {\sin ^2(c+d x)}}-\frac {B (b \cos (c+d x))^{4+n} \, _2F_1\left (\frac {1}{2},\frac {4+n}{2};\frac {6+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^4 d (4+n) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {16, 2827, 2722}
\begin {gather*} -\frac {A \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)}}-\frac {B \sin (c+d x) (b \cos (c+d x))^{n+4} \, _2F_1\left (\frac {1}{2},\frac {n+4}{2};\frac {n+6}{2};\cos ^2(c+d x)\right )}{b^4 d (n+4) \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 ^2(c+d x) (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx &=\frac {\int (b \cos (c+d x))^{2+n} (A+B \cos (c+d x)) \, dx}{b^2}\\ &=\frac {A \int (b \cos (c+d x))^{2+n} \, dx}{b^2}+\frac {B \int (b \cos (c+d x))^{3+n} \, dx}{b^3}\\ &=-\frac {A (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)}}-\frac {B (b \cos (c+d x))^{4+n} \, _2F_1\left (\frac {1}{2},\frac {4+n}{2};\frac {6+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^4 d (4+n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 120, normalized size = 0.85 \begin {gather*} -\frac {\cos ^2(c+d x) (b \cos (c+d x))^n \cot (c+d x) \left (A (4+n) \, _2F_1\left (\frac {1}{2},\frac {3+n}{2};\frac {5+n}{2};\cos ^2(c+d x)\right )+B (3+n) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {4+n}{2};\frac {6+n}{2};\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (3+n) (4+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.44, size = 0, normalized size = 0.00 \[\int \left (\cos ^{2}\left (d x +c \right )\right ) \left (b \cos \left (d x +c \right )\right )^{n} \left (A +B \cos \left (d x +c \right )\right )\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 )}^2\,{\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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