Optimal. Leaf size=20 \[ -\frac{(a+b \cos (x))^{n+1}}{b (n+1)} \]
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Rubi [A] time = 0.0245843, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2668, 32} \[ -\frac{(a+b \cos (x))^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 32
Rubi steps
\begin{align*} \int (a+b \cos (x))^n \sin (x) \, dx &=-\frac{\operatorname{Subst}\left (\int (a+x)^n \, dx,x,b \cos (x)\right )}{b}\\ &=-\frac{(a+b \cos (x))^{1+n}}{b (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0322533, size = 19, normalized size = 0.95 \[ -\frac{(a+b \cos (x))^{n+1}}{b n+b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 21, normalized size = 1.1 \begin{align*} -{\frac{ \left ( a+b\cos \left ( x \right ) \right ) ^{1+n}}{b \left ( 1+n \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07964, size = 59, normalized size = 2.95 \begin{align*} -\frac{{\left (b \cos \left (x\right ) + a\right )}{\left (b \cos \left (x\right ) + a\right )}^{n}}{b n + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.5132, size = 63, normalized size = 3.15 \begin{align*} \begin{cases} - \frac{\cos{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = -1 \\- a^{n} \cos{\left (x \right )} & \text{for}\: b = 0 \\- \frac{\log{\left (\frac{a}{b} + \cos{\left (x \right )} \right )}}{b} & \text{for}\: n = -1 \\- \frac{a \left (a + b \cos{\left (x \right )}\right )^{n}}{b n + b} - \frac{b \left (a + b \cos{\left (x \right )}\right )^{n} \cos{\left (x \right )}}{b n + b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10384, size = 27, normalized size = 1.35 \begin{align*} -\frac{{\left (b \cos \left (x\right ) + a\right )}^{n + 1}}{b{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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