Optimal. Leaf size=19 \[ \frac {(a+b \sin (x))^{n+1}}{b (n+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, 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} \[ \frac {(a+b \sin (x))^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2668
Rubi steps
\begin {align*} \int \cos (x) (a+b \sin (x))^n \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^n \, dx,x,b \sin (x)\right )}{b}\\ &=\frac {(a+b \sin (x))^{1+n}}{b (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 18, normalized size = 0.95 \[ \frac {(a+b \sin (x))^{n+1}}{b n+b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 22, normalized size = 1.16 \[ \frac {{\left (b \sin \relax (x) + a\right )} {\left (b \sin \relax (x) + a\right )}^{n}}{b n + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 19, normalized size = 1.00 \[ \frac {{\left (b \sin \relax (x) + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 20, normalized size = 1.05 \[ \frac {\left (a +b \sin \relax (x )\right )^{n +1}}{b \left (n +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 19, normalized size = 1.00 \[ \frac {{\left (b \sin \relax (x) + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.13, size = 19, normalized size = 1.00 \[ \frac {{\left (a+b\,\sin \relax (x)\right )}^{n+1}}{b\,\left (n+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.88, size = 56, normalized size = 2.95 \[ \begin {cases} \frac {\sin {\relax (x )}}{a} & \text {for}\: b = 0 \wedge n = -1 \\a^{n} \sin {\relax (x )} & \text {for}\: b = 0 \\\frac {\log {\left (\frac {a}{b} + \sin {\relax (x )} \right )}}{b} & \text {for}\: n = -1 \\\frac {a \left (a + b \sin {\relax (x )}\right )^{n}}{b n + b} + \frac {b \left (a + b \sin {\relax (x )}\right )^{n} \sin {\relax (x )}}{b n + b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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