Optimal. Leaf size=26 \[ \frac {(a+b \sin (c+d x))^{m+1}}{b d (m+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2668, 32} \[ \frac {(a+b \sin (c+d x))^{m+1}}{b d (m+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2668
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \sin (c+d x))^m \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^m \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {(a+b \sin (c+d x))^{1+m}}{b d (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 1.00 \[ \frac {(a+b \sin (c+d x))^{m+1}}{b d (m+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 33, normalized size = 1.27 \[ \frac {{\left (b \sin \left (d x + c\right ) + a\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{b d m + b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 26, normalized size = 1.00 \[ \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{m + 1}}{b d {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 27, normalized size = 1.04 \[ \frac {\left (a +b \sin \left (d x +c \right )\right )^{1+m}}{b d \left (1+m \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.29, size = 26, normalized size = 1.00 \[ \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{m + 1}}{b d {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.32, size = 26, normalized size = 1.00 \[ \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{m+1}}{b\,d\,\left (m+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.21, size = 99, normalized size = 3.81 \[ \begin {cases} \frac {x \cos {\relax (c )}}{a} & \text {for}\: b = 0 \wedge d = 0 \wedge m = -1 \\\frac {a^{m} \sin {\left (c + d x \right )}}{d} & \text {for}\: b = 0 \\x \left (a + b \sin {\relax (c )}\right )^{m} \cos {\relax (c )} & \text {for}\: d = 0 \\\frac {\log {\left (\frac {a}{b} + \sin {\left (c + d x \right )} \right )}}{b d} & \text {for}\: m = -1 \\\frac {a \left (a + b \sin {\left (c + d x \right )}\right )^{m}}{b d m + b d} + \frac {b \left (a + b \sin {\left (c + d x \right )}\right )^{m} \sin {\left (c + d x \right )}}{b d m + b d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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