Optimal. Leaf size=24 \[ \frac {(c \sin (a+b x))^{1+m}}{b c (1+m)} \]
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Rubi [A]
time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2644, 30}
\begin {gather*} \frac {(c \sin (a+b x))^{m+1}}{b c (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2644
Rubi steps
\begin {align*} \int \cos (a+b x) (c \sin (a+b x))^m \, dx &=\frac {\text {Subst}\left (\int x^m \, dx,x,c \sin (a+b x)\right )}{b c}\\ &=\frac {(c \sin (a+b x))^{1+m}}{b c (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.04 \begin {gather*} \frac {\sin (a+b x) (c \sin (a+b x))^m}{b (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.11, size = 25, normalized size = 1.04
method | result | size |
derivativedivides | \(\frac {\left (c \sin \left (b x +a \right )\right )^{1+m}}{b c \left (1+m \right )}\) | \(25\) |
default | \(\frac {\left (c \sin \left (b x +a \right )\right )^{1+m}}{b c \left (1+m \right )}\) | \(25\) |
norman | \(\frac {2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) {\mathrm e}^{m \ln \left (\frac {2 c \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\right )}}{b \left (1+m \right ) \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\) | \(66\) |
risch | \(\text {Expression too large to display}\) | \(1016\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 24, normalized size = 1.00 \begin {gather*} \frac {\left (c \sin \left (b x + a\right )\right )^{m + 1}}{b c {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 24, normalized size = 1.00 \begin {gather*} \frac {\left (c \sin \left (b x + a\right )\right )^{m} \sin \left (b x + a\right )}{b m + b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (17) = 34\).
time = 0.46, size = 56, normalized size = 2.33 \begin {gather*} \begin {cases} \frac {x \cos {\left (a \right )}}{c \sin {\left (a \right )}} & \text {for}\: b = 0 \wedge m = -1 \\x \left (c \sin {\left (a \right )}\right )^{m} \cos {\left (a \right )} & \text {for}\: b = 0 \\\frac {\log {\left (\sin {\left (a + b x \right )} \right )}}{b c} & \text {for}\: m = -1 \\\frac {\left (c \sin {\left (a + b x \right )}\right )^{m} \sin {\left (a + b x \right )}}{b m + b} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.24, size = 24, normalized size = 1.00 \begin {gather*} \frac {\left (c \sin \left (b x + a\right )\right )^{m + 1}}{b c {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.61, size = 25, normalized size = 1.04 \begin {gather*} \frac {\sin \left (a+b\,x\right )\,{\left (c\,\sin \left (a+b\,x\right )\right )}^m}{b\,\left (m+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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