Optimal. Leaf size=28 \[ \frac {\cos (c+d x)}{d (a \sin (c+d x)+a)}+\frac {x}{a} \]
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Rubi [A] time = 0.04, antiderivative size = 28, 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 = {2735, 2648} \[ \frac {\cos (c+d x)}{d (a \sin (c+d x)+a)}+\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {x}{a}-\int \frac {1}{a+a \sin (c+d x)} \, dx\\ &=\frac {x}{a}+\frac {\cos (c+d x)}{d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [B] time = 0.12, size = 72, normalized size = 2.57 \[ \frac {\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right ) \left ((c+d x-2) \sin \left (\frac {1}{2} (c+d x)\right )+(c+d x) \cos \left (\frac {1}{2} (c+d x)\right )\right )}{a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 54, normalized size = 1.93 \[ \frac {d x + {\left (d x + 1\right )} \cos \left (d x + c\right ) + {\left (d x - 1\right )} \sin \left (d x + c\right ) + 1}{a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.85, size = 32, normalized size = 1.14 \[ \frac {\frac {d x + c}{a} + \frac {2}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 41, normalized size = 1.46 \[ \frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {2}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 50, normalized size = 1.79 \[ \frac {2 \, {\left (\frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.74, size = 27, normalized size = 0.96 \[ \frac {x}{a}+\frac {2}{a\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.61, size = 80, normalized size = 2.86 \[ \begin {cases} \frac {d x \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {d x}{a d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {2}{a d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} & \text {for}\: d \neq 0 \\\frac {x \sin {\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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