Optimal. Leaf size=23 \[ -\frac {\cos (c+d x)}{d (a \sin (c+d x)+a)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2648} \[ -\frac {\cos (c+d x)}{d (a \sin (c+d x)+a)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2648
Rubi steps
\begin {align*} \int \frac {1}{a+a \sin (c+d x)} \, dx &=-\frac {\cos (c+d x)}{d (a+a \sin (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.04, size = 48, normalized size = 2.09 \[ \frac {2 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d (a \sin (c+d x)+a)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 42, normalized size = 1.83 \[ -\frac {\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1}{a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.34, size = 21, normalized size = 0.91 \[ -\frac {2}{a d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 22, normalized size = 0.96 \[ -\frac {2}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 27, normalized size = 1.17 \[ -\frac {2}{{\left (a + \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.43, size = 21, normalized size = 0.91 \[ -\frac {2}{a\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.83, size = 27, normalized size = 1.17 \[ \begin {cases} - \frac {2}{a d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} & \text {for}\: d \neq 0 \\\frac {x}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________