Optimal. Leaf size=21 \[ -\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 21, 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 = {2667, 32} \[ -\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 31, normalized size = 1.48 \[ -\frac {1}{a^2 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.62, size = 21, normalized size = 1.00 \[ -\frac {1}{a^{2} d \sin \left (d x + c\right ) + a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.42, size = 20, normalized size = 0.95 \[ -\frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )} a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 21, normalized size = 1.00 \[ -\frac {1}{d \left (a +a \sin \left (d x +c \right )\right ) a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.30, size = 20, normalized size = 0.95 \[ -\frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )} a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 18, normalized size = 0.86 \[ -\frac {1}{a^2\,d\,\left (\sin \left (c+d\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.12, size = 32, normalized size = 1.52 \[ \begin {cases} - \frac {1}{a^{2} d \sin {\left (c + d x \right )} + a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________