Optimal. Leaf size=21 \[ -\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, 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 = {2746, 32}
\begin {gather*} -\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\text {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 = 0.04, size = 31, normalized size = 1.48 \begin {gather*} -\frac {1}{a^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 21, normalized size = 1.00
method | result | size |
derivativedivides | \(-\frac {1}{d \left (a +a \sin \left (d x +c \right )\right ) a}\) | \(21\) |
default | \(-\frac {1}{d \left (a +a \sin \left (d x +c \right )\right ) a}\) | \(21\) |
risch | \(-\frac {2 i {\mathrm e}^{i \left (d x +c \right )}}{d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2}}\) | \(33\) |
norman | \(\frac {\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 20, normalized size = 0.95 \begin {gather*} -\frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )} a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.33, size = 21, normalized size = 1.00 \begin {gather*} -\frac {1}{a^{2} d \sin \left (d x + c\right ) + a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.55, size = 32, normalized size = 1.52 \begin {gather*} \begin {cases} - \frac {1}{a^{2} d \sin {\left (c + d x \right )} + a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \cos {\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 5.48, size = 20, normalized size = 0.95 \begin {gather*} -\frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )} a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 18, normalized size = 0.86 \begin {gather*} -\frac {1}{a^2\,d\,\left (\sin \left (c+d\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________