Optimal. Leaf size=13 \[ \frac{\sec (c+d x)}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0338714, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3175, 2606, 8} \[ \frac{\sec (c+d x)}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3175
Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \frac{\sin (c+d x)}{a-a \sin ^2(c+d x)} \, dx &=\frac{\int \sec (c+d x) \tan (c+d x) \, dx}{a}\\ &=\frac{\operatorname{Subst}(\int 1 \, dx,x,\sec (c+d x))}{a d}\\ &=\frac{\sec (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.0127608, size = 13, normalized size = 1. \[ \frac{\sec (c+d x)}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.03, size = 16, normalized size = 1.2 \begin{align*}{\frac{1}{da\cos \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.949293, size = 20, normalized size = 1.54 \begin{align*} \frac{1}{a d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.65355, size = 30, normalized size = 2.31 \begin{align*} \frac{1}{a d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.1869, size = 34, normalized size = 2.62 \begin{align*} \begin{cases} - \frac{2}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - a d} & \text{for}\: d \neq 0 \\\frac{x \sin{\left (c \right )}}{- a \sin ^{2}{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11697, size = 20, normalized size = 1.54 \begin{align*} \frac{1}{a d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]