Optimal. Leaf size=20 \[ \frac{\tan (c+d x)}{a d}-\frac{x}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0631069, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3171, 3175, 3767, 8} \[ \frac{\tan (c+d x)}{a d}-\frac{x}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3171
Rule 3175
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\sin ^2(c+d x)}{a-a \sin ^2(c+d x)} \, dx &=-\frac{x}{a}+\int \frac{1}{a-a \sin ^2(c+d x)} \, dx\\ &=-\frac{x}{a}+\frac{\int \sec ^2(c+d x) \, dx}{a}\\ &=-\frac{x}{a}-\frac{\operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{a d}\\ &=-\frac{x}{a}+\frac{\tan (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.0136556, size = 27, normalized size = 1.35 \[ \frac{\frac{\tan (c+d x)}{d}-\frac{\tan ^{-1}(\tan (c+d x))}{d}}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.038, size = 30, normalized size = 1.5 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{da}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.42814, size = 35, normalized size = 1.75 \begin{align*} -\frac{\frac{d x + c}{a} - \frac{\tan \left (d x + c\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.61839, size = 74, normalized size = 3.7 \begin{align*} -\frac{d x \cos \left (d x + c\right ) - \sin \left (d x + c\right )}{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 = 7.73582, size = 100, normalized size = 5. \begin{align*} \begin{cases} - \frac{d x \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - a d} + \frac{d x}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - a d} - \frac{2 \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - a d} & \text{for}\: d \neq 0 \\\frac{x \sin ^{2}{\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 = 1.17535, size = 35, normalized size = 1.75 \begin{align*} -\frac{\frac{d x + c}{a} - \frac{\tan \left (d x + c\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]