Optimal. Leaf size=13 \[ \frac{\tan (c+d x)}{a d} \]
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Rubi [A] time = 0.0224128, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3175, 3767, 8} \[ \frac{\tan (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{a-a \sin ^2(c+d x)} \, dx &=\frac{\int \sec ^2(c+d x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{a d}\\ &=\frac{\tan (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.00612, size = 13, normalized size = 1. \[ \frac{\tan (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 14, normalized size = 1.1 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.947621, size = 18, normalized size = 1.38 \begin{align*} \frac{\tan \left (d x + c\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59973, size = 45, normalized size = 3.46 \begin{align*} \frac{\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.
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Sympy [A] time = 2.67555, size = 41, normalized size = 3.15 \begin{align*} \begin{cases} - \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}{- a \sin ^{2}{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12077, size = 18, normalized size = 1.38 \begin{align*} \frac{\tan \left (d x + c\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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