Optimal. Leaf size=24 \[ \frac{\tan (c+d x)}{d \sqrt{a \sec ^2(c+d x)}} \]
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Rubi [A] time = 0.0292873, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3657, 4122, 191} \[ \frac{\tan (c+d x)}{d \sqrt{a \sec ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4122
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+a \tan ^2(c+d x)}} \, dx &=\int \frac{1}{\sqrt{a \sec ^2(c+d x)}} \, dx\\ &=\frac{a \operatorname{Subst}\left (\int \frac{1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\tan (c+d x)}{d \sqrt{a \sec ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0407648, size = 24, normalized size = 1. \[ \frac{\tan (c+d x)}{d \sqrt{a \sec ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 25, normalized size = 1. \begin{align*}{\frac{\tan \left ( dx+c \right ) }{d}{\frac{1}{\sqrt{a+a \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.80395, size = 18, normalized size = 0.75 \begin{align*} \frac{\sin \left (d x + c\right )}{\sqrt{a} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59211, size = 92, normalized size = 3.83 \begin{align*} \frac{\sqrt{a \tan \left (d x + c\right )^{2} + a} \tan \left (d x + c\right )}{a d \tan \left (d x + c\right )^{2} + a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \tan ^{2}{\left (c + d x \right )} + a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \tan \left (d x + c\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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