Optimal. Leaf size=36 \[ \frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec ^2(c+d x)}}\right )}{d} \]
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Rubi [A] time = 0.0349328, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3657, 4122, 217, 206} \[ \frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec ^2(c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4122
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \sqrt{a+a \tan ^2(c+d x)} \, dx &=\int \sqrt{a \sec ^2(c+d x)} \, dx\\ &=\frac{a \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+a x^2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\tan (c+d x)}{\sqrt{a \sec ^2(c+d x)}}\right )}{d}\\ &=\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec ^2(c+d x)}}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.0314185, size = 31, normalized size = 0.86 \[ \frac{\cos (c+d x) \sqrt{a \sec ^2(c+d x)} \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 34, normalized size = 0.9 \begin{align*}{\frac{1}{d}\sqrt{a}\ln \left ( \sqrt{a}\tan \left ( dx+c \right ) +\sqrt{a+a \left ( \tan \left ( dx+c \right ) \right ) ^{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.6991, size = 88, normalized size = 2.44 \begin{align*} \frac{\sqrt{a}{\left (\log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37526, size = 234, normalized size = 6.5 \begin{align*} \left [\frac{\sqrt{a} \log \left (2 \, a \tan \left (d x + c\right )^{2} + 2 \, \sqrt{a \tan \left (d x + c\right )^{2} + a} \sqrt{a} \tan \left (d x + c\right ) + a\right )}{2 \, d}, -\frac{\sqrt{-a} \arctan \left (\frac{\sqrt{a \tan \left (d x + c\right )^{2} + a} \sqrt{-a}}{a \tan \left (d x + c\right )}\right )}{d}\right ] \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 \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 [B] time = 1.43536, size = 89, normalized size = 2.47 \begin{align*} -\frac{{\left (\log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1\right ) - \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1\right )\right )} \sqrt{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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