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 = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 206
Rule 217
Rule 3657
Rule 4122
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.43, size = 90, normalized size = 2.50 \[ \left [\frac {\sqrt {a} \log \left (2 \, a \tan \left (d x + c\right )^{2} + 2 \, \sqrt {a \tan \left (d x + c\right )^{2} + a} \sqrt {a} \tan \left (d x + c\right ) + a\right )}{2 \, d}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {a \tan \left (d x + c\right )^{2} + a} \sqrt {-a}}{a \tan \left (d x + c\right )}\right )}{d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.33, size = 66, normalized size = 1.83 \[ -\frac {{\left (\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1\right ) - \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1\right )\right )} \sqrt {a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 34, normalized size = 0.94 \[ \frac {\sqrt {a}\, \ln \left (\sqrt {a}\, \tan \left (d x +c \right )+\sqrt {a +a \left (\tan ^{2}\left (d x +c \right )\right )}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.14, size = 65, normalized size = 1.81 \[ \frac {\sqrt {a} {\left (\log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.86, size = 41, normalized size = 1.14 \[ \left \{\begin {array}{cl} 0 & \text {\ if\ \ }a=0\\ \frac {\sqrt {a}\,\ln \left (\sqrt {a}\,\mathrm {tan}\left (c+d\,x\right )+\sqrt {a\,{\mathrm {tan}\left (c+d\,x\right )}^2+a}\right )}{d} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \tan ^{2}{\left (c + d x \right )} + a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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