Optimal. Leaf size=68 \[ \frac{2 \sqrt{x}}{a}-\frac{4 b \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )}{\sqrt{a+b}}\right )}{a d \sqrt{a-b} \sqrt{a+b}} \]
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Rubi [A] time = 0.0822412, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4204, 3783, 2659, 208} \[ \frac{2 \sqrt{x}}{a}-\frac{4 b \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )}{\sqrt{a+b}}\right )}{a d \sqrt{a-b} \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 4204
Rule 3783
Rule 2659
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \left (a+b \sec \left (c+d \sqrt{x}\right )\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{a+b \sec (c+d x)} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \sqrt{x}}{a}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{2 \sqrt{x}}{a}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}+\left (1-\frac{a}{b}\right ) x^2} \, dx,x,\tan \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )\right )}{a d}\\ &=\frac{2 \sqrt{x}}{a}-\frac{4 b \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )}{\sqrt{a+b}}\right )}{a \sqrt{a-b} \sqrt{a+b} d}\\ \end{align*}
Mathematica [A] time = 0.21409, size = 69, normalized size = 1.01 \[ \frac{2 \left (\frac{2 b \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )}{\sqrt{a^2-b^2}}\right )}{d \sqrt{a^2-b^2}}+\frac{c}{d}+\sqrt{x}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 71, normalized size = 1. \begin{align*} 4\,{\frac{\arctan \left ( \tan \left ( c/2+1/2\,d\sqrt{x} \right ) \right ) }{da}}-4\,{\frac{b}{da\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tan \left ( c/2+1/2\,d\sqrt{x} \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8606, size = 625, normalized size = 9.19 \begin{align*} \left [\frac{2 \,{\left (a^{2} - b^{2}\right )} d \sqrt{x} + \sqrt{a^{2} - b^{2}} b \log \left (\frac{2 \, a b \cos \left (d \sqrt{x} + c\right ) -{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d \sqrt{x} + c\right )^{2} + 2 \, a^{2} - b^{2} - 2 \,{\left (\sqrt{a^{2} - b^{2}} b \cos \left (d \sqrt{x} + c\right ) + \sqrt{a^{2} - b^{2}} a\right )} \sin \left (d \sqrt{x} + c\right )}{a^{2} \cos \left (d \sqrt{x} + c\right )^{2} + 2 \, a b \cos \left (d \sqrt{x} + c\right ) + b^{2}}\right )}{{\left (a^{3} - a b^{2}\right )} d}, \frac{2 \,{\left ({\left (a^{2} - b^{2}\right )} d \sqrt{x} - \sqrt{-a^{2} + b^{2}} b \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}} b \cos \left (d \sqrt{x} + c\right ) + \sqrt{-a^{2} + b^{2}} a}{{\left (a^{2} - b^{2}\right )} \sin \left (d \sqrt{x} + c\right )}\right )\right )}}{{\left (a^{3} - a b^{2}\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 \frac{1}{\sqrt{x} \left (a + b \sec{\left (c + d \sqrt{x} \right )}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28732, size = 140, normalized size = 2.06 \begin{align*} -\frac{4 \,{\left (\pi \left \lfloor \frac{d \sqrt{x} + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac{a \tan \left (\frac{1}{2} \, d \sqrt{x} + \frac{1}{2} \, c\right ) - b \tan \left (\frac{1}{2} \, d \sqrt{x} + \frac{1}{2} \, c\right )}{\sqrt{-a^{2} + b^{2}}}\right )\right )} b}{\sqrt{-a^{2} + b^{2}} a d} + \frac{2 \,{\left (d \sqrt{x} + c\right )}}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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