Optimal. Leaf size=66 \[ \frac{x^2}{2 a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} \left (c+d x^2\right )\right )}{\sqrt{a+b}}\right )}{a d \sqrt{a-b} \sqrt{a+b}} \]
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Rubi [A] time = 0.108982, antiderivative size = 66, 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 = {4204, 3783, 2659, 208} \[ \frac{x^2}{2 a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} \left (c+d x^2\right )\right )}{\sqrt{a+b}}\right )}{a d \sqrt{a-b} \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 4204
Rule 3783
Rule 2659
Rule 208
Rubi steps
\begin{align*} \int \frac{x}{a+b \sec \left (c+d x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{a+b \sec (c+d x)} \, dx,x,x^2\right )\\ &=\frac{x^2}{2 a}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx,x,x^2\right )}{2 a}\\ &=\frac{x^2}{2 a}-\frac{\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 x^2\right )\right )\right )}{a d}\\ &=\frac{x^2}{2 a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} \left (c+d x^2\right )\right )}{\sqrt{a+b}}\right )}{a \sqrt{a-b} \sqrt{a+b} d}\\ \end{align*}
Mathematica [A] time = 0.153877, size = 67, normalized size = 1.02 \[ \frac{\frac{2 b \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} \left (c+d x^2\right )\right )}{\sqrt{a^2-b^2}}\right )}{d \sqrt{a^2-b^2}}+\frac{c}{d}+x^2}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 70, normalized size = 1.1 \begin{align*}{\frac{1}{da}\arctan \left ( \tan \left ({\frac{d{x}^{2}}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{b}{da}{\it Artanh} \left ({(a-b)\tan \left ({\frac{d{x}^{2}}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76205, size = 535, normalized size = 8.11 \begin{align*} \left [\frac{2 \,{\left (a^{2} - b^{2}\right )} d x^{2} + \sqrt{a^{2} - b^{2}} b \log \left (\frac{2 \, a b \cos \left (d x^{2} + c\right ) -{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x^{2} + c\right )^{2} - 2 \, \sqrt{a^{2} - b^{2}}{\left (b \cos \left (d x^{2} + c\right ) + a\right )} \sin \left (d x^{2} + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x^{2} + c\right )^{2} + 2 \, a b \cos \left (d x^{2} + c\right ) + b^{2}}\right )}{4 \,{\left (a^{3} - a b^{2}\right )} d}, \frac{{\left (a^{2} - b^{2}\right )} d x^{2} - \sqrt{-a^{2} + b^{2}} b \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \cos \left (d x^{2} + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x^{2} + c\right )}\right )}{2 \,{\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{x}{a + b \sec{\left (c + d x^{2} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25145, size = 140, normalized size = 2.12 \begin{align*} -\frac{{\left (\pi \left \lfloor \frac{d x^{2} + 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 x^{2} + \frac{1}{2} \, c\right ) - b \tan \left (\frac{1}{2} \, d x^{2} + \frac{1}{2} \, c\right )}{\sqrt{-a^{2} + b^{2}}}\right )\right )} b}{\sqrt{-a^{2} + b^{2}} a d} + \frac{d x^{2} + c}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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