Optimal. Leaf size=56 \[ -\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right )}{\sqrt{c}} \]
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Rubi [A] time = 0.0926184, antiderivative size = 56, 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 = {4944, 266, 63, 208} \[ -\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 4944
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)}{x^2 \sqrt{c+a^2 c x^2}} \, dx &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{c x}+a \int \frac{1}{x \sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{c x}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{c x}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a^2}+\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c+a^2 c x^2}\right )}{a c}\\ &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{c x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{c+a^2 c x^2}}{\sqrt{c}}\right )}{\sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0921398, size = 62, normalized size = 1.11 \[ \frac{a \left (\log (x)-\log \left (\sqrt{c} \sqrt{a^2 c x^2+c}+c\right )\right )}{\sqrt{c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.369, size = 139, normalized size = 2.5 \begin{align*} -{\frac{\arctan \left ( ax \right ) }{cx}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{a}{c}\ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{a}{c}\ln \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-1 \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \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 = 2.39954, size = 163, normalized size = 2.91 \begin{align*} \frac{a \sqrt{c} x \log \left (-\frac{a^{2} c x^{2} - 2 \, \sqrt{a^{2} c x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) - 2 \, \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )}{2 \, c x} \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{\operatorname{atan}{\left (a x \right )}}{x^{2} \sqrt{c \left (a^{2} x^{2} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23342, size = 136, normalized size = 2.43 \begin{align*}{\left (\frac{a \arctan \left (x{\left | a \right |}\right )}{\sqrt{c}{\left | a \right |}} - \frac{2 \, \arctan \left (-\frac{\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}}\right )}{\left | a \right |} + \frac{2 \, \sqrt{c}{\left | a \right |} \arctan \left (a x\right )}{{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} + c}\right )}^{2} - c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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