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 = 0.09, antiderivative size = 56, normalized size of antiderivative = 1.00, 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 63
Rule 208
Rule 266
Rule 4944
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*}
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Mathematica [A] time = 0.10, 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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fricas [A] time = 0.60, size = 68, normalized size = 1.21 \[ \frac {a \sqrt {c} x \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {a^{2} c x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )}{2 \, c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.58, size = 139, normalized size = 2.48 \[ -\frac {\arctan \left (a x \right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{c x}+\frac {a \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c}-\frac {a \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 36, normalized size = 0.64 \[ -\frac {a \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) + \frac {\sqrt {a^{2} x^{2} + 1} \arctan \left (a x\right )}{x}}{\sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {atan}\left (a\,x\right )}{x^2\,\sqrt {c\,a^2\,x^2+c}} \,d x \]
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 \[ \int \frac {\operatorname {atan}{\left (a x \right )}}{x^{2} \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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