Optimal. Leaf size=40 \[ -a c \log \left (a^2 x^2+1\right )+a^2 c x \tan ^{-1}(a x)+a c \log (x)-\frac {c \tan ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4950, 4852, 266, 36, 29, 31, 4846, 260} \[ -a c \log \left (a^2 x^2+1\right )+a^2 c x \tan ^{-1}(a x)+a c \log (x)-\frac {c \tan ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 260
Rule 266
Rule 4846
Rule 4852
Rule 4950
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)}{x^2} \, dx &=c \int \frac {\tan ^{-1}(a x)}{x^2} \, dx+\left (a^2 c\right ) \int \tan ^{-1}(a x) \, dx\\ &=-\frac {c \tan ^{-1}(a x)}{x}+a^2 c x \tan ^{-1}(a x)+(a c) \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx-\left (a^3 c\right ) \int \frac {x}{1+a^2 x^2} \, dx\\ &=-\frac {c \tan ^{-1}(a x)}{x}+a^2 c x \tan ^{-1}(a x)-\frac {1}{2} a c \log \left (1+a^2 x^2\right )+\frac {1}{2} (a c) \operatorname {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {c \tan ^{-1}(a x)}{x}+a^2 c x \tan ^{-1}(a x)-\frac {1}{2} a c \log \left (1+a^2 x^2\right )+\frac {1}{2} (a c) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (a^3 c\right ) \operatorname {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac {c \tan ^{-1}(a x)}{x}+a^2 c x \tan ^{-1}(a x)+a c \log (x)-a c \log \left (1+a^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 40, normalized size = 1.00 \[ -a c \log \left (a^2 x^2+1\right )+a^2 c x \tan ^{-1}(a x)+a c \log (x)-\frac {c \tan ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 45, normalized size = 1.12 \[ -\frac {a c x \log \left (a^{2} x^{2} + 1\right ) - a c x \log \relax (x) - {\left (a^{2} c x^{2} - c\right )} \arctan \left (a x\right )}{x} \]
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 [A] time = 0.03, size = 43, normalized size = 1.08 \[ a^{2} c x \arctan \left (a x \right )-\frac {c \arctan \left (a x \right )}{x}-a c \ln \left (a^{2} x^{2}+1\right )+a c \ln \left (a x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 40, normalized size = 1.00 \[ -{\left (c \log \left (a^{2} x^{2} + 1\right ) - c \log \relax (x)\right )} a + {\left (a^{2} c x - \frac {c}{x}\right )} \arctan \left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 42, normalized size = 1.05 \[ a^2\,c\,x\,\mathrm {atan}\left (a\,x\right )-\frac {c\,\mathrm {atan}\left (a\,x\right )}{x}-c\,\left (a\,\ln \left (a^2\,x^2+1\right )-a\,\ln \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.75, size = 41, normalized size = 1.02 \[ \begin {cases} a^{2} c x \operatorname {atan}{\left (a x \right )} + a c \log {\relax (x )} - a c \log {\left (x^{2} + \frac {1}{a^{2}} \right )} - \frac {c \operatorname {atan}{\left (a x \right )}}{x} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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