Optimal. Leaf size=35 \[ -\frac {a+b \tan ^{-1}(c x)}{x}-\frac {1}{2} b c \log \left (c^2 x^2+1\right )+b c \log (x) \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4852, 266, 36, 29, 31} \[ -\frac {a+b \tan ^{-1}(c x)}{x}-\frac {1}{2} b c \log \left (c^2 x^2+1\right )+b c \log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 4852
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx &=-\frac {a+b \tan ^{-1}(c x)}{x}+(b c) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {a+b \tan ^{-1}(c x)}{x}+\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a+b \tan ^{-1}(c x)}{x}+\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac {a+b \tan ^{-1}(c x)}{x}+b c \log (x)-\frac {1}{2} b c \log \left (1+c^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 38, normalized size = 1.09 \[ -\frac {a}{x}-\frac {1}{2} b c \log \left (c^2 x^2+1\right )+b c \log (x)-\frac {b \tan ^{-1}(c x)}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 37, normalized size = 1.06 \[ -\frac {b c x \log \left (c^{2} x^{2} + 1\right ) - 2 \, b c x \log \relax (x) + 2 \, b \arctan \left (c x\right ) + 2 \, a}{2 \, x} \]
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 = 0.01, size = 39, normalized size = 1.11 \[ -\frac {a}{x}-\frac {b \arctan \left (c x \right )}{x}+c b \ln \left (c x \right )-\frac {b c \ln \left (c^{2} x^{2}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 39, normalized size = 1.11 \[ -\frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b - \frac {a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 36, normalized size = 1.03 \[ b\,c\,\ln \relax (x)-\frac {a}{x}-\frac {b\,\mathrm {atan}\left (c\,x\right )}{x}-\frac {b\,c\,\ln \left (c^2\,x^2+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.63, size = 37, normalized size = 1.06 \[ \begin {cases} - \frac {a}{x} + b c \log {\relax (x )} - \frac {b c \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2} - \frac {b \operatorname {atan}{\left (c x \right )}}{x} & \text {for}\: c \neq 0 \\- \frac {a}{x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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