Optimal. Leaf size=47 \[ -\frac{\log (a+b x)}{b}+\frac{\log \left ((a+b x)^2+1\right )}{2 b}-\frac{\cot ^{-1}(a+b x)}{b (a+b x)} \]
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Rubi [A] time = 0.0316369, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5044, 4853, 266, 36, 29, 31} \[ -\frac{\log (a+b x)}{b}+\frac{\log \left ((a+b x)^2+1\right )}{2 b}-\frac{\cot ^{-1}(a+b x)}{b (a+b x)} \]
Antiderivative was successfully verified.
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Rule 5044
Rule 4853
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}(a+b x)}{(a+b x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)}{x^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\cot ^{-1}(a+b x)}{b (a+b x)}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (1+x^2\right )} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\cot ^{-1}(a+b x)}{b (a+b x)}-\frac{\operatorname{Subst}\left (\int \frac{1}{x (1+x)} \, dx,x,(a+b x)^2\right )}{2 b}\\ &=-\frac{\cot ^{-1}(a+b x)}{b (a+b x)}-\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,(a+b x)^2\right )}{2 b}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,(a+b x)^2\right )}{2 b}\\ &=-\frac{\cot ^{-1}(a+b x)}{b (a+b x)}-\frac{\log (a+b x)}{b}+\frac{\log \left (1+(a+b x)^2\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0174963, size = 40, normalized size = 0.85 \[ -\frac{\log (a+b x)-\frac{1}{2} \log \left ((a+b x)^2+1\right )+\frac{\cot ^{-1}(a+b x)}{a+b x}}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 46, normalized size = 1. \begin{align*} -{\frac{{\rm arccot} \left (bx+a\right )}{b \left ( bx+a \right ) }}-{\frac{\ln \left ( bx+a \right ) }{b}}+{\frac{\ln \left ( 1+ \left ( bx+a \right ) ^{2} \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00666, size = 72, normalized size = 1.53 \begin{align*} \frac{\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b} - \frac{\log \left (b x + a\right )}{b} - \frac{\operatorname{arccot}\left (b x + a\right )}{{\left (b x + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22231, size = 150, normalized size = 3.19 \begin{align*} \frac{{\left (b x + a\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \,{\left (b x + a\right )} \log \left (b x + a\right ) - 2 \, \operatorname{arccot}\left (b x + a\right )}{2 \,{\left (b^{2} x + a b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.96155, size = 150, normalized size = 3.19 \begin{align*} \begin{cases} - \frac{2 a \log{\left (\frac{a}{b} + x \right )}}{2 a b + 2 b^{2} x} + \frac{a \log{\left (\frac{a^{2}}{b^{2}} + \frac{2 a x}{b} + x^{2} + \frac{1}{b^{2}} \right )}}{2 a b + 2 b^{2} x} - \frac{2 b x \log{\left (\frac{a}{b} + x \right )}}{2 a b + 2 b^{2} x} + \frac{b x \log{\left (\frac{a^{2}}{b^{2}} + \frac{2 a x}{b} + x^{2} + \frac{1}{b^{2}} \right )}}{2 a b + 2 b^{2} x} - \frac{2 \operatorname{acot}{\left (a + b x \right )}}{2 a b + 2 b^{2} x} & \text{for}\: b \neq 0 \\\frac{x \operatorname{acot}{\left (a \right )}}{a^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10879, size = 49, normalized size = 1.04 \begin{align*} \frac{\log \left (\frac{1}{{\left (b x + a\right )}^{2}} + 1\right )}{2 \, b} - \frac{\arctan \left (\frac{1}{b x + a}\right )}{{\left (b x + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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