Optimal. Leaf size=33 \[ \frac{\log \left ((a+b x)^2+1\right )}{2 b}+\frac{(a+b x) \cot ^{-1}(a+b x)}{b} \]
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Rubi [A] time = 0.0115419, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5040, 4847, 260} \[ \frac{\log \left ((a+b x)^2+1\right )}{2 b}+\frac{(a+b x) \cot ^{-1}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 5040
Rule 4847
Rule 260
Rubi steps
\begin{align*} \int \cot ^{-1}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \cot ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \cot ^{-1}(a+b x)}{b}+\frac{\operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \cot ^{-1}(a+b x)}{b}+\frac{\log \left (1+(a+b x)^2\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.013004, size = 44, normalized size = 1.33 \[ \frac{\log \left (a^2+2 a b x+b^2 x^2+1\right )-2 a \tan ^{-1}(a+b x)}{2 b}+x \cot ^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 36, normalized size = 1.1 \begin{align*} x{\rm arccot} \left (bx+a\right )+{\frac{{\rm arccot} \left (bx+a\right )a}{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 = 0.956345, size = 39, normalized size = 1.18 \begin{align*} \frac{2 \,{\left (b x + a\right )} \operatorname{arccot}\left (b x + a\right ) + \log \left ({\left (b x + a\right )}^{2} + 1\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14645, size = 119, normalized size = 3.61 \begin{align*} \frac{2 \, b x \operatorname{arccot}\left (b x + a\right ) - 2 \, a \arctan \left (b x + a\right ) + \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.487655, size = 46, normalized size = 1.39 \begin{align*} \begin{cases} \frac{a \operatorname{acot}{\left (a + b x \right )}}{b} + x \operatorname{acot}{\left (a + b x \right )} + \frac{\log{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 b} & \text{for}\: b \neq 0 \\x \operatorname{acot}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11836, size = 68, normalized size = 2.06 \begin{align*} -\frac{1}{2} \, b{\left (\frac{2 \, a \arctan \left (b x + a\right )}{b^{2}} - \frac{\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{2}}\right )} + x \arctan \left (\frac{1}{b x + a}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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