Optimal. Leaf size=60 \[ -\frac{\left (1-a^2\right ) \tan ^{-1}(a+b x)}{2 b^2}-\frac{a \log \left ((a+b x)^2+1\right )}{2 b^2}+\frac{1}{2} x^2 \cot ^{-1}(a+b x)+\frac{x}{2 b} \]
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Rubi [A] time = 0.0553805, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {5048, 4863, 702, 635, 203, 260} \[ -\frac{\left (1-a^2\right ) \tan ^{-1}(a+b x)}{2 b^2}-\frac{a \log \left ((a+b x)^2+1\right )}{2 b^2}+\frac{1}{2} x^2 \cot ^{-1}(a+b x)+\frac{x}{2 b} \]
Antiderivative was successfully verified.
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Rule 5048
Rule 4863
Rule 702
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int x \cot ^{-1}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b}+\frac{x}{b}\right ) \cot ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{1}{2} x^2 \cot ^{-1}(a+b x)+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{b}+\frac{x}{b}\right )^2}{1+x^2} \, dx,x,a+b x\right )\\ &=\frac{1}{2} x^2 \cot ^{-1}(a+b x)+\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{b^2}-\frac{1-a^2+2 a x}{b^2 \left (1+x^2\right )}\right ) \, dx,x,a+b x\right )\\ &=\frac{x}{2 b}+\frac{1}{2} x^2 \cot ^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int \frac{1-a^2+2 a x}{1+x^2} \, dx,x,a+b x\right )}{2 b^2}\\ &=\frac{x}{2 b}+\frac{1}{2} x^2 \cot ^{-1}(a+b x)-\frac{a \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,a+b x\right )}{b^2}-\frac{\left (1-a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,a+b x\right )}{2 b^2}\\ &=\frac{x}{2 b}+\frac{1}{2} x^2 \cot ^{-1}(a+b x)-\frac{\left (1-a^2\right ) \tan ^{-1}(a+b x)}{2 b^2}-\frac{a \log \left (1+(a+b x)^2\right )}{2 b^2}\\ \end{align*}
Mathematica [C] time = 0.0328707, size = 90, normalized size = 1.5 \[ \frac{i a^2 \log (a+b x+i)+2 b^2 x^2 \cot ^{-1}(a+b x)-2 a \log (a+b x+i)-i (a-i)^2 \log (-a-b x+i)-i \log (a+b x+i)+2 b x}{4 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 66, normalized size = 1.1 \begin{align*}{\frac{{x}^{2}{\rm arccot} \left (bx+a\right )}{2}}-{\frac{{\rm arccot} \left (bx+a\right ){a}^{2}}{2\,{b}^{2}}}+{\frac{x}{2\,b}}+{\frac{a}{2\,{b}^{2}}}-{\frac{a\ln \left ( 1+ \left ( bx+a \right ) ^{2} \right ) }{2\,{b}^{2}}}-{\frac{\arctan \left ( bx+a \right ) }{2\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47358, size = 92, normalized size = 1.53 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{arccot}\left (b x + a\right ) + \frac{1}{2} \, b{\left (\frac{x}{b^{2}} + \frac{{\left (a^{2} - 1\right )} \arctan \left (\frac{b^{2} x + a b}{b}\right )}{b^{3}} - \frac{a \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2728, size = 143, normalized size = 2.38 \begin{align*} \frac{b^{2} x^{2} \operatorname{arccot}\left (b x + a\right ) + b x +{\left (a^{2} - 1\right )} \arctan \left (b x + a\right ) - a \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.833307, size = 78, normalized size = 1.3 \begin{align*} \begin{cases} - \frac{a^{2} \operatorname{acot}{\left (a + b x \right )}}{2 b^{2}} - \frac{a \log{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 b^{2}} + \frac{x^{2} \operatorname{acot}{\left (a + b x \right )}}{2} + \frac{x}{2 b} + \frac{\operatorname{acot}{\left (a + b x \right )}}{2 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \operatorname{acot}{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09278, size = 84, normalized size = 1.4 \begin{align*} \frac{1}{2} \, x^{2} \arctan \left (\frac{1}{b x + a}\right ) + \frac{1}{2} \, b{\left (\frac{x}{b^{2}} + \frac{{\left (a^{2} - 1\right )} \arctan \left (b x + a\right )}{b^{3}} - \frac{a \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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