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 = 0.06, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, 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 203
Rule 260
Rule 635
Rule 702
Rule 4863
Rule 5048
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*}
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Mathematica [C] time = 0.04, size = 90, normalized size = 1.50 \[ \frac {i a^2 \log (a+b x+i)+2 b^2 x^2 \cot ^{-1}(a+b x)-2 a \log (a+b x+i)-i (a-i)^2 \log (-a-b x+i)-i \log (a+b x+i)+2 b x}{4 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 55, normalized size = 0.92 \[ \frac {b^{2} x^{2} \operatorname {arccot}\left (b x + a\right ) + b x + {\left (a^{2} - 1\right )} \arctan \left (b x + a\right ) - a \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 210, normalized size = 3.50 \[ \frac {4 \, a \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 4 \, a \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - 4 \, a \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) + 2 \, \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + \arctan \left (\frac {1}{b x + a}\right ) + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )}{8 \, b^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 66, normalized size = 1.10 \[ \frac {x^{2} \mathrm {arccot}\left (b x +a \right )}{2}-\frac {\mathrm {arccot}\left (b x +a \right ) a^{2}}{2 b^{2}}+\frac {x}{2 b}+\frac {a}{2 b^{2}}-\frac {a \ln \left (1+\left (b x +a \right )^{2}\right )}{2 b^{2}}-\frac {\arctan \left (b x +a \right )}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 68, normalized size = 1.13 \[ \frac {1}{2} \, x^{2} \operatorname {arccot}\left (b x + a\right ) + \frac {1}{2} \, b {\left (\frac {x}{b^{2}} + \frac {{\left (a^{2} - 1\right )} \arctan \left (\frac {b^{2} x + a b}{b}\right )}{b^{3}} - \frac {a \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 61, normalized size = 1.02 \[ \frac {x^2\,\mathrm {acot}\left (a+b\,x\right )}{2}+\frac {\frac {\mathrm {acot}\left (a+b\,x\right )}{2}+\frac {b\,x}{2}-\frac {a^2\,\mathrm {acot}\left (a+b\,x\right )}{2}-\frac {a\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}{2}}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 78, normalized size = 1.30 \[ \begin {cases} - \frac {a^{2} \operatorname {acot}{\left (a + b x \right )}}{2 b^{2}} - \frac {a \log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 b^{2}} + \frac {x^{2} \operatorname {acot}{\left (a + b x \right )}}{2} + \frac {x}{2 b} + \frac {\operatorname {acot}{\left (a + b x \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {acot}{\relax (a )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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