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 = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 260
Rule 4847
Rule 5040
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.59, size = 43, normalized size = 1.30 \[ \frac {2 \, b x \operatorname {arccot}\left (b x + a\right ) - 2 \, a \arctan \left (b x + a\right ) + \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 111, normalized size = 3.36 \[ -\frac {\arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) - \arctan \left (\frac {1}{b x + a}\right )}{2 \, b \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 36, normalized size = 1.09 \[ x \,\mathrm {arccot}\left (b x +a \right )+\frac {\mathrm {arccot}\left (b x +a \right ) a}{b}+\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 29, normalized size = 0.88 \[ \frac {2 \, {\left (b x + a\right )} \operatorname {arccot}\left (b x + a\right ) + \log \left ({\left (b x + a\right )}^{2} + 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 42, normalized size = 1.27 \[ \frac {\frac {\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}{2}+a\,\mathrm {acot}\left (a+b\,x\right )}{b}+x\,\mathrm {acot}\left (a+b\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 46, normalized size = 1.39 \[ \begin {cases} \frac {a \operatorname {acot}{\left (a + b x \right )}}{b} + x \operatorname {acot}{\left (a + b x \right )} + \frac {\log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 b} & \text {for}\: b \neq 0 \\x \operatorname {acot}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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