Optimal. Leaf size=33 \[ \frac {(a+b x) \cot ^{-1}(a+b x)}{b}+\frac {\log \left (1+(a+b x)^2\right )}{2 b} \]
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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 = {5148, 4931, 266}
\begin {gather*} \frac {\log \left ((a+b x)^2+1\right )}{2 b}+\frac {(a+b x) \cot ^{-1}(a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 4931
Rule 5148
Rubi steps
\begin {align*} \int \cot ^{-1}(a+b x) \, dx &=\frac {\text {Subst}\left (\int \cot ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \cot ^{-1}(a+b x)}{b}+\frac {\text {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 = 0.01, size = 44, normalized size = 1.33 \begin {gather*} x \cot ^{-1}(a+b x)+\frac {-2 a \text {ArcTan}(a+b x)+\log \left (1+a^2+2 a b x+b^2 x^2\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 30, normalized size = 0.91
method | result | size |
derivativedivides | \(\frac {\left (b x +a \right ) \mathrm {arccot}\left (b x +a \right )+\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2}}{b}\) | \(30\) |
default | \(\frac {\left (b x +a \right ) \mathrm {arccot}\left (b x +a \right )+\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2}}{b}\) | \(30\) |
risch | \(\frac {i x \ln \left (1+i \left (b x +a \right )\right )}{2}-\frac {i x \ln \left (1-i \left (b x +a \right )\right )}{2}+\frac {\pi x}{2}-\frac {a \arctan \left (b x +a \right )}{b}+\frac {\ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 29, normalized size = 0.88 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 6.18, size = 43, normalized size = 1.30 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 46, normalized size = 1.39 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 111 vs.
\(2 (31) = 62\).
time = 0.44, size = 111, normalized size = 3.36 \begin {gather*} -\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 )} \end {gather*}
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 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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