Optimal. Leaf size=47 \[ -\frac {\log (a+b x)}{b}+\frac {\log \left ((a+b x)^2+1\right )}{2 b}-\frac {\cot ^{-1}(a+b x)}{b (a+b x)} \]
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Rubi [A] time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5044, 4853, 266, 36, 29, 31} \[ -\frac {\log (a+b x)}{b}+\frac {\log \left ((a+b x)^2+1\right )}{2 b}-\frac {\cot ^{-1}(a+b x)}{b (a+b x)} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 4853
Rule 5044
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a+b x)}{(a+b x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cot ^{-1}(x)}{x^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\cot ^{-1}(a+b x)}{b (a+b x)}-\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (1+x^2\right )} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\cot ^{-1}(a+b x)}{b (a+b x)}-\frac {\operatorname {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,(a+b x)^2\right )}{2 b}\\ &=-\frac {\cot ^{-1}(a+b x)}{b (a+b x)}-\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,(a+b x)^2\right )}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,(a+b x)^2\right )}{2 b}\\ &=-\frac {\cot ^{-1}(a+b x)}{b (a+b x)}-\frac {\log (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 = 40, normalized size = 0.85 \[ -\frac {\log (a+b x)-\frac {1}{2} \log \left ((a+b x)^2+1\right )+\frac {\cot ^{-1}(a+b x)}{a+b x}}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 59, normalized size = 1.26 \[ \frac {{\left (b x + a\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \, {\left (b x + a\right )} \log \left (b x + a\right ) - 2 \, \operatorname {arccot}\left (b x + a\right )}{2 \, {\left (b^{2} x + a b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 238, normalized size = 5.06 \[ -\frac {\arctan \left (\frac {1}{b x + a}\right )^{2} - \frac {\arctan \left (\frac {1}{b x + a}\right )^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - \log \left (\frac {4 \, {\left (\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1\right )}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - \arctan \left (\frac {1}{b x + a}\right )^{2} + 4 \, \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) + \log \left (\frac {4 \, {\left (\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1\right )}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1}\right )}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - 1}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 46, normalized size = 0.98 \[ -\frac {\mathrm {arccot}\left (b x +a \right )}{b \left (b x +a \right )}-\frac {\ln \left (b x +a \right )}{b}+\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 53, normalized size = 1.13 \[ \frac {\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b} - \frac {\log \left (b x + a\right )}{b} - \frac {\operatorname {arccot}\left (b x + a\right )}{{\left (b x + a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.77, size = 57, normalized size = 1.21 \[ \frac {\ln \left (-a^2-2\,a\,b\,x-b^2\,x^2-1\right )}{2\,b}-\frac {\ln \left (a+b\,x\right )}{b}-\frac {\mathrm {acot}\left (a+b\,x\right )}{x\,b^2+a\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.14, size = 151, normalized size = 3.21 \[ \begin {cases} \frac {a \log {\left (\frac {a}{b} + x \right )}}{- a b - b^{2} x} - \frac {a \log {\left (\frac {a}{b} + x - \frac {i}{b} \right )}}{- a b - b^{2} x} - \frac {i a \operatorname {acot}{\left (a + b x \right )}}{- a b - b^{2} x} + \frac {b x \log {\left (\frac {a}{b} + x \right )}}{- a b - b^{2} x} - \frac {b x \log {\left (\frac {a}{b} + x - \frac {i}{b} \right )}}{- a b - b^{2} x} - \frac {i b x \operatorname {acot}{\left (a + b x \right )}}{- a b - b^{2} x} + \frac {\operatorname {acot}{\left (a + b x \right )}}{- a b - b^{2} x} & \text {for}\: b \neq 0 \\\frac {x \operatorname {acot}{\relax (a )}}{a^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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