Integrand size = 10, antiderivative size = 62 \[ \int \frac {\cot ^{-1}(a+b x)}{x^2} \, dx=-\frac {\cot ^{-1}(a+b x)}{x}+\frac {a b \arctan (a+b x)}{1+a^2}-\frac {b \log (x)}{1+a^2}+\frac {b \log \left (1+(a+b x)^2\right )}{2 \left (1+a^2\right )} \]
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Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5154, 378, 720, 31, 649, 209, 266} \[ \int \frac {\cot ^{-1}(a+b x)}{x^2} \, dx=\frac {a b \arctan (a+b x)}{a^2+1}-\frac {b \log (x)}{a^2+1}+\frac {b \log \left ((a+b x)^2+1\right )}{2 \left (a^2+1\right )}-\frac {\cot ^{-1}(a+b x)}{x} \]
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Rule 31
Rule 209
Rule 266
Rule 378
Rule 649
Rule 720
Rule 5154
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^{-1}(a+b x)}{x}-b \int \frac {1}{x \left (1+(a+b x)^2\right )} \, dx \\ & = -\frac {\cot ^{-1}(a+b x)}{x}-b \text {Subst}\left (\int \frac {1}{(-a+x) \left (1+x^2\right )} \, dx,x,a+b x\right ) \\ & = -\frac {\cot ^{-1}(a+b x)}{x}-\frac {b \text {Subst}\left (\int \frac {1}{-a+x} \, dx,x,a+b x\right )}{1+a^2}-\frac {b \text {Subst}\left (\int \frac {-a-x}{1+x^2} \, dx,x,a+b x\right )}{1+a^2} \\ & = -\frac {\cot ^{-1}(a+b x)}{x}-\frac {b \log (x)}{1+a^2}+\frac {b \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,a+b x\right )}{1+a^2}+\frac {(a b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,a+b x\right )}{1+a^2} \\ & = -\frac {\cot ^{-1}(a+b x)}{x}+\frac {a b \arctan (a+b x)}{1+a^2}-\frac {b \log (x)}{1+a^2}+\frac {b \log \left (1+(a+b x)^2\right )}{2 \left (1+a^2\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.06 \[ \int \frac {\cot ^{-1}(a+b x)}{x^2} \, dx=-\frac {\cot ^{-1}(a+b x)}{x}+\frac {b (-2 \log (x)+(1-i a) \log (i-a-b x)+(1+i a) \log (i+a+b x))}{2 \left (1+a^2\right )} \]
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Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(b \left (-\frac {\operatorname {arccot}\left (b x +a \right )}{b x}-\frac {\ln \left (-b x \right )}{a^{2}+1}+\frac {\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2}+a \arctan \left (b x +a \right )}{a^{2}+1}\right )\) | \(61\) |
default | \(b \left (-\frac {\operatorname {arccot}\left (b x +a \right )}{b x}-\frac {\ln \left (-b x \right )}{a^{2}+1}+\frac {\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2}+a \arctan \left (b x +a \right )}{a^{2}+1}\right )\) | \(61\) |
parts | \(-\frac {\operatorname {arccot}\left (b x +a \right )}{x}-b \left (\frac {\ln \left (x \right )}{a^{2}+1}-\frac {b \left (\frac {\ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b}+\frac {a \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right )}{b}\right )}{a^{2}+1}\right )\) | \(83\) |
parallelrisch | \(-\frac {2 x \,\operatorname {arccot}\left (b x +a \right ) a^{2} b^{2}+2 b^{2} \ln \left (x \right ) a x -b^{2} \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a x +2 \,\operatorname {arccot}\left (b x +a \right ) a^{3} b +2 \,\operatorname {arccot}\left (b x +a \right ) a b}{2 x a b \left (a^{2}+1\right )}\) | \(91\) |
risch | \(-\frac {i \ln \left (1+i \left (b x +a \right )\right )}{2 x}-\frac {-i a^{2} \ln \left (1-i \left (b x +a \right )\right )-i \ln \left (1-i \left (b x +a \right )\right )+\pi \,a^{2}+\pi +2 b \ln \left (x \right ) x -x b \ln \left (\left (i a b +3 b \right ) x +i a^{2}+3 i+2 a \right )-i x b \ln \left (\left (i a b +3 b \right ) x +i a^{2}+3 i+2 a \right ) a -x b \ln \left (\left (i a b -3 b \right ) x +i a^{2}+3 i-2 a \right )+i x b \ln \left (\left (i a b -3 b \right ) x +i a^{2}+3 i-2 a \right ) a}{2 x \left (i+a \right ) \left (a -i\right )}\) | \(196\) |
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Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.03 \[ \int \frac {\cot ^{-1}(a+b x)}{x^2} \, dx=\frac {2 \, a b x \arctan \left (b x + a\right ) + b x \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \, b x \log \left (x\right ) - 2 \, {\left (a^{2} + 1\right )} \operatorname {arccot}\left (b x + a\right )}{2 \, {\left (a^{2} + 1\right )} x} \]
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Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.69 \[ \int \frac {\cot ^{-1}(a+b x)}{x^2} \, dx=\begin {cases} - \frac {i b \operatorname {acot}{\left (b x - i \right )}}{2} - \frac {\operatorname {acot}{\left (b x - i \right )}}{x} + \frac {i}{2 x} & \text {for}\: a = - i \\\frac {i b \operatorname {acot}{\left (b x + i \right )}}{2} - \frac {\operatorname {acot}{\left (b x + i \right )}}{x} - \frac {i}{2 x} & \text {for}\: a = i \\- \frac {2 a^{2} \operatorname {acot}{\left (a + b x \right )}}{2 a^{2} x + 2 x} - \frac {2 a b x \operatorname {acot}{\left (a + b x \right )}}{2 a^{2} x + 2 x} - \frac {2 b x \log {\left (x \right )}}{2 a^{2} x + 2 x} + \frac {b x \log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 a^{2} x + 2 x} - \frac {2 \operatorname {acot}{\left (a + b x \right )}}{2 a^{2} x + 2 x} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.24 \[ \int \frac {\cot ^{-1}(a+b x)}{x^2} \, dx=\frac {1}{2} \, b {\left (\frac {2 \, a \arctan \left (\frac {b^{2} x + a b}{b}\right )}{a^{2} + 1} + \frac {\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{2} + 1} - \frac {2 \, \log \left (x\right )}{a^{2} + 1}\right )} - \frac {\operatorname {arccot}\left (b x + a\right )}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (60) = 120\).
Time = 0.42 (sec) , antiderivative size = 498, normalized size of antiderivative = 8.03 \[ \int \frac {\cot ^{-1}(a+b x)}{x^2} \, dx=-\frac {{\left (2 \, a \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 2 \, a \log \left (\frac {4 \, {\left (4 \, a^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 4 \, a \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 )^{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 ) + \log \left (\frac {4 \, {\left (4 \, a^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 4 \, a \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 )^{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} - 2 \, a \arctan \left (\frac {1}{b x + a}\right ) - 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 (4 \, a^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 4 \, a \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 )^{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 )\right )} b}{2 \, {\left (2 \, a^{3} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) + a^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - a^{2} + 2 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) + \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - 1\right )}} \]
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Time = 1.42 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(a+b x)}{x^2} \, dx=-\frac {\mathrm {acot}\left (a+b\,x\right )}{x}-\frac {b\,x\,\ln \left (x\right )-\frac {b\,x\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}{2}+a\,b\,x\,\mathrm {acot}\left (a+b\,x\right )}{x\,\left (a^2+1\right )} \]
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