Integrand size = 10, antiderivative size = 95 \[ \int \frac {\cot ^{-1}(a+b x)}{x^3} \, dx=\frac {b}{2 \left (1+a^2\right ) x}-\frac {\cot ^{-1}(a+b x)}{2 x^2}+\frac {\left (1-a^2\right ) b^2 \arctan (a+b x)}{2 \left (1+a^2\right )^2}+\frac {a b^2 \log (x)}{\left (1+a^2\right )^2}-\frac {a b^2 \log \left (1+(a+b x)^2\right )}{2 \left (1+a^2\right )^2} \]
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Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5154, 378, 724, 815, 649, 209, 266} \[ \int \frac {\cot ^{-1}(a+b x)}{x^3} \, dx=\frac {\left (1-a^2\right ) b^2 \arctan (a+b x)}{2 \left (a^2+1\right )^2}+\frac {a b^2 \log (x)}{\left (a^2+1\right )^2}-\frac {a b^2 \log \left ((a+b x)^2+1\right )}{2 \left (a^2+1\right )^2}+\frac {b}{2 \left (a^2+1\right ) x}-\frac {\cot ^{-1}(a+b x)}{2 x^2} \]
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Rule 209
Rule 266
Rule 378
Rule 649
Rule 724
Rule 815
Rule 5154
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^{-1}(a+b x)}{2 x^2}-\frac {1}{2} b \int \frac {1}{x^2 \left (1+(a+b x)^2\right )} \, dx \\ & = -\frac {\cot ^{-1}(a+b x)}{2 x^2}-\frac {1}{2} b^2 \text {Subst}\left (\int \frac {1}{(-a+x)^2 \left (1+x^2\right )} \, dx,x,a+b x\right ) \\ & = \frac {b}{2 \left (1+a^2\right ) x}-\frac {\cot ^{-1}(a+b x)}{2 x^2}-\frac {b^2 \text {Subst}\left (\int \frac {-a-x}{(-a+x) \left (1+x^2\right )} \, dx,x,a+b x\right )}{2 \left (1+a^2\right )} \\ & = \frac {b}{2 \left (1+a^2\right ) x}-\frac {\cot ^{-1}(a+b x)}{2 x^2}-\frac {b^2 \text {Subst}\left (\int \left (\frac {2 a}{\left (1+a^2\right ) (a-x)}+\frac {-1+a^2+2 a x}{\left (1+a^2\right ) \left (1+x^2\right )}\right ) \, dx,x,a+b x\right )}{2 \left (1+a^2\right )} \\ & = \frac {b}{2 \left (1+a^2\right ) x}-\frac {\cot ^{-1}(a+b x)}{2 x^2}+\frac {a b^2 \log (x)}{\left (1+a^2\right )^2}-\frac {b^2 \text {Subst}\left (\int \frac {-1+a^2+2 a x}{1+x^2} \, dx,x,a+b x\right )}{2 \left (1+a^2\right )^2} \\ & = \frac {b}{2 \left (1+a^2\right ) x}-\frac {\cot ^{-1}(a+b x)}{2 x^2}+\frac {a b^2 \log (x)}{\left (1+a^2\right )^2}-\frac {\left (a b^2\right ) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,a+b x\right )}{\left (1+a^2\right )^2}+\frac {\left (\left (1-a^2\right ) b^2\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,a+b x\right )}{2 \left (1+a^2\right )^2} \\ & = \frac {b}{2 \left (1+a^2\right ) x}-\frac {\cot ^{-1}(a+b x)}{2 x^2}+\frac {\left (1-a^2\right ) b^2 \arctan (a+b x)}{2 \left (1+a^2\right )^2}+\frac {a b^2 \log (x)}{\left (1+a^2\right )^2}-\frac {a b^2 \log \left (1+(a+b x)^2\right )}{2 \left (1+a^2\right )^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^{-1}(a+b x)}{x^3} \, dx=\frac {-2 \cot ^{-1}(a+b x)+\frac {b x \left (4 a b x \log (x)+i (i+a)^2 b x \log (i-a-b x)+(-i+a) (2 (i+a)+(-1-i a) b x \log (i+a+b x))\right )}{\left (1+a^2\right )^2}}{4 x^2} \]
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Time = 0.31 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(b^{2} \left (-\frac {\operatorname {arccot}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {a \ln \left (1+\left (b x +a \right )^{2}\right )+\left (a^{2}-1\right ) \arctan \left (b x +a \right )}{2 \left (a^{2}+1\right )^{2}}+\frac {1}{2 \left (a^{2}+1\right ) b x}+\frac {a \ln \left (-b x \right )}{\left (a^{2}+1\right )^{2}}\right )\) | \(83\) |
default | \(b^{2} \left (-\frac {\operatorname {arccot}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {a \ln \left (1+\left (b x +a \right )^{2}\right )+\left (a^{2}-1\right ) \arctan \left (b x +a \right )}{2 \left (a^{2}+1\right )^{2}}+\frac {1}{2 \left (a^{2}+1\right ) b x}+\frac {a \ln \left (-b x \right )}{\left (a^{2}+1\right )^{2}}\right )\) | \(83\) |
parts | \(-\frac {\operatorname {arccot}\left (b x +a \right )}{2 x^{2}}-\frac {b \left (-\frac {1}{\left (a^{2}+1\right ) x}-\frac {2 a b \ln \left (x \right )}{\left (a^{2}+1\right )^{2}}+\frac {b^{2} \left (\frac {a \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{b}+\frac {\left (a^{2}-1\right ) \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right )}{b}\right )}{\left (a^{2}+1\right )^{2}}\right )}{2}\) | \(103\) |
parallelrisch | \(\frac {x^{2} \operatorname {arccot}\left (b x +a \right ) a^{2} b^{2}+2 b^{2} a \ln \left (x \right ) x^{2}-b^{2} a \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) x^{2}-\operatorname {arccot}\left (b x +a \right ) b^{2} x^{2}-2 a \,b^{2} x^{2}-\operatorname {arccot}\left (b x +a \right ) a^{4}+a^{2} b x -2 \,\operatorname {arccot}\left (b x +a \right ) a^{2}+b x -\operatorname {arccot}\left (b x +a \right )}{2 x^{2} \left (a^{4}+2 a^{2}+1\right )}\) | \(135\) |
risch | \(-\frac {i \ln \left (1+i \left (b x +a \right )\right )}{4 x^{2}}+\frac {i a^{4} \ln \left (1-i \left (b x +a \right )\right )+2 i a^{2} \ln \left (1-i \left (b x +a \right )\right )+i \ln \left (1-i \left (b x +a \right )\right )+4 b^{2} a \ln \left (-x \right ) x^{2}-\pi \,a^{4}+2 a^{2} b x -2 \pi \,a^{2}+2 b x -\pi -2 x^{2} \ln \left (\left (-a^{4} b -4 i a^{3} b -10 a^{2} b +4 i a b -b \right ) x -a^{5}-14 a^{3}-3 i a^{4}+3 a +14 i a^{2}+i\right ) a \,b^{2}+i x^{2} \ln \left (\left (-a^{4} b -4 i a^{3} b -10 a^{2} b +4 i a b -b \right ) x -a^{5}-14 a^{3}-3 i a^{4}+3 a +14 i a^{2}+i\right ) a^{2} b^{2}-i x^{2} b^{2} \ln \left (\left (-a^{4} b -4 i a^{3} b -10 a^{2} b +4 i a b -b \right ) x -a^{5}-14 a^{3}-3 i a^{4}+3 a +14 i a^{2}+i\right )-2 x^{2} \ln \left (\left (-a^{4} b +4 i a^{3} b -10 a^{2} b -4 i a b -b \right ) x -a^{5}-14 a^{3}+3 i a^{4}+3 a -14 i a^{2}-i\right ) a \,b^{2}-i x^{2} \ln \left (\left (-a^{4} b +4 i a^{3} b -10 a^{2} b -4 i a b -b \right ) x -a^{5}-14 a^{3}+3 i a^{4}+3 a -14 i a^{2}-i\right ) a^{2} b^{2}+i x^{2} b^{2} \ln \left (\left (-a^{4} b +4 i a^{3} b -10 a^{2} b -4 i a b -b \right ) x -a^{5}-14 a^{3}+3 i a^{4}+3 a -14 i a^{2}-i\right )}{4 x^{2} \left (a -i\right )^{2} \left (i+a \right )^{2}}\) | \(538\) |
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Time = 0.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04 \[ \int \frac {\cot ^{-1}(a+b x)}{x^3} \, dx=-\frac {{\left (a^{2} - 1\right )} b^{2} x^{2} \arctan \left (b x + a\right ) + a b^{2} x^{2} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \, a b^{2} x^{2} \log \left (x\right ) - {\left (a^{2} + 1\right )} b x + {\left (a^{4} + 2 \, a^{2} + 1\right )} \operatorname {arccot}\left (b x + a\right )}{2 \, {\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2}} \]
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Result contains complex when optimal does not.
Time = 0.80 (sec) , antiderivative size = 381, normalized size of antiderivative = 4.01 \[ \int \frac {\cot ^{-1}(a+b x)}{x^3} \, dx=\begin {cases} - \frac {b^{2} \operatorname {acot}{\left (b x - i \right )}}{8} + \frac {b}{8 x} - \frac {\operatorname {acot}{\left (b x - i \right )}}{2 x^{2}} + \frac {i}{8 x^{2}} & \text {for}\: a = - i \\- \frac {b^{2} \operatorname {acot}{\left (b x + i \right )}}{8} + \frac {b}{8 x} - \frac {\operatorname {acot}{\left (b x + i \right )}}{2 x^{2}} - \frac {i}{8 x^{2}} & \text {for}\: a = i \\- \frac {a^{4} \operatorname {acot}{\left (a + b x \right )}}{2 a^{4} x^{2} + 4 a^{2} x^{2} + 2 x^{2}} + \frac {a^{2} b^{2} x^{2} \operatorname {acot}{\left (a + b x \right )}}{2 a^{4} x^{2} + 4 a^{2} x^{2} + 2 x^{2}} + \frac {a^{2} b x}{2 a^{4} x^{2} + 4 a^{2} x^{2} + 2 x^{2}} - \frac {2 a^{2} \operatorname {acot}{\left (a + b x \right )}}{2 a^{4} x^{2} + 4 a^{2} x^{2} + 2 x^{2}} + \frac {2 a b^{2} x^{2} \log {\left (x \right )}}{2 a^{4} x^{2} + 4 a^{2} x^{2} + 2 x^{2}} - \frac {a b^{2} x^{2} \log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 a^{4} x^{2} + 4 a^{2} x^{2} + 2 x^{2}} - \frac {b^{2} x^{2} \operatorname {acot}{\left (a + b x \right )}}{2 a^{4} x^{2} + 4 a^{2} x^{2} + 2 x^{2}} + \frac {b x}{2 a^{4} x^{2} + 4 a^{2} x^{2} + 2 x^{2}} - \frac {\operatorname {acot}{\left (a + b x \right )}}{2 a^{4} x^{2} + 4 a^{2} x^{2} + 2 x^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.18 \[ \int \frac {\cot ^{-1}(a+b x)}{x^3} \, dx=-\frac {1}{2} \, {\left (\frac {{\left (a^{2} - 1\right )} b \arctan \left (\frac {b^{2} x + a b}{b}\right )}{a^{4} + 2 \, a^{2} + 1} + \frac {a b \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{4} + 2 \, a^{2} + 1} - \frac {2 \, a b \log \left (x\right )}{a^{4} + 2 \, a^{2} + 1} - \frac {1}{{\left (a^{2} + 1\right )} x}\right )} b - \frac {\operatorname {arccot}\left (b x + a\right )}{2 \, x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1309 vs. \(2 (85) = 170\).
Time = 0.58 (sec) , antiderivative size = 1309, normalized size of antiderivative = 13.78 \[ \int \frac {\cot ^{-1}(a+b x)}{x^3} \, dx=\text {Too large to display} \]
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Time = 1.63 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.42 \[ \int \frac {\cot ^{-1}(a+b x)}{x^3} \, dx=\frac {\mathrm {atan}\left (\frac {2\,x\,b^2+2\,a\,b}{2\,\sqrt {b^2\,\left (a^2+1\right )-a^2\,b^2}}\right )\,\left (b^3-a^2\,b^3\right )}{\sqrt {b^2}\,\left (2\,a^4+4\,a^2+2\right )}-\frac {a\,b^2\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}{2\,{\left (a^2+1\right )}^2}-\frac {\mathrm {acot}\left (a+b\,x\right )\,\left (\frac {a^2}{2}+\frac {1}{2}\right )-\frac {b\,x}{2}+\frac {b^2\,x^2\,\mathrm {acot}\left (a+b\,x\right )}{2}-\frac {x^3\,\left (b^3-3\,a^2\,b^3\right )}{2\,\left (a^4+2\,a^2+1\right )}+\frac {a\,b^4\,x^4}{{\left (a^2+1\right )}^2}+a\,b\,x\,\mathrm {acot}\left (a+b\,x\right )}{a^2\,x^2+2\,a\,b\,x^3+b^2\,x^4+x^2}+\frac {a\,b^2\,\ln \left (x\right )}{{\left (a^2+1\right )}^2} \]
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