Integrand size = 15, antiderivative size = 59 \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=-\frac {a \arctan \left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c} \left (a^2+c\right )}+\frac {\log (x)}{a^2+c}-\frac {\log \left (c+(a+b x)^2\right )}{2 \left (a^2+c\right )} \]
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Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {378, 720, 31, 649, 209, 266} \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=-\frac {a \arctan \left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c} \left (a^2+c\right )}-\frac {\log \left ((a+b x)^2+c\right )}{2 \left (a^2+c\right )}+\frac {\log (x)}{a^2+c} \]
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Rule 31
Rule 209
Rule 266
Rule 378
Rule 649
Rule 720
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{(-a+x) \left (c+x^2\right )} \, dx,x,a+b x\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{-a+x} \, dx,x,a+b x\right )}{a^2+c}+\frac {\text {Subst}\left (\int \frac {-a-x}{c+x^2} \, dx,x,a+b x\right )}{a^2+c} \\ & = \frac {\log (x)}{a^2+c}-\frac {\text {Subst}\left (\int \frac {x}{c+x^2} \, dx,x,a+b x\right )}{a^2+c}-\frac {a \text {Subst}\left (\int \frac {1}{c+x^2} \, dx,x,a+b x\right )}{a^2+c} \\ & = -\frac {a \tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c} \left (a^2+c\right )}+\frac {\log (x)}{a^2+c}-\frac {\log \left (c+(a+b x)^2\right )}{2 \left (a^2+c\right )} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=-\frac {\frac {2 a \arctan \left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c}}-2 \log (b x)+\log \left (c+(a+b x)^2\right )}{2 \left (a^2+c\right )} \]
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Time = 0.84 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.25
method | result | size |
default | \(\frac {\ln \left (x \right )}{a^{2}+c}-\frac {b \left (\frac {\ln \left (b^{2} x^{2}+2 a b x +a^{2}+c \right )}{2 b}+\frac {a \arctan \left (\frac {2 b^{2} x +2 a b}{2 \sqrt {c}\, b}\right )}{\sqrt {c}\, b}\right )}{a^{2}+c}\) | \(74\) |
risch | \(\frac {\ln \left (x \right )}{a^{2}+c}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (c \,a^{2}+c^{2}\right ) \textit {\_Z}^{2}+2 c \textit {\_Z} \right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a^{2} b +3 b c \right ) \textit {\_R} +3 b \right ) x +\left (-a^{3}-a c \right ) \textit {\_R} +2 a \right )\right )}{2}\) | \(75\) |
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Time = 0.28 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.61 \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=\left [-\frac {a \sqrt {-c} \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 2 \, {\left (b x + a\right )} \sqrt {-c} - c}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right ) + c \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) - 2 \, c \log \left (x\right )}{2 \, {\left (a^{2} c + c^{2}\right )}}, -\frac {2 \, a \sqrt {c} \arctan \left (\frac {b x + a}{\sqrt {c}}\right ) + c \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) - 2 \, c \log \left (x\right )}{2 \, {\left (a^{2} c + c^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (49) = 98\).
Time = 1.85 (sec) , antiderivative size = 738, normalized size of antiderivative = 12.51 \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=\left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) \log {\left (x + \frac {- 4 a^{6} c \left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} + 4 a^{4} c^{2} \left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} - 6 a^{4} c \left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) + 20 a^{2} c^{3} \left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} - 12 a^{2} c^{2} \left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) + 10 a^{2} c + 12 c^{4} \left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} - 6 c^{3} \left (- \frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) - 6 c^{2}}{a^{3} b + 9 a b c} \right )} + \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) \log {\left (x + \frac {- 4 a^{6} c \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} + 4 a^{4} c^{2} \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} - 6 a^{4} c \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) + 20 a^{2} c^{3} \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} - 12 a^{2} c^{2} \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) + 10 a^{2} c + 12 c^{4} \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right )^{2} - 6 c^{3} \left (\frac {a \sqrt {- c}}{2 c \left (a^{2} + c\right )} - \frac {1}{2 \left (a^{2} + c\right )}\right ) - 6 c^{2}}{a^{3} b + 9 a b c} \right )} + \frac {\log {\left (x + \frac {- \frac {4 a^{6} c}{\left (a^{2} + c\right )^{2}} + \frac {4 a^{4} c^{2}}{\left (a^{2} + c\right )^{2}} - \frac {6 a^{4} c}{a^{2} + c} + \frac {20 a^{2} c^{3}}{\left (a^{2} + c\right )^{2}} - \frac {12 a^{2} c^{2}}{a^{2} + c} + 10 a^{2} c + \frac {12 c^{4}}{\left (a^{2} + c\right )^{2}} - \frac {6 c^{3}}{a^{2} + c} - 6 c^{2}}{a^{3} b + 9 a b c} \right )}}{a^{2} + c} \]
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Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=-\frac {a \arctan \left (\frac {b^{2} x + a b}{b \sqrt {c}}\right )}{{\left (a^{2} + c\right )} \sqrt {c}} - \frac {\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, {\left (a^{2} + c\right )}} + \frac {\log \left (x\right )}{a^{2} + c} \]
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Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=-\frac {a \arctan \left (\frac {b x + a}{\sqrt {c}}\right )}{{\left (a^{2} + c\right )} \sqrt {c}} - \frac {\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, {\left (a^{2} + c\right )}} + \frac {\log \left ({\left | x \right |}\right )}{a^{2} + c} \]
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Time = 10.58 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.93 \[ \int \frac {1}{x \left (c+(a+b x)^2\right )} \, dx=\frac {\ln \left (x\right )}{a^2+c}-\frac {\ln \left (2\,a\,b^3+3\,b^4\,x+\frac {b^3\,\left (c+a\,\sqrt {-c}\right )\,\left (a^3+b\,x\,a^2+c\,a-3\,b\,c\,x\right )}{c\,\left (a^2+c\right )}\right )\,\left (c+a\,\sqrt {-c}\right )}{2\,\left (a^2\,c+c^2\right )}-\frac {\ln \left (2\,a\,b^3+3\,b^4\,x+\frac {b^3\,\left (c-a\,\sqrt {-c}\right )\,\left (a^3+b\,x\,a^2+c\,a-3\,b\,c\,x\right )}{c\,\left (a^2+c\right )}\right )\,\left (c-a\,\sqrt {-c}\right )}{2\,\left (a^2\,c+c^2\right )} \]
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