Integrand size = 23, antiderivative size = 36 \[ \int \frac {a c+b c x^2}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {c}{a x}-\frac {\sqrt {b} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {21, 331, 211} \[ \int \frac {a c+b c x^2}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {b} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {c}{a x} \]
[In]
[Out]
Rule 21
Rule 211
Rule 331
Rubi steps \begin{align*} \text {integral}& = c \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx \\ & = -\frac {c}{a x}-\frac {(b c) \int \frac {1}{a+b x^2} \, dx}{a} \\ & = -\frac {c}{a x}-\frac {\sqrt {b} c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {a c+b c x^2}{x^2 \left (a+b x^2\right )^2} \, dx=c \left (-\frac {1}{a x}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}\right ) \]
[In]
[Out]
Time = 2.54 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89
method | result | size |
default | \(c \left (-\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a \sqrt {a b}}-\frac {1}{a x}\right )\) | \(32\) |
risch | \(-\frac {c}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{3} \textit {\_Z}^{2}+b \,c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{3}+2 b \,c^{2}\right ) x +a^{2} c \textit {\_R} \right )\right )}{2}\) | \(57\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.39 \[ \int \frac {a c+b c x^2}{x^2 \left (a+b x^2\right )^2} \, dx=\left [\frac {c x \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 2 \, c}{2 \, a x}, -\frac {c x \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + c}{a x}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (31) = 62\).
Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.83 \[ \int \frac {a c+b c x^2}{x^2 \left (a+b x^2\right )^2} \, dx=c \left (\frac {\sqrt {- \frac {b}{a^{3}}} \log {\left (- \frac {a^{2} \sqrt {- \frac {b}{a^{3}}}}{b} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{a^{3}}} \log {\left (\frac {a^{2} \sqrt {- \frac {b}{a^{3}}}}{b} + x \right )}}{2} - \frac {1}{a x}\right ) \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {a c+b c x^2}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {b c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {c}{a x} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {a c+b c x^2}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {b c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {c}{a x} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78 \[ \int \frac {a c+b c x^2}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {c}{a\,x}-\frac {\sqrt {b}\,c\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{a^{3/2}} \]
[In]
[Out]