Integrand size = 18, antiderivative size = 17 \[ \int \frac {\log \left (\frac {a}{x^2}\right )}{a x-x^3} \, dx=\frac {\operatorname {PolyLog}\left (2,1-\frac {a}{x^2}\right )}{2 a} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1607, 2378, 2370, 2352} \[ \int \frac {\log \left (\frac {a}{x^2}\right )}{a x-x^3} \, dx=\frac {\operatorname {PolyLog}\left (2,1-\frac {a}{x^2}\right )}{2 a} \]
[In]
[Out]
Rule 1607
Rule 2352
Rule 2370
Rule 2378
Rubi steps \begin{align*} \text {integral}& = \int \frac {\log \left (\frac {a}{x^2}\right )}{x \left (a-x^2\right )} \, dx \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {\log (a x)}{\left (a-\frac {1}{x}\right ) x} \, dx,x,\frac {1}{x^2}\right )\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {\log (a x)}{-1+a x} \, dx,x,\frac {1}{x^2}\right )\right ) \\ & = \frac {\text {Li}_2\left (1-\frac {a}{x^2}\right )}{2 a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int \frac {\log \left (\frac {a}{x^2}\right )}{a x-x^3} \, dx=\frac {\operatorname {PolyLog}\left (2,-\frac {a-x^2}{x^2}\right )}{2 a} \]
[In]
[Out]
Time = 0.67 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {\operatorname {dilog}\left (\frac {a}{x^{2}}\right )}{2 a}\) | \(12\) |
default | \(\frac {\operatorname {dilog}\left (\frac {a}{x^{2}}\right )}{2 a}\) | \(12\) |
risch | \(\frac {\operatorname {dilog}\left (\frac {a}{x^{2}}\right )}{2 a}\) | \(12\) |
parts | \(\frac {\ln \left (\frac {a}{x^{2}}\right ) \ln \left (x \right )}{a}-\frac {\ln \left (\frac {a}{x^{2}}\right ) \ln \left (-x^{2}+a \right )}{2 a}+\frac {\ln \left (x \right )^{2}}{a}-\frac {\ln \left (x \right ) \ln \left (-x^{2}+a \right )-\ln \left (x \right ) \ln \left (\frac {\sqrt {a}-x}{\sqrt {a}}\right )-\ln \left (x \right ) \ln \left (\frac {\sqrt {a}+x}{\sqrt {a}}\right )-\operatorname {dilog}\left (\frac {\sqrt {a}-x}{\sqrt {a}}\right )-\operatorname {dilog}\left (\frac {\sqrt {a}+x}{\sqrt {a}}\right )}{a}\) | \(114\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {\log \left (\frac {a}{x^2}\right )}{a x-x^3} \, dx=\frac {{\rm Li}_2\left (-\frac {a}{x^{2}} + 1\right )}{2 \, a} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 5.02 (sec) , antiderivative size = 92, normalized size of antiderivative = 5.41 \[ \int \frac {\log \left (\frac {a}{x^2}\right )}{a x-x^3} \, dx=- \frac {\begin {cases} \frac {\operatorname {Li}_{2}\left (\frac {a}{x^{2}}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\i \pi \log {\left (x \right )} + \frac {\operatorname {Li}_{2}\left (\frac {a}{x^{2}}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- i \pi \log {\left (\frac {1}{x} \right )} + \frac {\operatorname {Li}_{2}\left (\frac {a}{x^{2}}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- i \pi {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} + i \pi {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} + \frac {\operatorname {Li}_{2}\left (\frac {a}{x^{2}}\right )}{2} & \text {otherwise} \end {cases}}{a} - \frac {\log {\left (\frac {a}{x^{2}} \right )} \log {\left (\frac {a}{x^{2}} - 1 \right )}}{2 a} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (14) = 28\).
Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 4.76 \[ \int \frac {\log \left (\frac {a}{x^2}\right )}{a x-x^3} \, dx=-\frac {1}{2} \, {\left (\frac {\log \left (x^{2} - a\right )}{a} - \frac {2 \, \log \left (x\right )}{a}\right )} \log \left (\frac {a}{x^{2}}\right ) - \frac {\log \left (x^{2} - a\right ) \log \left (x\right ) - \log \left (x\right )^{2}}{a} + \frac {2 \, \log \left (x\right ) \log \left (-\frac {x^{2}}{a} + 1\right ) + {\rm Li}_2\left (\frac {x^{2}}{a}\right )}{2 \, a} \]
[In]
[Out]
\[ \int \frac {\log \left (\frac {a}{x^2}\right )}{a x-x^3} \, dx=\int { -\frac {\log \left (\frac {a}{x^{2}}\right )}{x^{3} - a x} \,d x } \]
[In]
[Out]
Time = 0.34 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \frac {\log \left (\frac {a}{x^2}\right )}{a x-x^3} \, dx=\frac {{\mathrm {Li}}_{\mathrm {2}}\left (\frac {a}{x^2}\right )}{2\,a} \]
[In]
[Out]