Integrand size = 11, antiderivative size = 42 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^2} \, dx=4 \sqrt {a} \text {arctanh}\left (\sqrt {a} x\right )+\frac {2 \log \left (1-a x^2\right )}{x}-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x} \] Output:
4*a^(1/2)*arctanh(a^(1/2)*x)+2*ln(-a*x^2+1)/x-polylog(2,a*x^2)/x
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.98 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^2} \, dx=\frac {4 \sqrt {a} x \text {arctanh}\left (\sqrt {a} x\right )+2 \log \left (1-a x^2\right )-\operatorname {PolyLog}\left (2,a x^2\right )}{x} \] Input:
Integrate[PolyLog[2, a*x^2]/x^2,x]
Output:
(4*Sqrt[a]*x*ArcTanh[Sqrt[a]*x] + 2*Log[1 - a*x^2] - PolyLog[2, a*x^2])/x
Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {7145, 25, 2905, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle 2 \int -\frac {\log \left (1-a x^2\right )}{x^2}dx-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {\log \left (1-a x^2\right )}{x^2}dx-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x}\) |
\(\Big \downarrow \) 2905 |
\(\displaystyle -2 \left (-2 a \int \frac {1}{1-a x^2}dx-\frac {\log \left (1-a x^2\right )}{x}\right )-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 \left (-2 \sqrt {a} \text {arctanh}\left (\sqrt {a} x\right )-\frac {\log \left (1-a x^2\right )}{x}\right )-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x}\) |
Input:
Int[PolyLog[2, a*x^2]/x^2,x]
Output:
-2*(-2*Sqrt[a]*ArcTanh[Sqrt[a]*x] - Log[1 - a*x^2]/x) - PolyLog[2, a*x^2]/ x
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^ (m_.), x_Symbol] :> Simp[(f*x)^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Simp[b*e*n*(p/(f*(m + 1))) Int[x^(n - 1)*((f*x)^(m + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]
Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbo l] :> Simp[(d*x)^(m + 1)*(PolyLog[n, a*(b*x^p)^q]/(d*(m + 1))), x] - Simp[p *(q/(m + 1)) Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]
Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.93
method | result | size |
default | \(4 \sqrt {a}\, \operatorname {arctanh}\left (\sqrt {a}\, x \right )+\frac {2 \ln \left (-a \,x^{2}+1\right )}{x}-\frac {\operatorname {polylog}\left (2, a \,x^{2}\right )}{x}\) | \(39\) |
parts | \(4 \sqrt {a}\, \operatorname {arctanh}\left (\sqrt {a}\, x \right )+\frac {2 \ln \left (-a \,x^{2}+1\right )}{x}-\frac {\operatorname {polylog}\left (2, a \,x^{2}\right )}{x}\) | \(39\) |
meijerg | \(\frac {a \left (-\frac {4 x \sqrt {-a}\, \left (\ln \left (1-\sqrt {a \,x^{2}}\right )-\ln \left (1+\sqrt {a \,x^{2}}\right )\right )}{\sqrt {a \,x^{2}}}+\frac {4 \sqrt {-a}\, \ln \left (-a \,x^{2}+1\right )}{x a}-\frac {2 \sqrt {-a}\, \operatorname {polylog}\left (2, a \,x^{2}\right )}{x a}\right )}{2 \sqrt {-a}}\) | \(92\) |
Input:
int(polylog(2,a*x^2)/x^2,x,method=_RETURNVERBOSE)
Output:
4*a^(1/2)*arctanh(a^(1/2)*x)+2*ln(-a*x^2+1)/x-polylog(2,a*x^2)/x
Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.24 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^2} \, dx=\left [\frac {2 \, \sqrt {a} x \log \left (\frac {a x^{2} + 2 \, \sqrt {a} x + 1}{a x^{2} - 1}\right ) - {\rm Li}_2\left (a x^{2}\right ) + 2 \, \log \left (-a x^{2} + 1\right )}{x}, -\frac {4 \, \sqrt {-a} x \arctan \left (\sqrt {-a} x\right ) + {\rm Li}_2\left (a x^{2}\right ) - 2 \, \log \left (-a x^{2} + 1\right )}{x}\right ] \] Input:
integrate(polylog(2,a*x^2)/x^2,x, algorithm="fricas")
Output:
[(2*sqrt(a)*x*log((a*x^2 + 2*sqrt(a)*x + 1)/(a*x^2 - 1)) - dilog(a*x^2) + 2*log(-a*x^2 + 1))/x, -(4*sqrt(-a)*x*arctan(sqrt(-a)*x) + dilog(a*x^2) - 2 *log(-a*x^2 + 1))/x]
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (34) = 68\).
Time = 12.78 (sec) , antiderivative size = 184, normalized size of antiderivative = 4.38 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^2} \, dx=\begin {cases} 0 & \text {for}\: a = 0 \\- \frac {\pi ^{2}}{6 x} & \text {for}\: a = \frac {1}{x^{2}} \\- \frac {4 a x^{3} \sqrt {\frac {1}{a}} \log {\left (x - \sqrt {\frac {1}{a}} \right )}}{x^{3} - \frac {x}{a}} - \frac {2 a x^{3} \sqrt {\frac {1}{a}} \operatorname {Li}_{1}\left (a x^{2}\right )}{x^{3} - \frac {x}{a}} - \frac {2 x^{2} \operatorname {Li}_{1}\left (a x^{2}\right )}{x^{3} - \frac {x}{a}} - \frac {x^{2} \operatorname {Li}_{2}\left (a x^{2}\right )}{x^{3} - \frac {x}{a}} + \frac {4 x \sqrt {\frac {1}{a}} \log {\left (x - \sqrt {\frac {1}{a}} \right )}}{x^{3} - \frac {x}{a}} + \frac {2 x \sqrt {\frac {1}{a}} \operatorname {Li}_{1}\left (a x^{2}\right )}{x^{3} - \frac {x}{a}} + \frac {2 \operatorname {Li}_{1}\left (a x^{2}\right )}{a x^{3} - x} + \frac {\operatorname {Li}_{2}\left (a x^{2}\right )}{a x^{3} - x} & \text {otherwise} \end {cases} \] Input:
integrate(polylog(2,a*x**2)/x**2,x)
Output:
Piecewise((0, Eq(a, 0)), (-pi**2/(6*x), Eq(a, x**(-2))), (-4*a*x**3*sqrt(1 /a)*log(x - sqrt(1/a))/(x**3 - x/a) - 2*a*x**3*sqrt(1/a)*polylog(1, a*x**2 )/(x**3 - x/a) - 2*x**2*polylog(1, a*x**2)/(x**3 - x/a) - x**2*polylog(2, a*x**2)/(x**3 - x/a) + 4*x*sqrt(1/a)*log(x - sqrt(1/a))/(x**3 - x/a) + 2*x *sqrt(1/a)*polylog(1, a*x**2)/(x**3 - x/a) + 2*polylog(1, a*x**2)/(a*x**3 - x) + polylog(2, a*x**2)/(a*x**3 - x), True))
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^2} \, dx=-2 \, \sqrt {a} \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right ) - \frac {{\rm Li}_2\left (a x^{2}\right ) - 2 \, \log \left (-a x^{2} + 1\right )}{x} \] Input:
integrate(polylog(2,a*x^2)/x^2,x, algorithm="maxima")
Output:
-2*sqrt(a)*log((a*x - sqrt(a))/(a*x + sqrt(a))) - (dilog(a*x^2) - 2*log(-a *x^2 + 1))/x
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^2} \, dx=\int { \frac {{\rm Li}_2\left (a x^{2}\right )}{x^{2}} \,d x } \] Input:
integrate(polylog(2,a*x^2)/x^2,x, algorithm="giac")
Output:
integrate(dilog(a*x^2)/x^2, x)
Time = 0.21 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^2} \, dx=4\,\sqrt {a}\,\mathrm {atanh}\left (\sqrt {a}\,x\right )-\frac {\mathrm {polylog}\left (2,a\,x^2\right )}{x}+\frac {2\,\ln \left (1-a\,x^2\right )}{x} \] Input:
int(polylog(2, a*x^2)/x^2,x)
Output:
4*a^(1/2)*atanh(a^(1/2)*x) - polylog(2, a*x^2)/x + (2*log(1 - a*x^2))/x
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^2} \, dx=\int \frac {\mathit {polylog}\left (2, a \,x^{2}\right )}{x^{2}}d x \] Input:
int(polylog(2,a*x^2)/x^2,x)
Output:
int(polylog(2,a*x**2)/x**2,x)