Integrand size = 11, antiderivative size = 49 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^3} \, dx=a \log (x)-\frac {1}{2} a \log \left (1-a x^2\right )+\frac {\log \left (1-a x^2\right )}{2 x^2}-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{2 x^2} \] Output:
a*ln(x)-1/2*a*ln(-a*x^2+1)+1/2*ln(-a*x^2+1)/x^2-1/2*polylog(2,a*x^2)/x^2
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^3} \, dx=a \log (x)-\frac {1}{2} a \log \left (1-a x^2\right )+\frac {\log \left (1-a x^2\right )}{2 x^2}-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{2 x^2} \] Input:
Integrate[PolyLog[2, a*x^2]/x^3,x]
Output:
a*Log[x] - (a*Log[1 - a*x^2])/2 + Log[1 - a*x^2]/(2*x^2) - PolyLog[2, a*x^ 2]/(2*x^2)
Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {7145, 25, 2904, 2842, 47, 14, 16}
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.
\(\displaystyle \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \int -\frac {\log \left (1-a x^2\right )}{x^3}dx-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{2 x^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\log \left (1-a x^2\right )}{x^3}dx-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{2 x^2}\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle -\frac {1}{2} \int \frac {\log \left (1-a x^2\right )}{x^4}dx^2-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{2 x^2}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle \frac {1}{2} \left (a \int \frac {1}{x^2 \left (1-a x^2\right )}dx^2+\frac {\log \left (1-a x^2\right )}{x^2}\right )-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{2 x^2}\) |
\(\Big \downarrow \) 47 |
\(\displaystyle \frac {1}{2} \left (a \left (a \int \frac {1}{1-a x^2}dx^2+\int \frac {1}{x^2}dx^2\right )+\frac {\log \left (1-a x^2\right )}{x^2}\right )-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{2 x^2}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {1}{2} \left (a \left (a \int \frac {1}{1-a x^2}dx^2+\log \left (x^2\right )\right )+\frac {\log \left (1-a x^2\right )}{x^2}\right )-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{2 x^2}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{2} \left (a \left (\log \left (x^2\right )-\log \left (1-a x^2\right )\right )+\frac {\log \left (1-a x^2\right )}{x^2}\right )-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{2 x^2}\) |
Input:
Int[PolyLog[2, a*x^2]/x^3,x]
Output:
(a*(Log[x^2] - Log[1 - a*x^2]) + Log[1 - a*x^2]/x^2)/2 - PolyLog[2, a*x^2] /(2*x^2)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_ ))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/( g*(q + 1))), x] - Simp[b*e*(n/(g*(q + 1))) Int[(f + g*x)^(q + 1)/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
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 = 43, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {\operatorname {polylog}\left (2, a \,x^{2}\right )}{2 x^{2}}+\frac {\ln \left (-a \,x^{2}+1\right )}{2 x^{2}}+a \left (\ln \left (x \right )-\frac {\ln \left (a \,x^{2}-1\right )}{2}\right )\) | \(43\) |
parts | \(-\frac {\operatorname {polylog}\left (2, a \,x^{2}\right )}{2 x^{2}}+\frac {\ln \left (-a \,x^{2}+1\right )}{2 x^{2}}+a \left (\ln \left (x \right )-\frac {\ln \left (a \,x^{2}-1\right )}{2}\right )\) | \(43\) |
meijerg | \(\frac {a \left (2 \ln \left (x \right )+\ln \left (-a \right )+\frac {\left (-4 a \,x^{2}+4\right ) \ln \left (-a \,x^{2}+1\right )}{4 a \,x^{2}}-\frac {\operatorname {polylog}\left (2, a \,x^{2}\right )}{a \,x^{2}}\right )}{2}\) | \(53\) |
parallelrisch | \(\frac {2 a^{2} \ln \left (x \right ) x^{2}-\ln \left (-a \,x^{2}+1\right ) x^{2} a^{2}-a \operatorname {polylog}\left (2, a \,x^{2}\right )+a \ln \left (-a \,x^{2}+1\right )}{2 x^{2} a}\) | \(58\) |
Input:
int(polylog(2,a*x^2)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/2*polylog(2,a*x^2)/x^2+1/2*ln(-a*x^2+1)/x^2+a*(ln(x)-1/2*ln(a*x^2-1))
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^3} \, dx=-\frac {a x^{2} \log \left (a x^{2} - 1\right ) - 2 \, a x^{2} \log \left (x\right ) + {\rm Li}_2\left (a x^{2}\right ) - \log \left (-a x^{2} + 1\right )}{2 \, x^{2}} \] Input:
integrate(polylog(2,a*x^2)/x^3,x, algorithm="fricas")
Output:
-1/2*(a*x^2*log(a*x^2 - 1) - 2*a*x^2*log(x) + dilog(a*x^2) - log(-a*x^2 + 1))/x^2
Time = 1.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.76 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^3} \, dx=a \log {\left (x \right )} + \frac {a \operatorname {Li}_{1}\left (a x^{2}\right )}{2} - \frac {\operatorname {Li}_{1}\left (a x^{2}\right )}{2 x^{2}} - \frac {\operatorname {Li}_{2}\left (a x^{2}\right )}{2 x^{2}} \] Input:
integrate(polylog(2,a*x**2)/x**3,x)
Output:
a*log(x) + a*polylog(1, a*x**2)/2 - polylog(1, a*x**2)/(2*x**2) - polylog( 2, a*x**2)/(2*x**2)
Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.69 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^3} \, dx=a \log \left (x\right ) - \frac {{\left (a x^{2} - 1\right )} \log \left (-a x^{2} + 1\right ) + {\rm Li}_2\left (a x^{2}\right )}{2 \, x^{2}} \] Input:
integrate(polylog(2,a*x^2)/x^3,x, algorithm="maxima")
Output:
a*log(x) - 1/2*((a*x^2 - 1)*log(-a*x^2 + 1) + dilog(a*x^2))/x^2
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^3} \, dx=\int { \frac {{\rm Li}_2\left (a x^{2}\right )}{x^{3}} \,d x } \] Input:
integrate(polylog(2,a*x^2)/x^3,x, algorithm="giac")
Output:
integrate(dilog(a*x^2)/x^3, x)
Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^3} \, dx=\frac {3\,a\,\ln \left (x\right )}{2}+\frac {\frac {\ln \left (1-a\,x^2\right )}{2}-\frac {\mathrm {polylog}\left (2,a\,x^2\right )}{2}}{x^2}-\frac {a\,\ln \left (a\,x^3-x\right )}{2} \] Input:
int(polylog(2, a*x^2)/x^3,x)
Output:
(3*a*log(x))/2 + (log(1 - a*x^2)/2 - polylog(2, a*x^2)/2)/x^2 - (a*log(a*x ^3 - x))/2
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^3} \, dx=\int \frac {\mathit {polylog}\left (2, a \,x^{2}\right )}{x^{3}}d x \] Input:
int(polylog(2,a*x^2)/x^3,x)
Output:
int(polylog(2,a*x**2)/x**3,x)