\(\int \frac {\log (d+e x^2)}{1-x^2} \, dx\) [359]

Optimal result

Integrand size = 18, antiderivative size = 217 \[ \int \frac {\log \left (d+e x^2\right )}{1-x^2} \, dx=2 \text {arctanh}(x) \log \left (\frac {2}{1+x}\right )-\text {arctanh}(x) \log \left (\frac {2 \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {-d}-\sqrt {e}\right ) (1+x)}\right )-\text {arctanh}(x) \log \left (\frac {2 \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {-d}+\sqrt {e}\right ) (1+x)}\right )+\text {arctanh}(x) \log \left (d+e x^2\right )-\operatorname {PolyLog}\left (2,1-\frac {2}{1+x}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2 \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {-d}-\sqrt {e}\right ) (1+x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2 \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {-d}+\sqrt {e}\right ) (1+x)}\right ) \] Output:

2*arctanh(x)*ln(2/(1+x))-arctanh(x)*ln(2*((-d)^(1/2)-e^(1/2)*x)/((-d)^(1/2 
)-e^(1/2))/(1+x))-arctanh(x)*ln(2*((-d)^(1/2)+e^(1/2)*x)/((-d)^(1/2)+e^(1/ 
2))/(1+x))+arctanh(x)*ln(e*x^2+d)-polylog(2,1-2/(1+x))+1/2*polylog(2,1-2*( 
(-d)^(1/2)-e^(1/2)*x)/((-d)^(1/2)-e^(1/2))/(1+x))+1/2*polylog(2,1-2*((-d)^ 
(1/2)+e^(1/2)*x)/((-d)^(1/2)+e^(1/2))/(1+x))
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.54 \[ \int \frac {\log \left (d+e x^2\right )}{1-x^2} \, dx=\frac {1}{2} \log (1-x) \log \left (\frac {\sqrt {-d}-\sqrt {e} x}{\sqrt {-d}-\sqrt {e}}\right )-\frac {1}{2} \log (1+x) \log \left (\frac {\sqrt {-d}-\sqrt {e} x}{\sqrt {-d}+\sqrt {e}}\right )-\frac {1}{2} \log (1+x) \log \left (\frac {\sqrt {-d}+\sqrt {e} x}{\sqrt {-d}-\sqrt {e}}\right )+\frac {1}{2} \log (1-x) \log \left (\frac {\sqrt {-d}+\sqrt {e} x}{\sqrt {-d}+\sqrt {e}}\right )-\frac {1}{2} \log (1-x) \log \left (d+e x^2\right )+\frac {1}{2} \log (1+x) \log \left (d+e x^2\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} (1-x)}{\sqrt {-d}-\sqrt {e}}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-x)}{\sqrt {-d}+\sqrt {e}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} (1+x)}{\sqrt {-d}-\sqrt {e}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+x)}{\sqrt {-d}+\sqrt {e}}\right ) \] Input:

Integrate[Log[d + e*x^2]/(1 - x^2),x]
 

Output:

(Log[1 - x]*Log[(Sqrt[-d] - Sqrt[e]*x)/(Sqrt[-d] - Sqrt[e])])/2 - (Log[1 + 
 x]*Log[(Sqrt[-d] - Sqrt[e]*x)/(Sqrt[-d] + Sqrt[e])])/2 - (Log[1 + x]*Log[ 
(Sqrt[-d] + Sqrt[e]*x)/(Sqrt[-d] - Sqrt[e])])/2 + (Log[1 - x]*Log[(Sqrt[-d 
] + Sqrt[e]*x)/(Sqrt[-d] + Sqrt[e])])/2 - (Log[1 - x]*Log[d + e*x^2])/2 + 
(Log[1 + x]*Log[d + e*x^2])/2 + PolyLog[2, -((Sqrt[e]*(1 - x))/(Sqrt[-d] - 
 Sqrt[e]))]/2 + PolyLog[2, (Sqrt[e]*(1 - x))/(Sqrt[-d] + Sqrt[e])]/2 - Pol 
yLog[2, -((Sqrt[e]*(1 + x))/(Sqrt[-d] - Sqrt[e]))]/2 - PolyLog[2, (Sqrt[e] 
*(1 + x))/(Sqrt[-d] + Sqrt[e])]/2
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2920, 6554, 2009}

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.

\(\displaystyle \int \frac {\log \left (d+e x^2\right )}{1-x^2} \, dx\)

\(\Big \downarrow \) 2920

\(\displaystyle \text {arctanh}(x) \log \left (d+e x^2\right )-2 e \int \frac {x \text {arctanh}(x)}{e x^2+d}dx\)

\(\Big \downarrow \) 6554

\(\displaystyle \text {arctanh}(x) \log \left (d+e x^2\right )-2 e \int \left (\frac {\text {arctanh}(x)}{2 \sqrt {e} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {\text {arctanh}(x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \text {arctanh}(x) \log \left (d+e x^2\right )-2 e \left (\frac {\text {arctanh}(x) \log \left (\frac {2 \left (\sqrt {-d}-\sqrt {e} x\right )}{(x+1) \left (\sqrt {-d}-\sqrt {e}\right )}\right )}{2 e}+\frac {\text {arctanh}(x) \log \left (\frac {2 \left (\sqrt {-d}+\sqrt {e} x\right )}{(x+1) \left (\sqrt {-d}+\sqrt {e}\right )}\right )}{2 e}-\frac {\text {arctanh}(x) \log \left (\frac {2}{x+1}\right )}{e}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {-d}-\sqrt {e}\right ) (x+1)}\right )}{4 e}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d}+\sqrt {e}\right ) (x+1)}\right )}{4 e}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{x+1}\right )}{2 e}\right )\)

Input:

Int[Log[d + e*x^2]/(1 - x^2),x]
 

Output:

ArcTanh[x]*Log[d + e*x^2] - 2*e*(-((ArcTanh[x]*Log[2/(1 + x)])/e) + (ArcTa 
nh[x]*Log[(2*(Sqrt[-d] - Sqrt[e]*x))/((Sqrt[-d] - Sqrt[e])*(1 + x))])/(2*e 
) + (ArcTanh[x]*Log[(2*(Sqrt[-d] + Sqrt[e]*x))/((Sqrt[-d] + Sqrt[e])*(1 + 
x))])/(2*e) + PolyLog[2, 1 - 2/(1 + x)]/(2*e) - PolyLog[2, 1 - (2*(Sqrt[-d 
] - Sqrt[e]*x))/((Sqrt[-d] - Sqrt[e])*(1 + x))]/(4*e) - PolyLog[2, 1 - (2* 
(Sqrt[-d] + Sqrt[e]*x))/((Sqrt[-d] + Sqrt[e])*(1 + x))]/(4*e))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2920

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.) 
*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(f + g*x^2), x]}, Simp[u*(a + b* 
Log[c*(d + e*x^n)^p]), x] - Simp[b*e*n*p   Int[u*(x^(n - 1)/(d + e*x^n)), x 
], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]
 

rule 6554

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Int[ExpandIntegrand[a + b*ArcTanh[c*x], x^m/(d + e*x^2), x], 
x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a, 0] 
)
 

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.30

method	result	size
risch	\(\frac {\ln \left (1+x \right ) \ln \left (e \,x^{2}+d \right )}{2}-\frac {\ln \left (1+x \right ) \ln \left (\frac {-e \left (1+x \right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )}{2}-\frac {\ln \left (1+x \right ) \ln \left (\frac {e \left (1+x \right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-e \left (1+x \right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {e \left (1+x \right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )}{2}-\frac {\ln \left (x -1\right ) \ln \left (e \,x^{2}+d \right )}{2}+\frac {\ln \left (x -1\right ) \ln \left (\frac {-e \left (x -1\right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )}{2}+\frac {\ln \left (x -1\right ) \ln \left (\frac {e \left (x -1\right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {-e \left (x -1\right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {e \left (x -1\right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )}{2}\)	\(282\)
default	\(\frac {\ln \left (1+x \right ) \ln \left (e \,x^{2}+d \right )}{2}-e \left (\frac {\ln \left (1+x \right ) \left (\ln \left (\frac {-e \left (1+x \right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )+\ln \left (\frac {e \left (1+x \right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-e \left (1+x \right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e \left (1+x \right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )}{2 e}\right )-\frac {\ln \left (x -1\right ) \ln \left (e \,x^{2}+d \right )}{2}+e \left (\frac {\ln \left (x -1\right ) \left (\ln \left (\frac {-e \left (x -1\right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )+\ln \left (\frac {e \left (x -1\right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-e \left (x -1\right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e \left (x -1\right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )}{2 e}\right )\)	\(289\)
parts	\(\operatorname {arctanh}\left (x \right ) \ln \left (e \,x^{2}+d \right )-2 e \left (\frac {\operatorname {arctanh}\left (x \right ) \ln \left (e \,x^{2}+d \right )}{2 e}-\frac {\frac {\ln \left (1+x \right ) \ln \left (e \,x^{2}+d \right )}{2}-e \left (\frac {\ln \left (1+x \right ) \left (\ln \left (\frac {-e \left (1+x \right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )+\ln \left (\frac {e \left (1+x \right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-e \left (1+x \right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e \left (1+x \right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )}{2 e}\right )-\frac {\ln \left (x -1\right ) \ln \left (e \,x^{2}+d \right )}{2}+e \left (\frac {\ln \left (x -1\right ) \left (\ln \left (\frac {-e \left (x -1\right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )+\ln \left (\frac {e \left (x -1\right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-e \left (x -1\right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e \left (x -1\right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )}{2 e}\right )}{2 e}\right )\)	\(325\)

Input:

int(ln(e*x^2+d)/(-x^2+1),x,method=_RETURNVERBOSE)
 

Output:

1/2*ln(1+x)*ln(e*x^2+d)-1/2*ln(1+x)*ln((-e*(1+x)+(-d*e)^(1/2)+e)/(e+(-d*e) 
^(1/2)))-1/2*ln(1+x)*ln((e*(1+x)+(-d*e)^(1/2)-e)/(-e+(-d*e)^(1/2)))-1/2*di 
log((-e*(1+x)+(-d*e)^(1/2)+e)/(e+(-d*e)^(1/2)))-1/2*dilog((e*(1+x)+(-d*e)^ 
(1/2)-e)/(-e+(-d*e)^(1/2)))-1/2*ln(x-1)*ln(e*x^2+d)+1/2*ln(x-1)*ln((-e*(x- 
1)+(-d*e)^(1/2)-e)/(-e+(-d*e)^(1/2)))+1/2*ln(x-1)*ln((e*(x-1)+(-d*e)^(1/2) 
+e)/(e+(-d*e)^(1/2)))+1/2*dilog((-e*(x-1)+(-d*e)^(1/2)-e)/(-e+(-d*e)^(1/2) 
))+1/2*dilog((e*(x-1)+(-d*e)^(1/2)+e)/(e+(-d*e)^(1/2)))
 

Fricas [F]

\[ \int \frac {\log \left (d+e x^2\right )}{1-x^2} \, dx=\int { -\frac {\log \left (e x^{2} + d\right )}{x^{2} - 1} \,d x } \] Input:

integrate(log(e*x^2+d)/(-x^2+1),x, algorithm="fricas")
 

Output:

integral(-log(e*x^2 + d)/(x^2 - 1), x)
 

Sympy [F]

\[ \int \frac {\log \left (d+e x^2\right )}{1-x^2} \, dx=- \int \frac {\log {\left (d + e x^{2} \right )}}{x^{2} - 1}\, dx \] Input:

integrate(ln(e*x**2+d)/(-x**2+1),x)
 

Output:

-Integral(log(d + e*x**2)/(x**2 - 1), x)
 

Maxima [F]

\[ \int \frac {\log \left (d+e x^2\right )}{1-x^2} \, dx=\int { -\frac {\log \left (e x^{2} + d\right )}{x^{2} - 1} \,d x } \] Input:

integrate(log(e*x^2+d)/(-x^2+1),x, algorithm="maxima")
 

Output:

-integrate(log(e*x^2 + d)/(x^2 - 1), x)
 

Giac [F]

\[ \int \frac {\log \left (d+e x^2\right )}{1-x^2} \, dx=\int { -\frac {\log \left (e x^{2} + d\right )}{x^{2} - 1} \,d x } \] Input:

integrate(log(e*x^2+d)/(-x^2+1),x, algorithm="giac")
 

Output:

integrate(-log(e*x^2 + d)/(x^2 - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (d+e x^2\right )}{1-x^2} \, dx=-\int \frac {\ln \left (e\,x^2+d\right )}{x^2-1} \,d x \] Input:

int(-log(d + e*x^2)/(x^2 - 1),x)
 

Output:

-int(log(d + e*x^2)/(x^2 - 1), x)
 

Reduce [F]

\[ \int \frac {\log \left (d+e x^2\right )}{1-x^2} \, dx=-\left (\int \frac {\mathrm {log}\left (e \,x^{2}+d \right )}{x^{2}-1}d x \right ) \] Input:

int(log(e*x^2+d)/(-x^2+1),x)
                                                                                    
                                                                                    
 

Output:

 - int(log(d + e*x**2)/(x**2 - 1),x)