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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {212, 2520, 6139, 6057, 2449, 2352, 2497} \[ \int \frac {\log \left (d+e x^2\right )}{1-x^2} \, dx=\text {arctanh}(x) \log \left (d+e x^2\right )-\text {arctanh}(x) \log \left (\frac {2 \left (\sqrt {-d}-\sqrt {e} x\right )}{(x+1) \left (\sqrt {-d}-\sqrt {e}\right )}\right )-\text {arctanh}(x) \log \left (\frac {2 \left (\sqrt {-d}+\sqrt {e} x\right )}{(x+1) \left (\sqrt {-d}+\sqrt {e}\right )}\right )+2 \text {arctanh}(x) \log \left (\frac {2}{x+1}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2 \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {-d}-\sqrt {e}\right ) (x+1)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2 \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d}+\sqrt {e}\right ) (x+1)}\right )-\operatorname {PolyLog}\left (2,1-\frac {2}{x+1}\right ) \]

[In]

Int[Log[d + e*x^2]/(1 - x^2),x]

[Out]

2*ArcTanh[x]*Log[2/(1 + x)] - ArcTanh[x]*Log[(2*(Sqrt[-d] - Sqrt[e]*x))/((Sqrt[-d] - Sqrt[e])*(1 + x))] - ArcT
anh[x]*Log[(2*(Sqrt[-d] + Sqrt[e]*x))/((Sqrt[-d] + Sqrt[e])*(1 + x))] + ArcTanh[x]*Log[d + e*x^2] - PolyLog[2,
1 - 2/(1 + x)] + PolyLog[2, 1 - (2*(Sqrt[-d] - Sqrt[e]*x))/((Sqrt[-d] - Sqrt[e])*(1 + x))]/2 + PolyLog[2, 1 -
(2*(Sqrt[-d] + Sqrt[e]*x))/((Sqrt[-d] + Sqrt[e])*(1 + x))]/2

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 2520

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[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 6057

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x]))*(Log[2/
(1 + c*x)]/e), x] + (Dist[b*(c/e), Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Dist[b*(c/e), Int[Log[2*c*((d
+ e*x)/((c*d + e)*(1 + c*x)))]/(1 - c^2*x^2), x], x] + Simp[(a + b*ArcTanh[c*x])*(Log[2*c*((d + e*x)/((c*d + e
)*(1 + c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]

Rule 6139

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])

Rubi steps \begin{align*} \text {integral}& = \tanh ^{-1}(x) \log \left (d+e x^2\right )-(2 e) \int \frac {x \tanh ^{-1}(x)}{d+e x^2} \, dx \\ & = \tanh ^{-1}(x) \log \left (d+e x^2\right )-(2 e) \int \left (-\frac {\tanh ^{-1}(x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\tanh ^{-1}(x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx \\ & = \tanh ^{-1}(x) \log \left (d+e x^2\right )+\sqrt {e} \int \frac {\tanh ^{-1}(x)}{\sqrt {-d}-\sqrt {e} x} \, dx-\sqrt {e} \int \frac {\tanh ^{-1}(x)}{\sqrt {-d}+\sqrt {e} x} \, dx \\ & = 2 \tanh ^{-1}(x) \log \left (\frac {2}{1+x}\right )-\tanh ^{-1}(x) \log \left (\frac {2 \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {-d}-\sqrt {e}\right ) (1+x)}\right )-\tanh ^{-1}(x) \log \left (\frac {2 \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {-d}+\sqrt {e}\right ) (1+x)}\right )+\tanh ^{-1}(x) \log \left (d+e x^2\right )-2 \int \frac {\log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx+\int \frac {\log \left (\frac {2 \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {-d}-\sqrt {e}\right ) (1+x)}\right )}{1-x^2} \, dx+\int \frac {\log \left (\frac {2 \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {-d}+\sqrt {e}\right ) (1+x)}\right )}{1-x^2} \, dx \\ & = 2 \tanh ^{-1}(x) \log \left (\frac {2}{1+x}\right )-\tanh ^{-1}(x) \log \left (\frac {2 \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {-d}-\sqrt {e}\right ) (1+x)}\right )-\tanh ^{-1}(x) \log \left (\frac {2 \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {-d}+\sqrt {e}\right ) (1+x)}\right )+\tanh ^{-1}(x) \log \left (d+e x^2\right )+\frac {1}{2} \text {Li}_2\left (1-\frac {2 \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {-d}-\sqrt {e}\right ) (1+x)}\right )+\frac {1}{2} \text {Li}_2\left (1-\frac {2 \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {-d}+\sqrt {e}\right ) (1+x)}\right )-2 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+x}\right ) \\ & = 2 \tanh ^{-1}(x) \log \left (\frac {2}{1+x}\right )-\tanh ^{-1}(x) \log \left (\frac {2 \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {-d}-\sqrt {e}\right ) (1+x)}\right )-\tanh ^{-1}(x) \log \left (\frac {2 \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {-d}+\sqrt {e}\right ) (1+x)}\right )+\tanh ^{-1}(x) \log \left (d+e x^2\right )-\text {Li}_2\left (1-\frac {2}{1+x}\right )+\frac {1}{2} \text {Li}_2\left (1-\frac {2 \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {-d}-\sqrt {e}\right ) (1+x)}\right )+\frac {1}{2} \text {Li}_2\left (1-\frac {2 \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {-d}+\sqrt {e}\right ) (1+x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.54 \[ \int \frac {\log \left (d+e x^2\right )}{1-x^2} \, dx=\frac {1}{2} \log (1-x) \log \left (\frac {\sqrt {-d}-\sqrt {e} x}{\sqrt {-d}-\sqrt {e}}\right )-\frac {1}{2} \log (1+x) \log \left (\frac {\sqrt {-d}-\sqrt {e} x}{\sqrt {-d}+\sqrt {e}}\right )-\frac {1}{2} \log (1+x) \log \left (\frac {\sqrt {-d}+\sqrt {e} x}{\sqrt {-d}-\sqrt {e}}\right )+\frac {1}{2} \log (1-x) \log \left (\frac {\sqrt {-d}+\sqrt {e} x}{\sqrt {-d}+\sqrt {e}}\right )-\frac {1}{2} \log (1-x) \log \left (d+e x^2\right )+\frac {1}{2} \log (1+x) \log \left (d+e x^2\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} (1-x)}{\sqrt {-d}-\sqrt {e}}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-x)}{\sqrt {-d}+\sqrt {e}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} (1+x)}{\sqrt {-d}-\sqrt {e}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+x)}{\sqrt {-d}+\sqrt {e}}\right ) \]

[In]

Integrate[Log[d + e*x^2]/(1 - x^2),x]

[Out]

(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 + Po
lyLog[2, -((Sqrt[e]*(1 - x))/(Sqrt[-d] - Sqrt[e]))]/2 + PolyLog[2, (Sqrt[e]*(1 - x))/(Sqrt[-d] + Sqrt[e])]/2 -
PolyLog[2, -((Sqrt[e]*(1 + x))/(Sqrt[-d] - Sqrt[e]))]/2 - PolyLog[2, (Sqrt[e]*(1 + x))/(Sqrt[-d] + Sqrt[e])]/
2

Maple [A] (verified)

Time = 0.73 (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	\(\operatorname {arctanh}\left (x \right ) \ln \left (e \,x^{2}+d \right )-2 e \left (\frac {\operatorname {arctanh}\left (x \right ) \ln \left (e \,x^{2}+d \right )}{2 e}-\frac {-\frac {\ln \left (-1+x \right ) \ln \left (e \,x^{2}+d \right )}{2}+e \left (\frac {\ln \left (-1+x \right ) \left (\ln \left (\frac {-e \left (-1+x \right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )+\ln \left (\frac {e \left (-1+x \right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-e \left (-1+x \right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e \left (-1+x \right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )}{2 e}\right )+\frac {\ln \left (x +1\right ) \ln \left (e \,x^{2}+d \right )}{2}-e \left (\frac {\ln \left (x +1\right ) \left (\ln \left (\frac {-e \left (x +1\right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )+\ln \left (\frac {e \left (x +1\right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-e \left (x +1\right )+\sqrt {-d e}+e}{e +\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e \left (x +1\right )+\sqrt {-d e}-e}{-e +\sqrt {-d e}}\right )}{2 e}\right )}{2 e}\right )\)	\(325\)

[In]

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

[Out]

-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*dilog((-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]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(-log(d + e*x^2)/(x^2 - 1),x)

[Out]

-int(log(d + e*x^2)/(x^2 - 1), x)