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

Optimal. Leaf size=217 \[ \frac{1}{2} \text{PolyLog}\left (2,1-\frac{2 \left (\sqrt{-d}-\sqrt{e} x\right )}{(x+1) \left (\sqrt{-d}-\sqrt{e}\right )}\right )+\frac{1}{2} \text{PolyLog}\left (2,1-\frac{2 \left (\sqrt{-d}+\sqrt{e} x\right )}{(x+1) \left (\sqrt{-d}+\sqrt{e}\right )}\right )-\text{PolyLog}\left (2,1-\frac{2}{x+1}\right )+\tanh ^{-1}(x) \log \left (d+e x^2\right )-\tanh ^{-1}(x) \log \left (\frac{2 \left (\sqrt{-d}-\sqrt{e} x\right )}{(x+1) \left (\sqrt{-d}-\sqrt{e}\right )}\right )-\tanh ^{-1}(x) \log \left (\frac{2 \left (\sqrt{-d}+\sqrt{e} x\right )}{(x+1) \left (\sqrt{-d}+\sqrt{e}\right )}\right )+2 \log \left (\frac{2}{x+1}\right ) \tanh ^{-1}(x) \]

[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

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Rubi [A]  time = 0.253017, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {206, 2470, 5992, 5920, 2402, 2315, 2447} \[ \frac{1}{2} \text{PolyLog}\left (2,1-\frac{2 \left (\sqrt{-d}-\sqrt{e} x\right )}{(x+1) \left (\sqrt{-d}-\sqrt{e}\right )}\right )+\frac{1}{2} \text{PolyLog}\left (2,1-\frac{2 \left (\sqrt{-d}+\sqrt{e} x\right )}{(x+1) \left (\sqrt{-d}+\sqrt{e}\right )}\right )-\text{PolyLog}\left (2,1-\frac{2}{x+1}\right )+\tanh ^{-1}(x) \log \left (d+e x^2\right )-\tanh ^{-1}(x) \log \left (\frac{2 \left (\sqrt{-d}-\sqrt{e} x\right )}{(x+1) \left (\sqrt{-d}-\sqrt{e}\right )}\right )-\tanh ^{-1}(x) \log \left (\frac{2 \left (\sqrt{-d}+\sqrt{e} x\right )}{(x+1) \left (\sqrt{-d}+\sqrt{e}\right )}\right )+2 \log \left (\frac{2}{x+1}\right ) \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2470

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 5992

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])

Rule 5920

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 2402

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 2315

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

Rule 2447

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]]

Rubi steps

\begin{align*} \int \frac{\log \left (d+e x^2\right )}{1-x^2} \, dx &=\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 \operatorname{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 [C]  time = 0.123114, size = 468, normalized size = 2.16 \[ \frac{1}{2} \left (-\text{PolyLog}\left (2,\frac{\sqrt{d}-i \sqrt{e} x}{\sqrt{d}-i \sqrt{e}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{d}-i \sqrt{e} x}{\sqrt{d}+i \sqrt{e}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{d}+i \sqrt{e} x}{\sqrt{d}-i \sqrt{e}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{d}+i \sqrt{e} x}{\sqrt{d}+i \sqrt{e}}\right )-\log (1-x) \log \left (d+e x^2\right )+\log (x+1) \log \left (d+e x^2\right )+\log (1-x) \log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right )-\log \left (\frac{\sqrt{e} (x-1)}{-\sqrt{e}+i \sqrt{d}}\right ) \log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right )-\log (x+1) \log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right )+\log \left (-\frac{i \sqrt{e} (x+1)}{\sqrt{d}-i \sqrt{e}}\right ) \log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right )+\log (1-x) \log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )-\log \left (\frac{\sqrt{e} (x-1)}{-\sqrt{e}-i \sqrt{d}}\right ) \log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )-\log (x+1) \log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )+\log \left (\frac{i \sqrt{e} (x+1)}{\sqrt{d}+i \sqrt{e}}\right ) \log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[d + e*x^2]/(1 - x^2),x]

[Out]

(Log[1 - x]*Log[((-I)*Sqrt[d])/Sqrt[e] + x] - Log[(Sqrt[e]*(-1 + x))/(I*Sqrt[d] - Sqrt[e])]*Log[((-I)*Sqrt[d])
/Sqrt[e] + x] - Log[1 + x]*Log[((-I)*Sqrt[d])/Sqrt[e] + x] + Log[((-I)*Sqrt[e]*(1 + x))/(Sqrt[d] - I*Sqrt[e])]
*Log[((-I)*Sqrt[d])/Sqrt[e] + x] + Log[1 - x]*Log[(I*Sqrt[d])/Sqrt[e] + x] - Log[(Sqrt[e]*(-1 + x))/((-I)*Sqrt
[d] - Sqrt[e])]*Log[(I*Sqrt[d])/Sqrt[e] + x] - Log[1 + x]*Log[(I*Sqrt[d])/Sqrt[e] + x] + Log[(I*Sqrt[e]*(1 + x
))/(Sqrt[d] + I*Sqrt[e])]*Log[(I*Sqrt[d])/Sqrt[e] + x] - Log[1 - x]*Log[d + e*x^2] + Log[1 + x]*Log[d + e*x^2]
- PolyLog[2, (Sqrt[d] - I*Sqrt[e]*x)/(Sqrt[d] - I*Sqrt[e])] + PolyLog[2, (Sqrt[d] - I*Sqrt[e]*x)/(Sqrt[d] + I
*Sqrt[e])] + PolyLog[2, (Sqrt[d] + I*Sqrt[e]*x)/(Sqrt[d] - I*Sqrt[e])] - PolyLog[2, (Sqrt[d] + I*Sqrt[e]*x)/(S
qrt[d] + I*Sqrt[e])])/2

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Maple [A]  time = 0.096, size = 282, normalized size = 1.3 \begin{align*} -{\frac{\ln \left ( x-1 \right ) \ln \left ( e{x}^{2}+d \right ) }{2}}+{\frac{\ln \left ( x-1 \right ) }{2}\ln \left ({ \left ( - \left ( x-1 \right ) e+\sqrt{-de}-e \right ) \left ( -e+\sqrt{-de} \right ) ^{-1}} \right ) }+{\frac{\ln \left ( x-1 \right ) }{2}\ln \left ({ \left ( \left ( x-1 \right ) e+\sqrt{-de}+e \right ) \left ( e+\sqrt{-de} \right ) ^{-1}} \right ) }+{\frac{1}{2}{\it dilog} \left ({ \left ( - \left ( x-1 \right ) e+\sqrt{-de}-e \right ) \left ( -e+\sqrt{-de} \right ) ^{-1}} \right ) }+{\frac{1}{2}{\it dilog} \left ({ \left ( \left ( x-1 \right ) e+\sqrt{-de}+e \right ) \left ( e+\sqrt{-de} \right ) ^{-1}} \right ) }+{\frac{\ln \left ( 1+x \right ) \ln \left ( e{x}^{2}+d \right ) }{2}}-{\frac{\ln \left ( 1+x \right ) }{2}\ln \left ({ \left ( - \left ( 1+x \right ) e+\sqrt{-de}+e \right ) \left ( e+\sqrt{-de} \right ) ^{-1}} \right ) }-{\frac{\ln \left ( 1+x \right ) }{2}\ln \left ({ \left ( \left ( 1+x \right ) e+\sqrt{-de}-e \right ) \left ( -e+\sqrt{-de} \right ) ^{-1}} \right ) }-{\frac{1}{2}{\it dilog} \left ({ \left ( - \left ( 1+x \right ) e+\sqrt{-de}+e \right ) \left ( e+\sqrt{-de} \right ) ^{-1}} \right ) }-{\frac{1}{2}{\it dilog} \left ({ \left ( \left ( 1+x \right ) e+\sqrt{-de}-e \right ) \left ( -e+\sqrt{-de} \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{\log \left (e x^{2} + d\right )}{x^{2} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\log \left (e x^{2} + d\right )}{x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\log{\left (d + e x^{2} \right )}}{x^{2} - 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\log \left (e x^{2} + d\right )}{x^{2} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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