### 3.40 $$\int \frac{\log (\frac{2 x (d \sqrt{-\frac{e}{d}}+e x)}{d+e x^2})}{d+e x^2} \, dx$$

Optimal. Leaf size=49 $-\frac{\sqrt{-\frac{e}{d}} \text{PolyLog}\left (2,1-\frac{2 x \left (d \sqrt{-\frac{e}{d}}+e x\right )}{d+e x^2}\right )}{2 e}$

[Out]

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

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Rubi [A]  time = 0.099256, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.026, Rules used = {2447} $-\frac{\sqrt{-\frac{e}{d}} \text{PolyLog}\left (2,1-\frac{2 x \left (d \sqrt{-\frac{e}{d}}+e x\right )}{d+e x^2}\right )}{2 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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 (\frac{2 x \left (d \sqrt{-\frac{e}{d}}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx &=-\frac{\sqrt{-\frac{e}{d}} \text{Li}_2\left (1-\frac{2 x \left (d \sqrt{-\frac{e}{d}}+e x\right )}{d+e x^2}\right )}{2 e}\\ \end{align*}

Mathematica [B]  time = 0.432738, size = 625, normalized size = 12.76 $\frac{2 \text{PolyLog}\left (2,\frac{\sqrt{-d}+\sqrt{e} x}{\frac{\sqrt{e}}{\sqrt{-\frac{e}{d}}}+\sqrt{-d}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}+1\right )-2 \text{PolyLog}\left (2,\frac{d-\sqrt{-d} \sqrt{e} x}{2 d}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{-d} \sqrt{e} x+d}{2 d}\right )-2 \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right )-2 \text{PolyLog}\left (2,\frac{\sqrt{-d} \sqrt{e}-e x}{d \sqrt{-\frac{e}{d}}+\sqrt{-d} \sqrt{e}}\right )+2 \log \left (\frac{2 x \left (d \sqrt{-\frac{e}{d}}+e x\right )}{d+e x^2}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )-2 \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{2 x \left (d \sqrt{-\frac{e}{d}}+e x\right )}{d+e x^2}\right )+\log ^2\left (\sqrt{-d}-\sqrt{e} x\right )-\log ^2\left (\sqrt{-d}+\sqrt{e} x\right )-2 \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )+2 \log \left (\frac{d-\sqrt{-d} \sqrt{e} x}{2 d}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )-2 \log \left (\frac{d \sqrt{-\frac{e}{d}}+e x}{d \sqrt{-\frac{e}{d}}+\sqrt{-d} \sqrt{e}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )+2 \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}\right ) \log \left (\sqrt{-d}+\sqrt{e} x\right )-2 \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{\sqrt{-d} \sqrt{e} x+d}{2 d}\right )+2 \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{d x \left (-\frac{e}{d}\right )^{3/2}+e}{\sqrt{-d} \sqrt{e} \sqrt{-\frac{e}{d}}+e}\right )}{4 \sqrt{-d} \sqrt{e}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*Log[(Sqrt[e]*x)/Sqrt[-d]]*Log[Sqrt[-d] - Sqrt[e]*x] + Log[Sqrt[-d] - Sqrt[e]*x]^2 + 2*Log[(d*Sqrt[e]*x)/(-
d)^(3/2)]*Log[Sqrt[-d] + Sqrt[e]*x] - Log[Sqrt[-d] + Sqrt[e]*x]^2 + 2*Log[Sqrt[-d] - Sqrt[e]*x]*Log[(d - Sqrt[
-d]*Sqrt[e]*x)/(2*d)] - 2*Log[Sqrt[-d] + Sqrt[e]*x]*Log[(d + Sqrt[-d]*Sqrt[e]*x)/(2*d)] - 2*Log[Sqrt[-d] - Sqr
t[e]*x]*Log[(d*Sqrt[-(e/d)] + e*x)/(Sqrt[-d]*Sqrt[e] + d*Sqrt[-(e/d)])] + 2*Log[Sqrt[-d] + Sqrt[e]*x]*Log[(e +
d*(-(e/d))^(3/2)*x)/(e + Sqrt[-d]*Sqrt[e]*Sqrt[-(e/d)])] + 2*Log[Sqrt[-d] - Sqrt[e]*x]*Log[(2*x*(d*Sqrt[-(e/d
)] + e*x))/(d + e*x^2)] - 2*Log[Sqrt[-d] + Sqrt[e]*x]*Log[(2*x*(d*Sqrt[-(e/d)] + e*x))/(d + e*x^2)] + 2*PolyLo
g[2, (Sqrt[-d] + Sqrt[e]*x)/(Sqrt[-d] + Sqrt[e]/Sqrt[-(e/d)])] + 2*PolyLog[2, 1 + (Sqrt[e]*x)/Sqrt[-d]] - 2*Po
lyLog[2, (d - Sqrt[-d]*Sqrt[e]*x)/(2*d)] + 2*PolyLog[2, (d + Sqrt[-d]*Sqrt[e]*x)/(2*d)] - 2*PolyLog[2, 1 + (d*
Sqrt[e]*x)/(-d)^(3/2)] - 2*PolyLog[2, (Sqrt[-d]*Sqrt[e] - e*x)/(Sqrt[-d]*Sqrt[e] + d*Sqrt[-(e/d)])])/(4*Sqrt[-
d]*Sqrt[e])

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{e{x}^{2}+d}\ln \left ( 2\,{\frac{x}{e{x}^{2}+d} \left ( ex+d\sqrt{-{\frac{e}{d}}} \right ) } \right ) }\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.94846, size = 95, normalized size = 1.94 \begin{align*} -\frac{\sqrt{-\frac{e}{d}}{\rm Li}_2\left (-\frac{2 \,{\left (e x^{2} + d x \sqrt{-\frac{e}{d}}\right )}}{e x^{2} + d} + 1\right )}{2 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError