### 3.45 $$\int \frac{\log (-\frac{2 x (\sqrt{d} \sqrt{-e}-e x)}{d+e x^2})}{d+e x^2} \, dx$$

Optimal. Leaf size=50 $-\frac{\text{PolyLog}\left (2,\frac{2 x \left (\sqrt{d} \sqrt{-e}-e x\right )}{d+e x^2}+1\right )}{2 \sqrt{d} \sqrt{-e}}$

[Out]

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

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Rubi [A]  time = 0.0783351, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 41, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.024, Rules used = {2447} $-\frac{\text{PolyLog}\left (2,\frac{2 x \left (\sqrt{d} \sqrt{-e}-e x\right )}{d+e x^2}+1\right )}{2 \sqrt{d} \sqrt{-e}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [B]  time = 0.251433, size = 645, normalized size = 12.9 $\frac{2 \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}+1\right )-2 \text{PolyLog}\left (2,\frac{d-\sqrt{-d} \sqrt{e} x}{2 d}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{-d} \sqrt{e} x+d}{2 d}\right )-2 \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right )-2 \text{PolyLog}\left (2,\frac{e x-\sqrt{-d} \sqrt{e}}{\sqrt{d} \sqrt{-e}-\sqrt{-d} \sqrt{e}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{-d} \sqrt{e}+e x}{\sqrt{d} \sqrt{-e}+\sqrt{-d} \sqrt{e}}\right )+2 \log \left (\frac{2 x \left (e x-\sqrt{d} \sqrt{-e}\right )}{d+e x^2}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )-2 \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{2 x \left (e x-\sqrt{d} \sqrt{-e}\right )}{d+e x^2}\right )+\log ^2\left (\sqrt{-d}-\sqrt{e} x\right )-\log ^2\left (\sqrt{-d}+\sqrt{e} x\right )-2 \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )+2 \log \left (\frac{d-\sqrt{-d} \sqrt{e} x}{2 d}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )-2 \log \left (\frac{\sqrt{d} \sqrt{-e}-e x}{\sqrt{d} \sqrt{-e}-\sqrt{-d} \sqrt{e}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )+2 \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}\right ) \log \left (\sqrt{-d}+\sqrt{e} x\right )-2 \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{\sqrt{-d} \sqrt{e} x+d}{2 d}\right )+2 \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{\sqrt{d} \sqrt{-e}-e x}{\sqrt{d} \sqrt{-e}+\sqrt{-d} \sqrt{e}}\right )}{4 \sqrt{-d} \sqrt{e}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[(-2*x*(Sqrt[d]*Sqrt[-e] - 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[(Sqrt[d]*Sqrt[-e] - e*x)/(Sqrt[d]*Sqrt[-e] - Sqrt[-d]*Sqrt[e])] + 2*Log[Sqrt[-d] + Sqrt[e]*x]*Log[
(Sqrt[d]*Sqrt[-e] - e*x)/(Sqrt[d]*Sqrt[-e] + Sqrt[-d]*Sqrt[e])] + 2*Log[Sqrt[-d] - Sqrt[e]*x]*Log[(2*x*(-(Sqrt
[d]*Sqrt[-e]) + e*x))/(d + e*x^2)] - 2*Log[Sqrt[-d] + Sqrt[e]*x]*Log[(2*x*(-(Sqrt[d]*Sqrt[-e]) + e*x))/(d + e*
x^2)] + 2*PolyLog[2, 1 + (Sqrt[e]*x)/Sqrt[-d]] - 2*PolyLog[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] - Sqrt[-d]*Sqrt[e])] + 2*PolyLog[2, (Sqrt[-d]*Sqrt[e] + e*x)/(Sqrt[d]*Sqrt[-e] + Sq
rt[-d]*Sqrt[e])])/(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 \left ( -ex+\sqrt{d}\sqrt{-e} \right ) }{e{x}^{2}+d}} \right ) }\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.84806, size = 109, normalized size = 2.18 \begin{align*} \frac{\sqrt{-e}{\rm Li}_2\left (-\frac{2 \,{\left (e x^{2} - \sqrt{d} \sqrt{-e} x\right )}}{e x^{2} + d} + 1\right )}{2 \, \sqrt{d} e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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