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

Optimal. Leaf size=53 \[ -\frac{\text{PolyLog}\left (2,\frac{2 \sqrt{e} x \left (\sqrt{-d}-\sqrt{e} x\right )}{d+e x^2}+1\right )}{2 \sqrt{-d} \sqrt{e}} \]

[Out]

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

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Rubi [A]  time = 0.0964644, antiderivative size = 53, 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} \[ -\frac{\text{PolyLog}\left (2,\frac{2 \sqrt{e} x \left (\sqrt{-d}-\sqrt{e} x\right )}{d+e x^2}+1\right )}{2 \sqrt{-d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [B]  time = 0.208992, size = 320, normalized size = 6.04 \[ \frac{2 \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}+1\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{-d} \sqrt{e} x+d}{2 d}\right )-2 \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right )-2 \log \left (\frac{2 \left (e x^2-\sqrt{-d} \sqrt{e} x\right )}{d+e x^2}\right ) \log \left (\sqrt{-d}+\sqrt{e} x\right )+2 \log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (\frac{2 \left (e x^2-\sqrt{-d} \sqrt{e} x\right )}{d+e x^2}\right )-\log ^2\left (\sqrt{-d}+\sqrt{e} x\right )+2 \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}\right ) \log \left (\sqrt{-d}+\sqrt{e} x\right )-2 \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )+2 \log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (\frac{d-\sqrt{-d} \sqrt{e} x}{2 d}\right )}{4 \sqrt{-d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[(2*x*((d*Sqrt[e])/Sqrt[-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] + 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[(2*(-(Sqrt[-d]*Sqrt[e]*x) + e*x^2))/(d + e*x^2)] - 2*Log[Sqrt[-d] + Sqrt[e]*x]*Log[(
2*(-(Sqrt[-d]*Sqrt[e]*x) + e*x^2))/(d + e*x^2)] + 2*PolyLog[2, 1 + (Sqrt[e]*x)/Sqrt[-d]] + 2*PolyLog[2, (d + S
qrt[-d]*Sqrt[e]*x)/(2*d)] - 2*PolyLog[2, 1 + (d*Sqrt[e]*x)/(-d)^(3/2)])/(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+{\frac{d\sqrt{e}}{\sqrt{-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^(1/2)/(-d)^(1/2))/(e*x^2+d))/(e*x^2+d),x)

[Out]

int(ln(2*x*(e*x+d*e^(1/2)/(-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^(1/2)/(-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 = 1.94567, size = 109, normalized size = 2.06 \begin{align*} \frac{\sqrt{-d}{\rm Li}_2\left (-\frac{2 \,{\left (e x^{2} - \sqrt{-d} \sqrt{e} x\right )}}{e x^{2} + d} + 1\right )}{2 \, d \sqrt{e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-d)*dilog(-2*(e*x^2 - sqrt(-d)*sqrt(e)*x)/(e*x^2 + d) + 1)/(d*sqrt(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*e**(1/2)/(-d)**(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*} \mathit{sage}_{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage2