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

Optimal. Leaf size=49 \[ \frac {\text {Li}_2\left (1-\frac {2 x \left (e x+\sqrt {d} \sqrt {-e}\right )}{e x^2+d}\right )}{2 \sqrt {d} \sqrt {-e}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2447} \[ \frac {\text {PolyLog}\left (2,1-\frac {2 x \left (\sqrt {d} \sqrt {-e}+e x\right )}{d+e x^2}\right )}{2 \sqrt {d} \sqrt {-e}} \]

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 = 0.35, size = 641, normalized size = 13.08 \[ \frac {2 \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )-2 \text {Li}_2\left (\frac {d-\sqrt {-d} \sqrt {e} x}{2 d}\right )+2 \text {Li}_2\left (\frac {d+\sqrt {-d} \sqrt {e} x}{2 d}\right )-2 \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}+1\right )-2 \text {Li}_2\left (\frac {\sqrt {-d} \sqrt {e}-e x}{\sqrt {d} \sqrt {-e}+\sqrt {-d} \sqrt {e}}\right )+2 \text {Li}_2\left (\frac {e x+\sqrt {-d} \sqrt {e}}{\sqrt {-d} \sqrt {e}-\sqrt {d} \sqrt {-e}}\right )+2 \log \left (\frac {2 x \left (\sqrt {d} \sqrt {-e}+e x\right )}{d+e x^2}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {2 x \left (\sqrt {d} \sqrt {-e}+e x\right )}{d+e x^2}\right )+\log ^2\left (\sqrt {-d}-\sqrt {e} x\right )-\log ^2\left (\sqrt {-d}+\sqrt {e} x\right )-2 \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )+2 \log \left (\frac {d-\sqrt {-d} \sqrt {e} x}{2 d}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )-2 \log \left (\frac {\sqrt {d} \sqrt {-e}+e x}{\sqrt {d} \sqrt {-e}+\sqrt {-d} \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )+2 \log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {\sqrt {-d} \sqrt {e} x+d}{2 d}\right )+2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {\sqrt {d} \sqrt {-e}+e x}{\sqrt {d} \sqrt {-e}-\sqrt {-d} \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}} \]

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 + Sq
rt[-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]) + Sqrt[-d]
*Sqrt[e])])/(4*Sqrt[-d]*Sqrt[e])

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 43, normalized size = 0.88 \[ -\frac {\sqrt {-e} {\rm Li}_2\left (-\frac {2 \, {\left (e x^{2} + \sqrt {d} \sqrt {-e} x\right )}}{e x^{2} + d} + 1\right )}{2 \, \sqrt {d} e} \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\frac {2 \, {\left (e x + \sqrt {d} \sqrt {-e}\right )} x}{e x^{2} + d}\right )}{e x^{2} + d}\,{d x} \]

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)

________________________________________________________________________________________

maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (\frac {2 \left (e x +\sqrt {-e}\, \sqrt {d}\right ) x}{e \,x^{2}+d}\right )}{e \,x^{2}+d}\, dx \]

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)

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\ln \left (\frac {2\,x\,\left (e\,x+\sqrt {d}\,\sqrt {-e}\right )}{e\,x^2+d}\right )}{e\,x^2+d} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]

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

________________________________________________________________________________________