\(\int \frac {\log (\frac {2 x (\sqrt {d} \sqrt {-e}+e x)}{d+e x^2})}{d+e x^2} \, dx\) [44]

Optimal result

Integrand size = 40, antiderivative size = 49 \[ \int \frac {\log \left (\frac {2 x \left (\sqrt {d} \sqrt {-e}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx=\frac {\operatorname {PolyLog}\left (2,1-\frac {2 x \left (\sqrt {d} \sqrt {-e}+e x\right )}{d+e x^2}\right )}{2 \sqrt {d} \sqrt {-e}} \]

[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] (verified)

Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2497} \[ \int \frac {\log \left (\frac {2 x \left (\sqrt {d} \sqrt {-e}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx=\frac {\operatorname {PolyLog}\left (2,1-\frac {2 x \left (e x+\sqrt {d} \sqrt {-e}\right )}{e x^2+d}\right )}{2 \sqrt {d} \sqrt {-e}} \]

[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 2497

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*} \text {integral}& = \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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(641\) vs. \(2(49)=98\).

Time = 0.25 (sec) , antiderivative size = 641, normalized size of antiderivative = 13.08 \[ \int \frac {\log \left (\frac {2 x \left (\sqrt {d} \sqrt {-e}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx=\frac {-2 \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )+\log ^2\left (\sqrt {-d}-\sqrt {e} x\right )+2 \log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )-\log ^2\left (\sqrt {-d}+\sqrt {e} x\right )+2 \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {d-\sqrt {-d} \sqrt {e} x}{2 d}\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {d+\sqrt {-d} \sqrt {e} x}{2 d}\right )+2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {\sqrt {d} \sqrt {-e}+e x}{\sqrt {d} \sqrt {-e}-\sqrt {-d} \sqrt {e}}\right )-2 \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {\sqrt {d} \sqrt {-e}+e x}{\sqrt {d} \sqrt {-e}+\sqrt {-d} \sqrt {e}}\right )+2 \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {2 x \left (\sqrt {d} \sqrt {-e}+e x\right )}{d+e x^2}\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {2 x \left (\sqrt {d} \sqrt {-e}+e x\right )}{d+e x^2}\right )+2 \operatorname {PolyLog}\left (2,1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )-2 \operatorname {PolyLog}\left (2,\frac {d-\sqrt {-d} \sqrt {e} x}{2 d}\right )+2 \operatorname {PolyLog}\left (2,\frac {d+\sqrt {-d} \sqrt {e} x}{2 d}\right )-2 \operatorname {PolyLog}\left (2,1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )-2 \operatorname {PolyLog}\left (2,\frac {\sqrt {-d} \sqrt {e}-e x}{\sqrt {d} \sqrt {-e}+\sqrt {-d} \sqrt {e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {-d} \sqrt {e}+e x}{-\sqrt {d} \sqrt {-e}+\sqrt {-d} \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}} \]

[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])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.61 (sec) , antiderivative size = 233, normalized size of antiderivative = 4.76

method	result	size
risch	\(\frac {\ln \left (2\right ) \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{\sqrt {d e}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+d \right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x \left (e x +\sqrt {d}\, \sqrt {-e}\right )}{e \,x^{2}+d}\right )+e \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{\underline {\hspace {1.25 ex}}\alpha e}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{d}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{d}\right )-2 \operatorname {dilog}\left (\frac {\underline {\hspace {1.25 ex}}\alpha e +\sqrt {d}\, \sqrt {-e}+\left (x -\underline {\hspace {1.25 ex}}\alpha \right ) e}{\underline {\hspace {1.25 ex}}\alpha e +\sqrt {d}\, \sqrt {-e}}\right )-2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\underline {\hspace {1.25 ex}}\alpha e +\sqrt {d}\, \sqrt {-e}+\left (x -\underline {\hspace {1.25 ex}}\alpha \right ) e}{\underline {\hspace {1.25 ex}}\alpha e +\sqrt {d}\, \sqrt {-e}}\right )-2 \operatorname {dilog}\left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )-2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )}{\underline {\hspace {1.25 ex}}\alpha }}{4 e}\)	\(233\)

[In]

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

[Out]

ln(2)/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/4/e*sum(1/_alpha*(2*ln(x-_alpha)*ln(x*(e*x+d^(1/2)*(-e)^(1/2))/(e*
x^2+d))+e*(1/_alpha/e*ln(x-_alpha)^2+2*_alpha/d*ln(x-_alpha)*ln(1/2*(x+_alpha)/_alpha)+2*_alpha/d*dilog(1/2*(x
+_alpha)/_alpha))-2*dilog((_alpha*e+d^(1/2)*(-e)^(1/2)+(x-_alpha)*e)/(_alpha*e+d^(1/2)*(-e)^(1/2)))-2*ln(x-_al
pha)*ln((_alpha*e+d^(1/2)*(-e)^(1/2)+(x-_alpha)*e)/(_alpha*e+d^(1/2)*(-e)^(1/2)))-2*dilog(x/_alpha)-2*ln(x-_al
pha)*ln(x/_alpha)),_alpha=RootOf(_Z^2*e+d))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int \frac {\log \left (\frac {2 x \left (\sqrt {d} \sqrt {-e}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx=-\frac {\sqrt {-e} {\rm Li}_2\left (-\frac {2 \, {\left (e x^{2} + \sqrt {d} \sqrt {-e} x\right )}}{e x^{2} + d} + 1\right )}{2 \, \sqrt {d} e} \]

[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)

Sympy [F(-2)]

Exception generated. \[ \int \frac {\log \left (\frac {2 x \left (\sqrt {d} \sqrt {-e}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx=\text {Exception raised: AttributeError} \]

[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 >> 'NoneType' object has no attribute 'primitive'

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log \left (\frac {2 x \left (\sqrt {d} \sqrt {-e}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]

[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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F(-2)]

Exception generated. \[ \int \frac {\log \left (\frac {2 x \left (\sqrt {d} \sqrt {-e}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx=\text {Exception raised: TypeError} \]

[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]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (\frac {2 x \left (\sqrt {d} \sqrt {-e}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx=\int \frac {\ln \left (\frac {2\,x\,\left (e\,x+\sqrt {d}\,\sqrt {-e}\right )}{e\,x^2+d}\right )}{e\,x^2+d} \,d x \]

[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)