\(\int \frac {\log (\frac {2 x (d \sqrt {-\frac {e}{d}}+e x)}{d+e x^2})}{d+e x^2} \, dx\) [40]

Optimal result

Integrand size = 39, antiderivative size = 49 \[ \int \frac {\log \left (\frac {2 x \left (d \sqrt {-\frac {e}{d}}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx=-\frac {\sqrt {-\frac {e}{d}} \operatorname {PolyLog}\left (2,1-\frac {2 x \left (d \sqrt {-\frac {e}{d}}+e x\right )}{d+e x^2}\right )}{2 e} \]

[Out]

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

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2497} \[ \int \frac {\log \left (\frac {2 x \left (d \sqrt {-\frac {e}{d}}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx=-\frac {\sqrt {-\frac {e}{d}} \operatorname {PolyLog}\left (2,1-\frac {2 x \left (\sqrt {-\frac {e}{d}} d+e x\right )}{e x^2+d}\right )}{2 e} \]

[In]

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

[Out]

-1/2*(Sqrt[-(e/d)]*PolyLog[2, 1 - (2*x*(d*Sqrt[-(e/d)] + e*x))/(d + e*x^2)])/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 {\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] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(625\) vs. \(2(49)=98\).

Time = 0.31 (sec) , antiderivative size = 625, normalized size of antiderivative = 12.76 \[ \int \frac {\log \left (\frac {2 x \left (d \sqrt {-\frac {e}{d}}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx=\frac {-2 \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )+\log ^2\left (\sqrt {-d}-\sqrt {e} x\right )+2 \log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )-\log ^2\left (\sqrt {-d}+\sqrt {e} x\right )+2 \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {d-\sqrt {-d} \sqrt {e} x}{2 d}\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {d+\sqrt {-d} \sqrt {e} x}{2 d}\right )-2 \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {d \sqrt {-\frac {e}{d}}+e x}{\sqrt {-d} \sqrt {e}+d \sqrt {-\frac {e}{d}}}\right )+2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {e+d \left (-\frac {e}{d}\right )^{3/2} x}{e+\sqrt {-d} \sqrt {e} \sqrt {-\frac {e}{d}}}\right )+2 \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {2 x \left (d \sqrt {-\frac {e}{d}}+e x\right )}{d+e x^2}\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {2 x \left (d \sqrt {-\frac {e}{d}}+e x\right )}{d+e x^2}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {-d}+\sqrt {e} x}{\sqrt {-d}+\frac {\sqrt {e}}{\sqrt {-\frac {e}{d}}}}\right )+2 \operatorname {PolyLog}\left (2,1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )-2 \operatorname {PolyLog}\left (2,\frac {d-\sqrt {-d} \sqrt {e} x}{2 d}\right )+2 \operatorname {PolyLog}\left (2,\frac {d+\sqrt {-d} \sqrt {e} x}{2 d}\right )-2 \operatorname {PolyLog}\left (2,1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )-2 \operatorname {PolyLog}\left (2,\frac {\sqrt {-d} \sqrt {e}-e x}{\sqrt {-d} \sqrt {e}+d \sqrt {-\frac {e}{d}}}\right )}{4 \sqrt {-d} \sqrt {e}} \]

[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])

Maple [C] (verified)

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

Time = 1.49 (sec) , antiderivative size = 238, normalized size of antiderivative = 4.86

method	result	size
risch	\(\frac {\ln \left (2\right ) \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{\sqrt {d e}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+d \right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x \left (e x +d \sqrt {-\frac {e}{d}}\right )}{e \,x^{2}+d}\right )+e \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{\underline {\hspace {1.25 ex}}\alpha e}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{d}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{d}\right )-2 \operatorname {dilog}\left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )-2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )-2 \operatorname {dilog}\left (\frac {\underline {\hspace {1.25 ex}}\alpha e +d \sqrt {-\frac {e}{d}}+\left (x -\underline {\hspace {1.25 ex}}\alpha \right ) e}{\underline {\hspace {1.25 ex}}\alpha e +d \sqrt {-\frac {e}{d}}}\right )-2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\underline {\hspace {1.25 ex}}\alpha e +d \sqrt {-\frac {e}{d}}+\left (x -\underline {\hspace {1.25 ex}}\alpha \right ) e}{\underline {\hspace {1.25 ex}}\alpha e +d \sqrt {-\frac {e}{d}}}\right )}{\underline {\hspace {1.25 ex}}\alpha }}{4 e}\)	\(238\)

[In]

int(ln(2*x*(e*x+d*(-e/d)^(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*(-e/d)^(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+_al
pha)/_alpha))-2*dilog(x/_alpha)-2*ln(x-_alpha)*ln(x/_alpha)-2*dilog((_alpha*e+d*(-e/d)^(1/2)+(x-_alpha)*e)/(_a
lpha*e+d*(-e/d)^(1/2)))-2*ln(x-_alpha)*ln((_alpha*e+d*(-e/d)^(1/2)+(x-_alpha)*e)/(_alpha*e+d*(-e/d)^(1/2)))),_
alpha=RootOf(_Z^2*e+d))

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int \frac {\log \left (\frac {2 x \left (d \sqrt {-\frac {e}{d}}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx=-\frac {\sqrt {-\frac {e}{d}} {\rm Li}_2\left (-\frac {2 \, {\left (e x^{2} + d x \sqrt {-\frac {e}{d}}\right )}}{e x^{2} + d} + 1\right )}{2 \, e} \]

[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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

Maxima [F(-2)]

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

[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 >> 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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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