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

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

[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]
time = 0.07, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2497} \begin {gather*} -\frac {\sqrt {-\frac {e}{d}} \text {PolyLog}\left (2,1-\frac {2 x \left (d \sqrt {-\frac {e}{d}}+e x\right )}{d+e x^2}\right )}{2 e} \end {gather*}

Antiderivative was successfully verified.

[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*} \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}} \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] Leaf count is larger than twice the leaf count of optimal. \(625\) vs. \(2(49)=98\).
time = 0.31, size = 625, normalized size = 12.76 \begin {gather*} \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 \text {Li}_2\left (\frac {\sqrt {-d}+\sqrt {e} x}{\sqrt {-d}+\frac {\sqrt {e}}{\sqrt {-\frac {e}{d}}}}\right )+2 \text {Li}_2\left (1+\frac {\sqrt {e} x}{\sqrt {-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 {-d} \sqrt {e} x}{2 d}\right )-2 \text {Li}_2\left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )-2 \text {Li}_2\left (\frac {\sqrt {-d} \sqrt {e}-e x}{\sqrt {-d} \sqrt {e}+d \sqrt {-\frac {e}{d}}}\right )}{4 \sqrt {-d} \sqrt {e}} \end {gather*}

Warning: Unable to verify antiderivative.

[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 [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (\frac {2 x \left (e x +d \sqrt {-\frac {e}{d}}\right )}{e \,x^{2}+d}\right )}{e \,x^{2}+d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 47, normalized size = 0.96 \begin {gather*} -\frac {1}{2} \, \sqrt {-\frac {e}{d}} {\rm Li}_2\left (-\frac {2 \, {\left (x^{2} e + d x \sqrt {-\frac {e}{d}}\right )}}{x^{2} e + d} + 1\right ) e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________