3.1.41 \(\int \frac {\log (-\frac {2 x (d \sqrt {-\frac {e}{d}}-e x)}{d+e x^2})}{d+e x^2} \, dx\) [41]

Optimal. Leaf size=50 \[ \frac {\sqrt {-\frac {e}{d}} \text {Li}_2\left (1+\frac {2 x \left (d \sqrt {-\frac {e}{d}}-e x\right )}{d+e x^2}\right )}{2 e} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 50, 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 = {2497} \begin {gather*} \frac {\sqrt {-\frac {e}{d}} \text {PolyLog}\left (2,\frac {2 x \left (d \sqrt {-\frac {e}{d}}-e x\right )}{d+e x^2}+1\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]

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(642\) vs. \(2(50)=100\).
time = 0.27, size = 642, normalized size = 12.84 \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 {\sqrt {e} \left (1+\sqrt {-\frac {e}{d}} x\right )}{\sqrt {e}-\sqrt {-d} \sqrt {-\frac {e}{d}}}\right )-2 \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {\sqrt {e} \left (1+\sqrt {-\frac {e}{d}} x\right )}{\sqrt {e}+\sqrt {-d} \sqrt {-\frac {e}{d}}}\right )+2 \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {2 e x \left (\frac {1}{\sqrt {-\frac {e}{d}}}+x\right )}{d+e x^2}\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {2 e x \left (\frac {1}{\sqrt {-\frac {e}{d}}}+x\right )}{d+e x^2}\right )-2 \text {Li}_2\left (\frac {\sqrt {-\frac {e}{d}} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e}+\sqrt {-d} \sqrt {-\frac {e}{d}}}\right )+2 \text {Li}_2\left (\frac {\sqrt {-\frac {e}{d}} \left (\sqrt {-d}+\sqrt {e} x\right )}{-\sqrt {e}+\sqrt {-d} \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 )}{4 \sqrt {-d} \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[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[(Sqrt[e]*(1 + Sqrt[-(e/d)]*x))/(Sqrt[e] - Sqrt[-d]*Sqrt[-(e/d)])] - 2*Log[Sqrt[-d] - Sqrt[e]*x]*Lo
g[(Sqrt[e]*(1 + Sqrt[-(e/d)]*x))/(Sqrt[e] + Sqrt[-d]*Sqrt[-(e/d)])] + 2*Log[Sqrt[-d] - Sqrt[e]*x]*Log[(2*e*x*(
1/Sqrt[-(e/d)] + x))/(d + e*x^2)] - 2*Log[Sqrt[-d] + Sqrt[e]*x]*Log[(2*e*x*(1/Sqrt[-(e/d)] + x))/(d + e*x^2)]
- 2*PolyLog[2, (Sqrt[-(e/d)]*(Sqrt[-d] - Sqrt[e]*x))/(Sqrt[e] + Sqrt[-d]*Sqrt[-(e/d)])] + 2*PolyLog[2, (Sqrt[-
(e/d)]*(Sqrt[-d] + Sqrt[e]*x))/(-Sqrt[e] + Sqrt[-d]*Sqrt[-(e/d)])] + 2*PolyLog[2, 1 + (Sqrt[e]*x)/Sqrt[-d]] -
2*PolyLog[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)])/(4*Sqrt[-d]*Sqrt[e])

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Maple [F]
time = 0.21, 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)

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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.

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Fricas [A]
time = 0.37, size = 48, 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)

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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

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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)

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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)

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