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

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

[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 = 0.10, 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 = {2447} \[ -\frac {\sqrt {-\frac {e}{d}} \text {PolyLog}\left (2,1-\frac {2 x \left (d \sqrt {-\frac {e}{d}}+e x\right )}{d+e x^2}\right )}{2 e} \]

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 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 (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]  time = 0.48, size = 625, normalized size = 12.76 \[ \frac {2 \text {Li}_2\left (\frac {\sqrt {e} x+\sqrt {-d}}{\sqrt {-d}+\frac {\sqrt {e}}{\sqrt {-\frac {e}{d}}}}\right )+2 \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )-2 \text {Li}_2\left (\frac {d-\sqrt {-d} \sqrt {e} x}{2 d}\right )+2 \text {Li}_2\left (\frac {d+\sqrt {-d} \sqrt {e} x}{2 d}\right )-2 \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}+1\right )-2 \text {Li}_2\left (\frac {\sqrt {-d} \sqrt {e}-e x}{\sqrt {-\frac {e}{d}} d+\sqrt {-d} \sqrt {e}}\right )+2 \log \left (\frac {2 x \left (d \sqrt {-\frac {e}{d}}+e x\right )}{d+e x^2}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {2 x \left (d \sqrt {-\frac {e}{d}}+e x\right )}{d+e x^2}\right )+\log ^2\left (\sqrt {-d}-\sqrt {e} x\right )-\log ^2\left (\sqrt {-d}+\sqrt {e} x\right )-2 \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )+2 \log \left (\frac {d-\sqrt {-d} \sqrt {e} x}{2 d}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )-2 \log \left (\frac {d \sqrt {-\frac {e}{d}}+e x}{d \sqrt {-\frac {e}{d}}+\sqrt {-d} \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )+2 \log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {\sqrt {-d} \sqrt {e} x+d}{2 d}\right )+2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {d x \left (-\frac {e}{d}\right )^{3/2}+e}{\sqrt {-d} \sqrt {e} \sqrt {-\frac {e}{d}}+e}\right )}{4 \sqrt {-d} \sqrt {e}} \]

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])

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fricas [A]  time = 0.44, size = 44, normalized size = 0.90 \[ -\frac {\sqrt {-\frac {e}{d}} {\rm Li}_2\left (-\frac {2 \, {\left (e x^{2} + d x \sqrt {-\frac {e}{d}}\right )}}{e x^{2} + d} + 1\right )}{2 \, e} \]

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

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(2*x*(e*x+d*(-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 \[ \text {Exception raised: RuntimeError} \]

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

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

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

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