3.1.42 \(\int \frac {\log (\frac {2 x (\frac {d \sqrt {e}}{\sqrt {-d}}+e x)}{d+e x^2})}{d+e x^2} \, dx\) [42]

Optimal. Leaf size=53 \[ -\frac {\text {Li}_2\left (1+\frac {2 \sqrt {e} x \left (\sqrt {-d}-\sqrt {e} x\right )}{d+e x^2}\right )}{2 \sqrt {-d} \sqrt {e}} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2497} \begin {gather*} -\frac {\text {PolyLog}\left (2,\frac {2 \sqrt {e} x \left (\sqrt {-d}-\sqrt {e} x\right )}{d+e x^2}+1\right )}{2 \sqrt {-d} \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*PolyLog[2, 1 + (2*Sqrt[e]*x*(Sqrt[-d] - Sqrt[e]*x))/(d + e*x^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*} \int \frac {\log \left (\frac {2 x \left (\frac {d \sqrt {e}}{\sqrt {-d}}+e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx &=-\frac {\text {Li}_2\left (1+\frac {2 \sqrt {e} x \left (\sqrt {-d}-\sqrt {e} x\right )}{d+e x^2}\right )}{2 \sqrt {-d} \sqrt {e}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(320\) vs. \(2(53)=106\).
time = 0.15, size = 320, normalized size = 6.04 \begin {gather*} \frac {-2 \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}\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 )-\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 {2 \left (-\sqrt {-d} \sqrt {e} x+e x^2\right )}{d+e x^2}\right )-2 \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {2 \left (-\sqrt {-d} \sqrt {e} x+e x^2\right )}{d+e x^2}\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 (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])/Sqrt[-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] + 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[(2*(-(Sqrt[-d]*Sqrt[e]*x) + e*x^2))/(d + e*x^2)] - 2*Log[Sqrt[-d] + Sqrt[e]*x]*Log[(
2*(-(Sqrt[-d]*Sqrt[e]*x) + e*x^2))/(d + e*x^2)] + 2*PolyLog[2, 1 + (Sqrt[e]*x)/Sqrt[-d]] + 2*PolyLog[2, (d + S
qrt[-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 = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (\frac {2 x \left (e x +\frac {d \sqrt {e}}{\sqrt {-d}}\right )}{e \,x^{2}+d}\right )}{e \,x^{2}+d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(ln(2*x*(e*x+d*e^(1/2)/(-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^(1/2)/(-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 = 0.34, size = 44, normalized size = 0.83 \begin {gather*} \frac {\sqrt {-d} {\rm Li}_2\left (-\frac {2 \, {\left (x^{2} e - \sqrt {-d} x e^{\frac {1}{2}}\right )}}{x^{2} e + d} + 1\right ) e^{\left (-\frac {1}{2}\right )}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError >> 'NoneType' object has no attribute 'primitive'

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

[Out]

integrate(log(2*(x*e + d*e^(1/2)/sqrt(-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-\sqrt {-d}\,\sqrt {e}\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)^(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)

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