3.4.37 \(\int \frac {\log (\frac {2 a}{a+b x})}{a^2-b^2 x^2} \, dx\) [337]

Optimal. Leaf size=24 \[ \frac {\text {Li}_2\left (1-\frac {2 a}{a+b x}\right )}{2 a b} \]

[Out]

1/2*polylog(2,1-2*a/(b*x+a))/a/b

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Rubi [A]
time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2449, 2352} \begin {gather*} \frac {\text {PolyLog}\left (2,1-\frac {2 a}{a+b x}\right )}{2 a b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[(2*a)/(a + b*x)]/(a^2 - b^2*x^2),x]

[Out]

PolyLog[2, 1 - (2*a)/(a + b*x)]/(2*a*b)

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rubi steps

\begin {align*} \int \frac {\log \left (\frac {2 a}{a+b x}\right )}{a^2-b^2 x^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\log (2 a x)}{1-2 a x} \, dx,x,\frac {1}{a+b x}\right )}{b}\\ &=\frac {\text {Li}_2\left (1-\frac {2 a}{a+b x}\right )}{2 a b}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 27, normalized size = 1.12 \begin {gather*} \frac {\text {Li}_2\left (\frac {-a+b x}{a+b x}\right )}{2 a b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[(2*a)/(a + b*x)]/(a^2 - b^2*x^2),x]

[Out]

PolyLog[2, (-a + b*x)/(a + b*x)]/(2*a*b)

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Maple [A]
time = 0.72, size = 20, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {\dilog \left (\frac {2 a}{b x +a}\right )}{2 b a}\)	\(20\)
default	\(\frac {\dilog \left (\frac {2 a}{b x +a}\right )}{2 b a}\)	\(20\)
risch	\(\frac {\dilog \left (\frac {2 a}{b x +a}\right )}{2 b a}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(2*a/(b*x+a))/(-b^2*x^2+a^2),x,method=_RETURNVERBOSE)

[Out]

1/2/b/a*dilog(2*a/(b*x+a))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (21) = 42\).
time = 0.30, size = 120, normalized size = 5.00 \begin {gather*} \frac {1}{4} \, b {\left (\frac {\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (b x - a\right )}{a b^{2}} + \frac {2 \, {\left (\log \left (b x + a\right ) \log \left (-\frac {b x + a}{2 \, a} + 1\right ) + {\rm Li}_2\left (\frac {b x + a}{2 \, a}\right )\right )}}{a b^{2}}\right )} + \frac {1}{2} \, {\left (\frac {\log \left (b x + a\right )}{a b} - \frac {\log \left (b x - a\right )}{a b}\right )} \log \left (\frac {2 \, a}{b x + a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(2*a/(b*x+a))/(-b^2*x^2+a^2),x, algorithm="maxima")

[Out]

1/4*b*((log(b*x + a)^2 - 2*log(b*x + a)*log(b*x - a))/(a*b^2) + 2*(log(b*x + a)*log(-1/2*(b*x + a)/a + 1) + di
log(1/2*(b*x + a)/a))/(a*b^2)) + 1/2*(log(b*x + a)/(a*b) - log(b*x - a)/(a*b))*log(2*a/(b*x + a))

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Fricas [A]
time = 0.38, size = 21, normalized size = 0.88 \begin {gather*} \frac {{\rm Li}_2\left (-\frac {2 \, a}{b x + a} + 1\right )}{2 \, a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(2*a/(b*x+a))/(-b^2*x^2+a^2),x, algorithm="fricas")

[Out]

1/2*dilog(-2*a/(b*x + a) + 1)/(a*b)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\log {\left (2 \right )}}{- a^{2} + b^{2} x^{2}}\, dx - \int \frac {\log {\left (\frac {a}{a + b x} \right )}}{- a^{2} + b^{2} x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(2*a/(b*x+a))/(-b**2*x**2+a**2),x)

[Out]

-Integral(log(2)/(-a**2 + b**2*x**2), x) - Integral(log(a/(a + b*x))/(-a**2 + b**2*x**2), x)

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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*a/(b*x+a))/(-b^2*x^2+a^2),x, algorithm="giac")

[Out]

integrate(-log(2*a/(b*x + a))/(b^2*x^2 - a^2), x)

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Mupad [B]
time = 0.28, size = 19, normalized size = 0.79 \begin {gather*} \frac {{\mathrm {Li}}_{\mathrm {2}}\left (\frac {2\,a}{a+b\,x}\right )}{2\,a\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log((2*a)/(a + b*x))/(a^2 - b^2*x^2),x)

[Out]

dilog((2*a)/(a + b*x))/(2*a*b)

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