3.4.70 \(\int \frac {\tanh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx\) [370]

Optimal. Leaf size=75 \[ -2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\text {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\text {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6165} \begin {gather*} \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]/(x*Sqrt[1 - a^2*x^2]),x]

[Out]

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

Rule 6165

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Simp[(-2/Sqrt[d])*(a +
b*ArcTanh[c*x])*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]], x] + (Simp[(b/Sqrt[d])*PolyLog[2, -Sqrt[1 - c*x]/Sqrt[1
+ c*x]], x] - Simp[(b/Sqrt[d])*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]], x]) /; FreeQ[{a, b, c, d, e}, x] && Eq
Q[c^2*d + e, 0] && GtQ[d, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.07, size = 57, normalized size = 0.76 \begin {gather*} \tanh ^{-1}(a x) \left (\log \left (1-e^{-\tanh ^{-1}(a x)}\right )-\log \left (1+e^{-\tanh ^{-1}(a x)}\right )\right )+\text {PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )-\text {PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTanh[a*x]/(x*Sqrt[1 - a^2*x^2]),x]

[Out]

ArcTanh[a*x]*(Log[1 - E^(-ArcTanh[a*x])] - Log[1 + E^(-ArcTanh[a*x])]) + PolyLog[2, -E^(-ArcTanh[a*x])] - Poly
Log[2, E^(-ArcTanh[a*x])]

________________________________________________________________________________________

Maple [A]
time = 1.97, size = 99, normalized size = 1.32

method	result	size
default	\(\arctanh \left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)/x/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)*ln(1+(a*x+1)/
(-a^2*x^2+1)^(1/2))-polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)/x/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(arctanh(a*x)/(sqrt(-a^2*x^2 + 1)*x), x)

________________________________________________________________________________________

Fricas [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(arctanh(a*x)/x/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*x^2 + 1)*arctanh(a*x)/(a^2*x^3 - x), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atanh}{\left (a x \right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)/x/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(atanh(a*x)/(x*sqrt(-(a*x - 1)*(a*x + 1))), x)

________________________________________________________________________________________

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(arctanh(a*x)/x/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(arctanh(a*x)/(sqrt(-a^2*x^2 + 1)*x), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {atanh}\left (a\,x\right )}{x\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a*x)/(x*(1 - a^2*x^2)^(1/2)),x)

[Out]

int(atanh(a*x)/(x*(1 - a^2*x^2)^(1/2)), x)

________________________________________________________________________________________