3.6.7 \(\int \frac {\tanh ^{-1}(a+b x)}{1-x^2} \, dx\) [507]

Optimal. Leaf size=203 \[ \frac {1}{4} \log \left (-\frac {b (1-x)}{1-a-b}\right ) \log (1-a-b x)-\frac {1}{4} \log \left (\frac {b (1+x)}{1-a+b}\right ) \log (1-a-b x)-\frac {1}{4} \log \left (\frac {b (1-x)}{1+a+b}\right ) \log (1+a+b x)+\frac {1}{4} \log \left (-\frac {b (1+x)}{1+a-b}\right ) \log (1+a+b x)+\frac {1}{4} \text {PolyLog}\left (2,\frac {1-a-b x}{1-a-b}\right )-\frac {1}{4} \text {PolyLog}\left (2,\frac {1-a-b x}{1-a+b}\right )+\frac {1}{4} \text {PolyLog}\left (2,\frac {1+a+b x}{1+a-b}\right )-\frac {1}{4} \text {PolyLog}\left (2,\frac {1+a+b x}{1+a+b}\right ) \]

[Out]

1/4*ln(-b*(1-x)/(1-a-b))*ln(-b*x-a+1)-1/4*ln(b*(1+x)/(1-a+b))*ln(-b*x-a+1)-1/4*ln(b*(1-x)/(1+a+b))*ln(b*x+a+1)
+1/4*ln(-b*(1+x)/(1+a-b))*ln(b*x+a+1)+1/4*polylog(2,(-b*x-a+1)/(1-a-b))-1/4*polylog(2,(-b*x-a+1)/(1-a+b))+1/4*
polylog(2,(b*x+a+1)/(1+a-b))-1/4*polylog(2,(b*x+a+1)/(1+a+b))

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Rubi [A]
time = 0.20, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6250, 2456, 2441, 2440, 2438} \begin {gather*} \frac {1}{4} \text {Li}_2\left (\frac {-a-b x+1}{-a-b+1}\right )-\frac {1}{4} \text {Li}_2\left (\frac {-a-b x+1}{-a+b+1}\right )+\frac {1}{4} \text {Li}_2\left (\frac {a+b x+1}{a-b+1}\right )-\frac {1}{4} \text {Li}_2\left (\frac {a+b x+1}{a+b+1}\right )+\frac {1}{4} \log \left (-\frac {b (1-x)}{-a-b+1}\right ) \log (-a-b x+1)-\frac {1}{4} \log \left (\frac {b (x+1)}{-a+b+1}\right ) \log (-a-b x+1)-\frac {1}{4} \log \left (\frac {b (1-x)}{a+b+1}\right ) \log (a+b x+1)+\frac {1}{4} \log \left (-\frac {b (x+1)}{a-b+1}\right ) \log (a+b x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a + b*x]/(1 - x^2),x]

[Out]

(Log[-((b*(1 - x))/(1 - a - b))]*Log[1 - a - b*x])/4 - (Log[(b*(1 + x))/(1 - a + b)]*Log[1 - a - b*x])/4 - (Lo
g[(b*(1 - x))/(1 + a + b)]*Log[1 + a + b*x])/4 + (Log[-((b*(1 + x))/(1 + a - b))]*Log[1 + a + b*x])/4 + PolyLo
g[2, (1 - a - b*x)/(1 - a - b)]/4 - PolyLog[2, (1 - a - b*x)/(1 - a + b)]/4 + PolyLog[2, (1 + a + b*x)/(1 + a
- b)]/4 - PolyLog[2, (1 + a + b*x)/(1 + a + b)]/4

Rule 2438

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

Rule 2440

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

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2456

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 6250

Int[ArcTanh[(c_) + (d_.)*(x_)]/((e_) + (f_.)*(x_)^(n_.)), x_Symbol] :> Dist[1/2, Int[Log[1 + c + d*x]/(e + f*x
^n), x], x] - Dist[1/2, Int[Log[1 - c - d*x]/(e + f*x^n), x], x] /; FreeQ[{c, d, e, f}, x] && RationalQ[n]

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a+b x)}{1-x^2} \, dx &=-\left (\frac {1}{2} \int \frac {\log (1-a-b x)}{1-x^2} \, dx\right )+\frac {1}{2} \int \frac {\log (1+a+b x)}{1-x^2} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {\log (1-a-b x)}{2 (1-x)}+\frac {\log (1-a-b x)}{2 (1+x)}\right ) \, dx\right )+\frac {1}{2} \int \left (\frac {\log (1+a+b x)}{2 (1-x)}+\frac {\log (1+a+b x)}{2 (1+x)}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\log (1-a-b x)}{1-x} \, dx\right )-\frac {1}{4} \int \frac {\log (1-a-b x)}{1+x} \, dx+\frac {1}{4} \int \frac {\log (1+a+b x)}{1-x} \, dx+\frac {1}{4} \int \frac {\log (1+a+b x)}{1+x} \, dx\\ &=\frac {1}{4} \log \left (-\frac {b (1-x)}{1-a-b}\right ) \log (1-a-b x)-\frac {1}{4} \log \left (\frac {b (1+x)}{1-a+b}\right ) \log (1-a-b x)-\frac {1}{4} \log \left (\frac {b (1-x)}{1+a+b}\right ) \log (1+a+b x)+\frac {1}{4} \log \left (-\frac {b (1+x)}{1+a-b}\right ) \log (1+a+b x)+\frac {1}{4} b \int \frac {\log \left (-\frac {b (1-x)}{1-a-b}\right )}{1-a-b x} \, dx+\frac {1}{4} b \int \frac {\log \left (\frac {b (1-x)}{1+a+b}\right )}{1+a+b x} \, dx-\frac {1}{4} b \int \frac {\log \left (-\frac {b (1+x)}{-1+a-b}\right )}{1-a-b x} \, dx-\frac {1}{4} b \int \frac {\log \left (\frac {b (1+x)}{-1-a+b}\right )}{1+a+b x} \, dx\\ &=\frac {1}{4} \log \left (-\frac {b (1-x)}{1-a-b}\right ) \log (1-a-b x)-\frac {1}{4} \log \left (\frac {b (1+x)}{1-a+b}\right ) \log (1-a-b x)-\frac {1}{4} \log \left (\frac {b (1-x)}{1+a+b}\right ) \log (1+a+b x)+\frac {1}{4} \log \left (-\frac {b (1+x)}{1+a-b}\right ) \log (1+a+b x)-\frac {1}{4} \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{1-a-b}\right )}{x} \, dx,x,1-a-b x\right )+\frac {1}{4} \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{-1+a-b}\right )}{x} \, dx,x,1-a-b x\right )-\frac {1}{4} \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{-1-a+b}\right )}{x} \, dx,x,1+a+b x\right )+\frac {1}{4} \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{1+a+b}\right )}{x} \, dx,x,1+a+b x\right )\\ &=\frac {1}{4} \log \left (-\frac {b (1-x)}{1-a-b}\right ) \log (1-a-b x)-\frac {1}{4} \log \left (\frac {b (1+x)}{1-a+b}\right ) \log (1-a-b x)-\frac {1}{4} \log \left (\frac {b (1-x)}{1+a+b}\right ) \log (1+a+b x)+\frac {1}{4} \log \left (-\frac {b (1+x)}{1+a-b}\right ) \log (1+a+b x)+\frac {1}{4} \text {Li}_2\left (\frac {1-a-b x}{1-a-b}\right )-\frac {1}{4} \text {Li}_2\left (\frac {1-a-b x}{1-a+b}\right )+\frac {1}{4} \text {Li}_2\left (\frac {1+a+b x}{1+a-b}\right )-\frac {1}{4} \text {Li}_2\left (\frac {1+a+b x}{1+a+b}\right )\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 203, normalized size = 1.00 \begin {gather*} \frac {1}{4} \log \left (-\frac {b (1-x)}{1-a-b}\right ) \log (1-a-b x)-\frac {1}{4} \log \left (\frac {b (1+x)}{1-a+b}\right ) \log (1-a-b x)-\frac {1}{4} \log \left (\frac {b (1-x)}{1+a+b}\right ) \log (1+a+b x)+\frac {1}{4} \log \left (-\frac {b (1+x)}{1+a-b}\right ) \log (1+a+b x)+\frac {1}{4} \text {PolyLog}\left (2,\frac {1-a-b x}{1-a-b}\right )-\frac {1}{4} \text {PolyLog}\left (2,\frac {1-a-b x}{1-a+b}\right )+\frac {1}{4} \text {PolyLog}\left (2,\frac {1+a+b x}{1+a-b}\right )-\frac {1}{4} \text {PolyLog}\left (2,\frac {1+a+b x}{1+a+b}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[-((b*(1 - x))/(1 - a - b))]*Log[1 - a - b*x])/4 - (Log[(b*(1 + x))/(1 - a + b)]*Log[1 - a - b*x])/4 - (Lo
g[(b*(1 - x))/(1 + a + b)]*Log[1 + a + b*x])/4 + (Log[-((b*(1 + x))/(1 + a - b))]*Log[1 + a + b*x])/4 + PolyLo
g[2, (1 - a - b*x)/(1 - a - b)]/4 - PolyLog[2, (1 - a - b*x)/(1 - a + b)]/4 + PolyLog[2, (1 + a + b*x)/(1 + a
- b)]/4 - PolyLog[2, (1 + a + b*x)/(1 + a + b)]/4

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Maple [A]
time = 12.85, size = 278, normalized size = 1.37

method	result	size
risch	\(\frac {\ln \left (-b x -a +1\right ) \ln \left (\frac {-b x +b}{b -1+a}\right )}{4}+\frac {\dilog \left (\frac {-b x +b}{b -1+a}\right )}{4}-\frac {\ln \left (-b x -a +1\right ) \ln \left (\frac {-b x -b}{-b -1+a}\right )}{4}-\frac {\dilog \left (\frac {-b x -b}{-b -1+a}\right )}{4}+\frac {\ln \left (b x +a +1\right ) \ln \left (\frac {b x +b}{-1+b -a}\right )}{4}+\frac {\dilog \left (\frac {b x +b}{-1+b -a}\right )}{4}-\frac {\ln \left (b x +a +1\right ) \ln \left (\frac {b x -b}{-1-b -a}\right )}{4}-\frac {\dilog \left (\frac {b x -b}{-1-b -a}\right )}{4}\)	\(184\)
derivativedivides	\(\frac {-\frac {\arctanh \left (b x +a \right ) b \ln \left (-b x +b \right )}{2}+\frac {\arctanh \left (b x +a \right ) b \ln \left (-b x -b \right )}{2}+\frac {b^{2} \left (\frac {\dilog \left (\frac {-b x -a +1}{1-a +b}\right )}{2 b}+\frac {\ln \left (-b x -b \right ) \ln \left (\frac {-b x -a +1}{1-a +b}\right )}{2 b}-\frac {\dilog \left (\frac {-b x -a -1}{-1+b -a}\right )}{2 b}-\frac {\ln \left (-b x -b \right ) \ln \left (\frac {-b x -a -1}{-1+b -a}\right )}{2 b}+\frac {\dilog \left (\frac {-b x -a -1}{-1-b -a}\right )}{2 b}+\frac {\ln \left (-b x +b \right ) \ln \left (\frac {-b x -a -1}{-1-b -a}\right )}{2 b}-\frac {\dilog \left (\frac {-b x -a +1}{1-a -b}\right )}{2 b}-\frac {\ln \left (-b x +b \right ) \ln \left (\frac {-b x -a +1}{1-a -b}\right )}{2 b}\right )}{2}}{b}\)	\(278\)
default	\(\frac {-\frac {\arctanh \left (b x +a \right ) b \ln \left (-b x +b \right )}{2}+\frac {\arctanh \left (b x +a \right ) b \ln \left (-b x -b \right )}{2}+\frac {b^{2} \left (\frac {\dilog \left (\frac {-b x -a +1}{1-a +b}\right )}{2 b}+\frac {\ln \left (-b x -b \right ) \ln \left (\frac {-b x -a +1}{1-a +b}\right )}{2 b}-\frac {\dilog \left (\frac {-b x -a -1}{-1+b -a}\right )}{2 b}-\frac {\ln \left (-b x -b \right ) \ln \left (\frac {-b x -a -1}{-1+b -a}\right )}{2 b}+\frac {\dilog \left (\frac {-b x -a -1}{-1-b -a}\right )}{2 b}+\frac {\ln \left (-b x +b \right ) \ln \left (\frac {-b x -a -1}{-1-b -a}\right )}{2 b}-\frac {\dilog \left (\frac {-b x -a +1}{1-a -b}\right )}{2 b}-\frac {\ln \left (-b x +b \right ) \ln \left (\frac {-b x -a +1}{1-a -b}\right )}{2 b}\right )}{2}}{b}\)	\(278\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(-1/2*arctanh(b*x+a)*b*ln(-b*x+b)+1/2*arctanh(b*x+a)*b*ln(-b*x-b)+1/2*b^2*(1/2/b*dilog((-b*x-a+1)/(1-a+b))
+1/2/b*ln(-b*x-b)*ln((-b*x-a+1)/(1-a+b))-1/2/b*dilog((-b*x-a-1)/(-1+b-a))-1/2/b*ln(-b*x-b)*ln((-b*x-a-1)/(-1+b
-a))+1/2/b*dilog((-b*x-a-1)/(-1-b-a))+1/2/b*ln(-b*x+b)*ln((-b*x-a-1)/(-1-b-a))-1/2/b*dilog((-b*x-a+1)/(1-a-b))
-1/2/b*ln(-b*x+b)*ln((-b*x-a+1)/(1-a-b))))

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Maxima [A]
time = 0.26, size = 198, normalized size = 0.98 \begin {gather*} \frac {1}{4} \, b {\left (\frac {\log \left (x - 1\right ) \log \left (\frac {b x - b}{a + b + 1} + 1\right ) + {\rm Li}_2\left (-\frac {b x - b}{a + b + 1}\right )}{b} - \frac {\log \left (x - 1\right ) \log \left (\frac {b x - b}{a + b - 1} + 1\right ) + {\rm Li}_2\left (-\frac {b x - b}{a + b - 1}\right )}{b} - \frac {\log \left (x + 1\right ) \log \left (\frac {b x + b}{a - b + 1} + 1\right ) + {\rm Li}_2\left (-\frac {b x + b}{a - b + 1}\right )}{b} + \frac {\log \left (x + 1\right ) \log \left (\frac {b x + b}{a - b - 1} + 1\right ) + {\rm Li}_2\left (-\frac {b x + b}{a - b - 1}\right )}{b}\right )} + \frac {1}{2} \, {\left (\log \left (x + 1\right ) - \log \left (x - 1\right )\right )} \operatorname {artanh}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*((log(x - 1)*log((b*x - b)/(a + b + 1) + 1) + dilog(-(b*x - b)/(a + b + 1)))/b - (log(x - 1)*log((b*x -
b)/(a + b - 1) + 1) + dilog(-(b*x - b)/(a + b - 1)))/b - (log(x + 1)*log((b*x + b)/(a - b + 1) + 1) + dilog(-(
b*x + b)/(a - b + 1)))/b + (log(x + 1)*log((b*x + b)/(a - b - 1) + 1) + dilog(-(b*x + b)/(a - b - 1)))/b) + 1/
2*(log(x + 1) - log(x - 1))*arctanh(b*x + a)

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\operatorname {atanh}{\left (a + b x \right )}}{x^{2} - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-atanh(a + b*x)/(x^2 - 1),x)

[Out]

-int(atanh(a + b*x)/(x^2 - 1), x)

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