3.1.30 \(\int \frac {\tanh ^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx\) [30]

Optimal. Leaf size=32 \[ -\frac {\text {PolyLog}(2,-a-b x)}{2 d}+\frac {\text {PolyLog}(2,a+b x)}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6242, 12, 6031} \begin {gather*} \frac {\text {Li}_2(a+b x)}{2 d}-\frac {\text {Li}_2(-a-b x)}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a + b*x]/((a*d)/b + d*x),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6031

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Simp[(b/2)*PolyLog[2, (-c)*x]
, x] + Simp[(b/2)*PolyLog[2, c*x], x]) /; FreeQ[{a, b, c}, x]

Rule 6242

Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[(f*(x/d))^m*(a + b*ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f,
0] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 52, normalized size = 1.62 \begin {gather*} b \left (-\frac {\text {PolyLog}\left (2,-\frac {a d+b d x}{d}\right )}{2 b d}+\frac {\text {PolyLog}\left (2,\frac {a d+b d x}{d}\right )}{2 b d}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a + b*x]/((a*d)/b + d*x),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(28)=56\).
time = 0.87, size = 62, normalized size = 1.94

method	result	size
risch	\(-\frac {\dilog \left (b x +a +1\right )}{2 d}+\frac {\dilog \left (-b x -a +1\right )}{2 d}\)	\(29\)
derivativedivides	\(\frac {\frac {b \ln \left (b x +a \right ) \arctanh \left (b x +a \right )}{d}-\frac {b \left (\frac {\dilog \left (b x +a +1\right )}{2}+\frac {\ln \left (b x +a \right ) \ln \left (b x +a +1\right )}{2}+\frac {\dilog \left (b x +a \right )}{2}\right )}{d}}{b}\)	\(62\)
default	\(\frac {\frac {b \ln \left (b x +a \right ) \arctanh \left (b x +a \right )}{d}-\frac {b \left (\frac {\dilog \left (b x +a +1\right )}{2}+\frac {\ln \left (b x +a \right ) \ln \left (b x +a +1\right )}{2}+\frac {\dilog \left (b x +a \right )}{2}\right )}{d}}{b}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(b*x+a)/(a*d/b+d*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (26) = 52\).
time = 0.26, size = 132, normalized size = 4.12 \begin {gather*} -\frac {1}{2} \, b {\left (\frac {\log \left (b x + a\right ) \log \left (b x + a - 1\right ) + {\rm Li}_2\left (-b x - a + 1\right )}{b d} - \frac {\log \left (b x + a + 1\right ) \log \left (-b x - a\right ) + {\rm Li}_2\left (b x + a + 1\right )}{b d}\right )} - \frac {b {\left (\frac {\log \left (b x + a + 1\right )}{b} - \frac {\log \left (b x + a - 1\right )}{b}\right )} \log \left (d x + \frac {a d}{b}\right )}{2 \, d} + \frac {\operatorname {artanh}\left (b x + a\right ) \log \left (d x + \frac {a d}{b}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(b*x+a)/(a*d/b+d*x),x, algorithm="maxima")

[Out]

-1/2*b*((log(b*x + a)*log(b*x + a - 1) + dilog(-b*x - a + 1))/(b*d) - (log(b*x + a + 1)*log(-b*x - a) + dilog(
b*x + a + 1))/(b*d)) - 1/2*b*(log(b*x + a + 1)/b - log(b*x + a - 1)/b)*log(d*x + a*d/b)/d + arctanh(b*x + a)*l
og(d*x + a*d/b)/d

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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)/(a*d/b+d*x),x, algorithm="fricas")

[Out]

integral(b*arctanh(b*x + a)/(b*d*x + a*d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(b*x+a)/(a*d/b+d*x),x)

[Out]

b*Integral(atanh(a + b*x)/(a + b*x), x)/d

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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)/(a*d/b+d*x),x, algorithm="giac")

[Out]

integrate(arctanh(b*x + a)/(d*x + a*d/b), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {atanh}\left (a+b\,x\right )}{d\,x+\frac {a\,d}{b}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a + b*x)/(d*x + (a*d)/b),x)

[Out]

int(atanh(a + b*x)/(d*x + (a*d)/b), x)

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