3.60 \(\int \frac{\tanh ^{-1}(a+b x)}{c+\frac{d}{\sqrt{x}}} \, dx\)

Optimal. Leaf size=661 \[ -\frac{d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{c^3}-\frac{d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )}{c^3}+\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )}{c^3}+\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log (-a-b x+1) \log \left (c \sqrt{x}+d\right )}{c^3}+\frac{d^2 \log (a+b x+1) \log \left (c \sqrt{x}+d\right )}{c^3}+\frac{d \sqrt{x} \log (-a-b x+1)}{c^2}-\frac{d \sqrt{x} \log (a+b x+1)}{c^2}-\frac{2 \sqrt{a+1} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} c^2}+\frac{(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac{(a+b x+1) \log (a+b x+1)}{2 b c} \]

[Out]

(-2*Sqrt[1 + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]])/(Sqrt[b]*c^2) + (2*Sqrt[1 - a]*d*ArcTanh[(Sqrt[b]*Sqr
t[x])/Sqrt[1 - a]])/(Sqrt[b]*c^2) - (d^2*Log[(c*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)
]*Log[d + c*Sqrt[x]])/c^3 + (d^2*Log[(c*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c + Sqrt[b]*d)]*Log[d +
c*Sqrt[x]])/c^3 - (d^2*Log[(c*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x
]])/c^3 + (d^2*Log[(c*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 +
(d*Sqrt[x]*Log[1 - a - b*x])/c^2 + ((1 - a - b*x)*Log[1 - a - b*x])/(2*b*c) - (d^2*Log[d + c*Sqrt[x]]*Log[1 -
a - b*x])/c^3 - (d*Sqrt[x]*Log[1 + a + b*x])/c^2 + ((1 + a + b*x)*Log[1 + a + b*x])/(2*b*c) + (d^2*Log[d + c*S
qrt[x]]*Log[1 + a + b*x])/c^3 - (d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d))])/c^
3 + (d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b]*d))])/c^3 - (d^2*PolyLog[2, (Sqrt[b]*
(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)])/c^3 + (d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a]*c
+ Sqrt[b]*d)])/c^3

________________________________________________________________________________________

Rubi [A]  time = 1.07218, antiderivative size = 661, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 16, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889, Rules used = {6115, 2408, 2476, 2448, 321, 205, 2454, 2389, 2295, 2462, 260, 2416, 2394, 2393, 2391, 208} \[ -\frac{d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{c^3}-\frac{d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )}{c^3}+\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )}{c^3}+\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log (-a-b x+1) \log \left (c \sqrt{x}+d\right )}{c^3}+\frac{d^2 \log (a+b x+1) \log \left (c \sqrt{x}+d\right )}{c^3}+\frac{d \sqrt{x} \log (-a-b x+1)}{c^2}-\frac{d \sqrt{x} \log (a+b x+1)}{c^2}-\frac{2 \sqrt{a+1} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} c^2}+\frac{(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac{(a+b x+1) \log (a+b x+1)}{2 b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]])/(Sqrt[b]*c^2) + (2*Sqrt[1 - a]*d*ArcTanh[(Sqrt[b]*Sqr
t[x])/Sqrt[1 - a]])/(Sqrt[b]*c^2) - (d^2*Log[(c*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)
]*Log[d + c*Sqrt[x]])/c^3 + (d^2*Log[(c*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c + Sqrt[b]*d)]*Log[d +
c*Sqrt[x]])/c^3 - (d^2*Log[(c*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x
]])/c^3 + (d^2*Log[(c*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 +
(d*Sqrt[x]*Log[1 - a - b*x])/c^2 + ((1 - a - b*x)*Log[1 - a - b*x])/(2*b*c) - (d^2*Log[d + c*Sqrt[x]]*Log[1 -
a - b*x])/c^3 - (d*Sqrt[x]*Log[1 + a + b*x])/c^2 + ((1 + a + b*x)*Log[1 + a + b*x])/(2*b*c) + (d^2*Log[d + c*S
qrt[x]]*Log[1 + a + b*x])/c^3 - (d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d))])/c^
3 + (d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b]*d))])/c^3 - (d^2*PolyLog[2, (Sqrt[b]*
(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)])/c^3 + (d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a]*c
+ Sqrt[b]*d)])/c^3

Rule 6115

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]

Rule 2408

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> W
ith[{k = Denominator[r]}, Dist[k, Subst[Int[x^(k - 1)*(f + g*x^(k*r))^q*(a + b*Log[c*(d + e*x^k)^n])^p, x], x,
x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && FractionQ[r] && IGtQ[p, 0]

Rule 2476

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

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2462

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[f +
g*x]*(a + b*Log[c*(d + e*x^n)^p]))/g, x] - Dist[(b*e*n*p)/g, Int[(x^(n - 1)*Log[f + g*x])/(d + e*x^n), x], x]
/; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[n]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2394

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 2393

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 2391

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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a+b x)}{c+\frac{d}{\sqrt{x}}} \, dx &=-\left (\frac{1}{2} \int \frac{\log (1-a-b x)}{c+\frac{d}{\sqrt{x}}} \, dx\right )+\frac{1}{2} \int \frac{\log (1+a+b x)}{c+\frac{d}{\sqrt{x}}} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{x \log \left (1-a-b x^2\right )}{c+\frac{d}{x}} \, dx,x,\sqrt{x}\right )+\operatorname{Subst}\left (\int \frac{x \log \left (1+a+b x^2\right )}{c+\frac{d}{x}} \, dx,x,\sqrt{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{d \log \left (1-a-b x^2\right )}{c^2}+\frac{x \log \left (1-a-b x^2\right )}{c}+\frac{d^2 \log \left (1-a-b x^2\right )}{c^2 (d+c x)}\right ) \, dx,x,\sqrt{x}\right )+\operatorname{Subst}\left (\int \left (-\frac{d \log \left (1+a+b x^2\right )}{c^2}+\frac{x \log \left (1+a+b x^2\right )}{c}+\frac{d^2 \log \left (1+a+b x^2\right )}{c^2 (d+c x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int x \log \left (1-a-b x^2\right ) \, dx,x,\sqrt{x}\right )}{c}+\frac{\operatorname{Subst}\left (\int x \log \left (1+a+b x^2\right ) \, dx,x,\sqrt{x}\right )}{c}+\frac{d \operatorname{Subst}\left (\int \log \left (1-a-b x^2\right ) \, dx,x,\sqrt{x}\right )}{c^2}-\frac{d \operatorname{Subst}\left (\int \log \left (1+a+b x^2\right ) \, dx,x,\sqrt{x}\right )}{c^2}-\frac{d^2 \operatorname{Subst}\left (\int \frac{\log \left (1-a-b x^2\right )}{d+c x} \, dx,x,\sqrt{x}\right )}{c^2}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\log \left (1+a+b x^2\right )}{d+c x} \, dx,x,\sqrt{x}\right )}{c^2}\\ &=\frac{d \sqrt{x} \log (1-a-b x)}{c^2}-\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1-a-b x)}{c^3}-\frac{d \sqrt{x} \log (1+a+b x)}{c^2}+\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1+a+b x)}{c^3}-\frac{\operatorname{Subst}(\int \log (1-a-b x) \, dx,x,x)}{2 c}+\frac{\operatorname{Subst}(\int \log (1+a+b x) \, dx,x,x)}{2 c}+\frac{(2 b d) \operatorname{Subst}\left (\int \frac{x^2}{1-a-b x^2} \, dx,x,\sqrt{x}\right )}{c^2}+\frac{(2 b d) \operatorname{Subst}\left (\int \frac{x^2}{1+a+b x^2} \, dx,x,\sqrt{x}\right )}{c^2}-\frac{\left (2 b d^2\right ) \operatorname{Subst}\left (\int \frac{x \log (d+c x)}{1-a-b x^2} \, dx,x,\sqrt{x}\right )}{c^3}-\frac{\left (2 b d^2\right ) \operatorname{Subst}\left (\int \frac{x \log (d+c x)}{1+a+b x^2} \, dx,x,\sqrt{x}\right )}{c^3}\\ &=\frac{d \sqrt{x} \log (1-a-b x)}{c^2}-\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1-a-b x)}{c^3}-\frac{d \sqrt{x} \log (1+a+b x)}{c^2}+\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1+a+b x)}{c^3}+\frac{\operatorname{Subst}(\int \log (x) \, dx,x,1-a-b x)}{2 b c}+\frac{\operatorname{Subst}(\int \log (x) \, dx,x,1+a+b x)}{2 b c}+\frac{(2 (1-a) d) \operatorname{Subst}\left (\int \frac{1}{1-a-b x^2} \, dx,x,\sqrt{x}\right )}{c^2}-\frac{(2 (1+a) d) \operatorname{Subst}\left (\int \frac{1}{1+a+b x^2} \, dx,x,\sqrt{x}\right )}{c^2}-\frac{\left (2 b d^2\right ) \operatorname{Subst}\left (\int \left (-\frac{\log (d+c x)}{2 \sqrt{b} \left (\sqrt{-1-a}-\sqrt{b} x\right )}+\frac{\log (d+c x)}{2 \sqrt{b} \left (\sqrt{-1-a}+\sqrt{b} x\right )}\right ) \, dx,x,\sqrt{x}\right )}{c^3}-\frac{\left (2 b d^2\right ) \operatorname{Subst}\left (\int \left (\frac{\log (d+c x)}{2 \sqrt{b} \left (\sqrt{1-a}-\sqrt{b} x\right )}-\frac{\log (d+c x)}{2 \sqrt{b} \left (\sqrt{1-a}+\sqrt{b} x\right )}\right ) \, dx,x,\sqrt{x}\right )}{c^3}\\ &=-\frac{2 \sqrt{1+a} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1+a}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} c^2}+\frac{d \sqrt{x} \log (1-a-b x)}{c^2}+\frac{(1-a-b x) \log (1-a-b x)}{2 b c}-\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1-a-b x)}{c^3}-\frac{d \sqrt{x} \log (1+a+b x)}{c^2}+\frac{(1+a+b x) \log (1+a+b x)}{2 b c}+\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1+a+b x)}{c^3}+\frac{\left (\sqrt{b} d^2\right ) \operatorname{Subst}\left (\int \frac{\log (d+c x)}{\sqrt{-1-a}-\sqrt{b} x} \, dx,x,\sqrt{x}\right )}{c^3}-\frac{\left (\sqrt{b} d^2\right ) \operatorname{Subst}\left (\int \frac{\log (d+c x)}{\sqrt{1-a}-\sqrt{b} x} \, dx,x,\sqrt{x}\right )}{c^3}-\frac{\left (\sqrt{b} d^2\right ) \operatorname{Subst}\left (\int \frac{\log (d+c x)}{\sqrt{-1-a}+\sqrt{b} x} \, dx,x,\sqrt{x}\right )}{c^3}+\frac{\left (\sqrt{b} d^2\right ) \operatorname{Subst}\left (\int \frac{\log (d+c x)}{\sqrt{1-a}+\sqrt{b} x} \, dx,x,\sqrt{x}\right )}{c^3}\\ &=-\frac{2 \sqrt{1+a} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1+a}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} c^2}-\frac{d^2 \log \left (\frac{c \left (\sqrt{-1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-1-a} c+\sqrt{b} d}\right ) \log \left (d+c \sqrt{x}\right )}{c^3}+\frac{d^2 \log \left (\frac{c \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c+\sqrt{b} d}\right ) \log \left (d+c \sqrt{x}\right )}{c^3}-\frac{d^2 \log \left (\frac{c \left (\sqrt{-1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-1-a} c-\sqrt{b} d}\right ) \log \left (d+c \sqrt{x}\right )}{c^3}+\frac{d^2 \log \left (\frac{c \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c-\sqrt{b} d}\right ) \log \left (d+c \sqrt{x}\right )}{c^3}+\frac{d \sqrt{x} \log (1-a-b x)}{c^2}+\frac{(1-a-b x) \log (1-a-b x)}{2 b c}-\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1-a-b x)}{c^3}-\frac{d \sqrt{x} \log (1+a+b x)}{c^2}+\frac{(1+a+b x) \log (1+a+b x)}{2 b c}+\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1+a+b x)}{c^3}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\log \left (\frac{c \left (\sqrt{-1-a}-\sqrt{b} x\right )}{\sqrt{-1-a} c+\sqrt{b} d}\right )}{d+c x} \, dx,x,\sqrt{x}\right )}{c^2}-\frac{d^2 \operatorname{Subst}\left (\int \frac{\log \left (\frac{c \left (\sqrt{1-a}-\sqrt{b} x\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{d+c x} \, dx,x,\sqrt{x}\right )}{c^2}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\log \left (\frac{c \left (\sqrt{-1-a}+\sqrt{b} x\right )}{\sqrt{-1-a} c-\sqrt{b} d}\right )}{d+c x} \, dx,x,\sqrt{x}\right )}{c^2}-\frac{d^2 \operatorname{Subst}\left (\int \frac{\log \left (\frac{c \left (\sqrt{1-a}+\sqrt{b} x\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{d+c x} \, dx,x,\sqrt{x}\right )}{c^2}\\ &=-\frac{2 \sqrt{1+a} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1+a}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} c^2}-\frac{d^2 \log \left (\frac{c \left (\sqrt{-1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-1-a} c+\sqrt{b} d}\right ) \log \left (d+c \sqrt{x}\right )}{c^3}+\frac{d^2 \log \left (\frac{c \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c+\sqrt{b} d}\right ) \log \left (d+c \sqrt{x}\right )}{c^3}-\frac{d^2 \log \left (\frac{c \left (\sqrt{-1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-1-a} c-\sqrt{b} d}\right ) \log \left (d+c \sqrt{x}\right )}{c^3}+\frac{d^2 \log \left (\frac{c \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c-\sqrt{b} d}\right ) \log \left (d+c \sqrt{x}\right )}{c^3}+\frac{d \sqrt{x} \log (1-a-b x)}{c^2}+\frac{(1-a-b x) \log (1-a-b x)}{2 b c}-\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1-a-b x)}{c^3}-\frac{d \sqrt{x} \log (1+a+b x)}{c^2}+\frac{(1+a+b x) \log (1+a+b x)}{2 b c}+\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1+a+b x)}{c^3}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{b} x}{\sqrt{-1-a} c-\sqrt{b} d}\right )}{x} \, dx,x,d+c \sqrt{x}\right )}{c^3}-\frac{d^2 \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{b} x}{\sqrt{1-a} c-\sqrt{b} d}\right )}{x} \, dx,x,d+c \sqrt{x}\right )}{c^3}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{b} x}{\sqrt{-1-a} c+\sqrt{b} d}\right )}{x} \, dx,x,d+c \sqrt{x}\right )}{c^3}-\frac{d^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{b} x}{\sqrt{1-a} c+\sqrt{b} d}\right )}{x} \, dx,x,d+c \sqrt{x}\right )}{c^3}\\ &=-\frac{2 \sqrt{1+a} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1+a}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} c^2}-\frac{d^2 \log \left (\frac{c \left (\sqrt{-1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-1-a} c+\sqrt{b} d}\right ) \log \left (d+c \sqrt{x}\right )}{c^3}+\frac{d^2 \log \left (\frac{c \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c+\sqrt{b} d}\right ) \log \left (d+c \sqrt{x}\right )}{c^3}-\frac{d^2 \log \left (\frac{c \left (\sqrt{-1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-1-a} c-\sqrt{b} d}\right ) \log \left (d+c \sqrt{x}\right )}{c^3}+\frac{d^2 \log \left (\frac{c \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c-\sqrt{b} d}\right ) \log \left (d+c \sqrt{x}\right )}{c^3}+\frac{d \sqrt{x} \log (1-a-b x)}{c^2}+\frac{(1-a-b x) \log (1-a-b x)}{2 b c}-\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1-a-b x)}{c^3}-\frac{d \sqrt{x} \log (1+a+b x)}{c^2}+\frac{(1+a+b x) \log (1+a+b x)}{2 b c}+\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1+a+b x)}{c^3}-\frac{d^2 \text{Li}_2\left (-\frac{\sqrt{b} \left (d+c \sqrt{x}\right )}{\sqrt{-1-a} c-\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{Li}_2\left (-\frac{\sqrt{b} \left (d+c \sqrt{x}\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{c^3}-\frac{d^2 \text{Li}_2\left (\frac{\sqrt{b} \left (d+c \sqrt{x}\right )}{\sqrt{-1-a} c+\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{Li}_2\left (\frac{\sqrt{b} \left (d+c \sqrt{x}\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{c^3}\\ \end{align*}

Mathematica [A]  time = 0.606834, size = 598, normalized size = 0.9 \[ \frac{-2 d^2 \left (\text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{b} d-\sqrt{-a-1} c}\right )+\text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )+\log \left (c \sqrt{x}+d\right ) \left (\log \left (\frac{c \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )+\log \left (\frac{c \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )\right )\right )+2 d^2 \left (\text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{b} d-\sqrt{1-a} c}\right )+\text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )+\log \left (c \sqrt{x}+d\right ) \left (\log \left (\frac{c \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )+\log \left (\frac{c \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )\right )\right )-\frac{c^2 (a+b x-1) \log (-a-b x+1)}{b}+\frac{c^2 (a+b x+1) \log (a+b x+1)}{b}-2 d^2 \log (-a-b x+1) \log \left (c \sqrt{x}+d\right )+2 d^2 \log (a+b x+1) \log \left (c \sqrt{x}+d\right )+2 c d \sqrt{x} \log (-a-b x+1)-2 c d \sqrt{x} \log (a+b x+1)+4 c d \left (\sqrt{x}-\frac{\sqrt{a+1} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b}}\right )-4 c d \left (\sqrt{x}-\frac{\sqrt{1-a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b}}\right )}{2 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*c*d*(Sqrt[x] - (Sqrt[1 + a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]])/Sqrt[b]) - 4*c*d*(Sqrt[x] - (Sqrt[1 - a]
*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[1 - a]])/Sqrt[b]) + 2*c*d*Sqrt[x]*Log[1 - a - b*x] - (c^2*(-1 + a + b*x)*Log[1
- a - b*x])/b - 2*d^2*Log[d + c*Sqrt[x]]*Log[1 - a - b*x] - 2*c*d*Sqrt[x]*Log[1 + a + b*x] + (c^2*(1 + a + b*
x)*Log[1 + a + b*x])/b + 2*d^2*Log[d + c*Sqrt[x]]*Log[1 + a + b*x] - 2*d^2*((Log[(c*(Sqrt[-1 - a] - Sqrt[b]*Sq
rt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)] + Log[(c*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d)]
)*Log[d + c*Sqrt[x]] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[-1 - a]*c) + Sqrt[b]*d)] + PolyLog[2, (Sqr
t[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)]) + 2*d^2*((Log[(c*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1
- a]*c + Sqrt[b]*d)] + Log[(c*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b]*d)])*Log[d + c*Sqrt[x
]] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[1 - a]*c) + Sqrt[b]*d)] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]
))/(Sqrt[1 - a]*c + Sqrt[b]*d)]))/(2*c^3)

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Maple [A]  time = 0.181, size = 970, normalized size = 1.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(b*x+a)/(c+d/x^(1/2)),x)

[Out]

arctanh(b*x+a)/c*x-2*arctanh(b*x+a)/c^2*d*x^(1/2)+2*arctanh(b*x+a)*d^2/c^3*ln(d+c*x^(1/2))-1/2/b/c*a*ln(b*(d+c
*x^(1/2))^2-2*(d+c*x^(1/2))*b*d+a*c^2+b*d^2-c^2)+2/c*a*d/(a*b*c^2-b*c^2)^(1/2)*arctan(1/2*(2*b*(d+c*x^(1/2))-2
*b*d)/(a*b*c^2-b*c^2)^(1/2))+1/2/b/c*ln(b*(d+c*x^(1/2))^2-2*(d+c*x^(1/2))*b*d+a*c^2+b*d^2-c^2)-2/c*d/(a*b*c^2-
b*c^2)^(1/2)*arctan(1/2*(2*b*(d+c*x^(1/2))-2*b*d)/(a*b*c^2-b*c^2)^(1/2))+1/2/b/c*ln(b*(d+c*x^(1/2))^2-2*(d+c*x
^(1/2))*b*d+a*c^2+b*d^2+c^2)-2/c*d/(a*b*c^2+b*c^2)^(1/2)*arctan(1/2*(2*b*(d+c*x^(1/2))-2*b*d)/(a*b*c^2+b*c^2)^
(1/2))+1/2/b/c*a*ln(b*(d+c*x^(1/2))^2-2*(d+c*x^(1/2))*b*d+a*c^2+b*d^2+c^2)-2/c*a*d/(a*b*c^2+b*c^2)^(1/2)*arcta
n(1/2*(2*b*(d+c*x^(1/2))-2*b*d)/(a*b*c^2+b*c^2)^(1/2))+1/c^3*d^2*ln(d+c*x^(1/2))*ln((-b*(d+c*x^(1/2))+b*d+(-a*
b*c^2+b*c^2)^(1/2))/(b*d+(-a*b*c^2+b*c^2)^(1/2)))+1/c^3*d^2*ln(d+c*x^(1/2))*ln((b*(d+c*x^(1/2))-b*d+(-a*b*c^2+
b*c^2)^(1/2))/(-b*d+(-a*b*c^2+b*c^2)^(1/2)))+1/c^3*d^2*dilog((-b*(d+c*x^(1/2))+b*d+(-a*b*c^2+b*c^2)^(1/2))/(b*
d+(-a*b*c^2+b*c^2)^(1/2)))+1/c^3*d^2*dilog((b*(d+c*x^(1/2))-b*d+(-a*b*c^2+b*c^2)^(1/2))/(-b*d+(-a*b*c^2+b*c^2)
^(1/2)))-1/c^3*d^2*ln(d+c*x^(1/2))*ln((-b*(d+c*x^(1/2))+b*d+(-a*b*c^2-b*c^2)^(1/2))/(b*d+(-a*b*c^2-b*c^2)^(1/2
)))-1/c^3*d^2*ln(d+c*x^(1/2))*ln((b*(d+c*x^(1/2))-b*d+(-a*b*c^2-b*c^2)^(1/2))/(-b*d+(-a*b*c^2-b*c^2)^(1/2)))-1
/c^3*d^2*dilog((-b*(d+c*x^(1/2))+b*d+(-a*b*c^2-b*c^2)^(1/2))/(b*d+(-a*b*c^2-b*c^2)^(1/2)))-1/c^3*d^2*dilog((b*
(d+c*x^(1/2))-b*d+(-a*b*c^2-b*c^2)^(1/2))/(-b*d+(-a*b*c^2-b*c^2)^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (b x + a\right )}{c + \frac{d}{\sqrt{x}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{c x \operatorname{artanh}\left (b x + a\right ) - d \sqrt{x} \operatorname{artanh}\left (b x + a\right )}{c^{2} x - d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(b*x+a)/(c+d/x^(1/2)),x, algorithm="fricas")

[Out]

integral((c*x*arctanh(b*x + a) - d*sqrt(x)*arctanh(b*x + a))/(c^2*x - d^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (b x + a\right )}{c + \frac{d}{\sqrt{x}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(b*x+a)/(c+d/x^(1/2)),x, algorithm="giac")

[Out]

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