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

Optimal. Leaf size=585 \[ \frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )}{d^2}+\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )}{d^2}-\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )}{d^2}-\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )}{d^2}+\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )}{d^2}-\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )}{d^2}+\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )}{d^2}-\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )}{d^2}+\frac{c \log (-a-b x+1) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{c \log (a+b x+1) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{\sqrt{x} \log (-a-b x+1)}{d}+\frac{\sqrt{x} \log (a+b x+1)}{d}+\frac{2 \sqrt{a+1} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b} d}-\frac{2 \sqrt{1-a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.01346, antiderivative size = 585, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 13, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.722, Rules used = {6115, 2408, 2466, 2448, 321, 205, 2462, 260, 2416, 2394, 2393, 2391, 208} \[ \frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )}{d^2}+\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )}{d^2}-\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )}{d^2}-\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )}{d^2}+\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )}{d^2}-\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )}{d^2}+\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )}{d^2}-\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )}{d^2}+\frac{c \log (-a-b x+1) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{c \log (a+b x+1) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{\sqrt{x} \log (-a-b x+1)}{d}+\frac{\sqrt{x} \log (a+b x+1)}{d}+\frac{2 \sqrt{a+1} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b} d}-\frac{2 \sqrt{1-a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2466

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

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

Mathematica [A]  time = 0.518536, size = 549, normalized size = 0.94 \[ \frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )+c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )-c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )-c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )+c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )-c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )+c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} d-\sqrt{b} c}\right )-c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} d-\sqrt{b} c}\right )+c \log (-a-b x+1) \log \left (c+d \sqrt{x}\right )-c \log (a+b x+1) \log \left (c+d \sqrt{x}\right )-d \sqrt{x} \log (-a-b x+1)+d \sqrt{x} \log (a+b x+1)+\frac{2 \sqrt{a+1} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b}}-\frac{2 \sqrt{1-a} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b}}}{d^2} \]

Antiderivative was successfully verified.

[In]

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

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Maple [A]  time = 0.239, size = 738, normalized size = 1.3 \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]

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

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

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