3.1.60 \(\int \frac {\tanh ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx\) [60]

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

[Out]

1/2*(-b*x-a+1)*ln(-b*x-a+1)/b/c+1/2*(b*x+a+1)*ln(b*x+a+1)/b/c-d^2*ln(-b*x-a+1)*ln(d+c*x^(1/2))/c^3+d^2*ln(b*x+
a+1)*ln(d+c*x^(1/2))/c^3-d^2*ln(d+c*x^(1/2))*ln(c*((-1-a)^(1/2)-b^(1/2)*x^(1/2))/(c*(-1-a)^(1/2)+d*b^(1/2)))/c
^3+d^2*ln(d+c*x^(1/2))*ln(c*((1-a)^(1/2)-b^(1/2)*x^(1/2))/(c*(1-a)^(1/2)+d*b^(1/2)))/c^3-d^2*ln(d+c*x^(1/2))*l
n(c*((-1-a)^(1/2)+b^(1/2)*x^(1/2))/(c*(-1-a)^(1/2)-d*b^(1/2)))/c^3+d^2*ln(d+c*x^(1/2))*ln(c*((1-a)^(1/2)+b^(1/
2)*x^(1/2))/(c*(1-a)^(1/2)-d*b^(1/2)))/c^3-d^2*polylog(2,-b^(1/2)*(d+c*x^(1/2))/(c*(-1-a)^(1/2)-d*b^(1/2)))/c^
3+d^2*polylog(2,-b^(1/2)*(d+c*x^(1/2))/(c*(1-a)^(1/2)-d*b^(1/2)))/c^3-d^2*polylog(2,b^(1/2)*(d+c*x^(1/2))/(c*(
-1-a)^(1/2)+d*b^(1/2)))/c^3+d^2*polylog(2,b^(1/2)*(d+c*x^(1/2))/(c*(1-a)^(1/2)+d*b^(1/2)))/c^3+2*d*arctanh(b^(
1/2)*x^(1/2)/(1-a)^(1/2))*(1-a)^(1/2)/c^2/b^(1/2)-2*d*arctan(b^(1/2)*x^(1/2)/(1+a)^(1/2))*(1+a)^(1/2)/c^2/b^(1
/2)+d*ln(-b*x-a+1)*x^(1/2)/c^2-d*ln(b*x+a+1)*x^(1/2)/c^2

________________________________________________________________________________________

Rubi [A]
time = 0.79, antiderivative size = 661, normalized size of antiderivative = 1.00, 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 = {6250, 2455, 2526, 2498, 327, 211, 2504, 2436, 2332, 2512, 266, 2463, 2441, 2440, 2438, 214} \begin {gather*} -\frac {2 \sqrt {a+1} d \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+1}}\right )}{\sqrt {b} c^2}-\frac {d^2 \text {Li}_2\left (-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-1} c-\sqrt {b} d}\right )}{c^3}+\frac {d^2 \text {Li}_2\left (-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )}{c^3}-\frac {d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-1} c+\sqrt {b} d}\right )}{c^3}+\frac {d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+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 {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} \end {gather*}

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 211

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

Rule 214

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

Rule 266

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 327

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 2332

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

Rule 2436

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 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 2455

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 2463

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 2498

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 2504

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 2512

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 2526

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

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Mathematica [A]
time = 0.44, size = 598, normalized size = 0.90 \begin {gather*} \frac {4 c d \left (\sqrt {x}-\frac {\sqrt {1+a} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\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 d \sqrt {x} \log (1-a-b x)-\frac {c^2 (-1+a+b x) \log (1-a-b x)}{b}-2 d^2 \log \left (d+c \sqrt {x}\right ) \log (1-a-b x)-2 c d \sqrt {x} \log (1+a+b x)+\frac {c^2 (1+a+b x) \log (1+a+b x)}{b}+2 d^2 \log \left (d+c \sqrt {x}\right ) \log (1+a+b x)-2 d^2 \left (\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 ) \log \left (d+c \sqrt {x}\right )+\text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{-\sqrt {-1-a} c+\sqrt {b} d}\right )+\text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right )\right )+2 d^2 \left (\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 ) \log \left (d+c \sqrt {x}\right )+\text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{-\sqrt {1-a} c+\sqrt {b} d}\right )+\text {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )\right )}{2 c^3} \end {gather*}

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 = 1.60, size = 1001, normalized size = 1.51 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))-4*b/c^2*(1/4/c*d^2/
b*ln(d+c*x^(1/2))*ln((d*b-b*(d+c*x^(1/2))+(-a*b*c^2-b*c^2)^(1/2))/(d*b+(-a*b*c^2-b*c^2)^(1/2)))+1/4/c*d^2/b*ln
(d+c*x^(1/2))*ln((-d*b+b*(d+c*x^(1/2))+(-a*b*c^2-b*c^2)^(1/2))/(-d*b+(-a*b*c^2-b*c^2)^(1/2)))+1/4/c*d^2/b*dilo
g((d*b-b*(d+c*x^(1/2))+(-a*b*c^2-b*c^2)^(1/2))/(d*b+(-a*b*c^2-b*c^2)^(1/2)))+1/4/c*d^2/b*dilog((-d*b+b*(d+c*x^
(1/2))+(-a*b*c^2-b*c^2)^(1/2))/(-d*b+(-a*b*c^2-b*c^2)^(1/2)))-1/4/c*d^2/b*ln(d+c*x^(1/2))*ln((d*b-b*(d+c*x^(1/
2))+(-a*b*c^2+b*c^2)^(1/2))/(d*b+(-a*b*c^2+b*c^2)^(1/2)))-1/4/c*d^2/b*ln(d+c*x^(1/2))*ln((-d*b+b*(d+c*x^(1/2))
+(-a*b*c^2+b*c^2)^(1/2))/(-d*b+(-a*b*c^2+b*c^2)^(1/2)))-1/4/c*d^2/b*dilog((d*b-b*(d+c*x^(1/2))+(-a*b*c^2+b*c^2
)^(1/2))/(d*b+(-a*b*c^2+b*c^2)^(1/2)))-1/4/c*d^2/b*dilog((-d*b+b*(d+c*x^(1/2))+(-a*b*c^2+b*c^2)^(1/2))/(-d*b+(
-a*b*c^2+b*c^2)^(1/2)))-1/8*c/b^2*a*ln(a*c^2+d^2*b-2*b*d*(d+c*x^(1/2))+b*(d+c*x^(1/2))^2+c^2)+1/2*c/b*a*d/(a*b
*c^2+b*c^2)^(1/2)*arctan(1/2*(-2*d*b+2*b*(d+c*x^(1/2)))/(a*b*c^2+b*c^2)^(1/2))-1/8*c/b^2*ln(a*c^2+d^2*b-2*b*d*
(d+c*x^(1/2))+b*(d+c*x^(1/2))^2+c^2)+1/2*c/b*d/(a*b*c^2+b*c^2)^(1/2)*arctan(1/2*(-2*d*b+2*b*(d+c*x^(1/2)))/(a*
b*c^2+b*c^2)^(1/2))+1/8*c/b^2*a*ln(a*c^2+d^2*b-2*b*d*(d+c*x^(1/2))+b*(d+c*x^(1/2))^2-c^2)-1/2*c/b*a*d/(a*b*c^2
-b*c^2)^(1/2)*arctan(1/2*(-2*d*b+2*b*(d+c*x^(1/2)))/(a*b*c^2-b*c^2)^(1/2))-1/8*c/b^2*ln(a*c^2+d^2*b-2*b*d*(d+c
*x^(1/2))+b*(d+c*x^(1/2))^2-c^2)+1/2*c/b*d/(a*b*c^2-b*c^2)^(1/2)*arctan(1/2*(-2*d*b+2*b*(d+c*x^(1/2)))/(a*b*c^
2-b*c^2)^(1/2)))

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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(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 = 0.00, 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)/(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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, 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)/(c+d/x^(1/2)),x, algorithm="giac")

[Out]

integrate(arctanh(b*x + a)/(c + d/sqrt(x)), 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 )}{c+\frac {d}{\sqrt {x}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a + b*x)/(c + d/x^(1/2)),x)

[Out]

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

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