\(\int \frac {\coth ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx\) [80]

Optimal result

Integrand size = 18, antiderivative size = 619 \[ \int \frac {\coth ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=\frac {2 \sqrt {1+a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} d}-\frac {2 \sqrt {1-a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} d}+\frac {c \log \left (\frac {d \left (\sqrt {-1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {c \log \left (\frac {d \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {c \log \left (-\frac {d \left (\sqrt {-1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {c \log \left (-\frac {d \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {\sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{d^2}+\frac {\sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{d^2}+\frac {c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-1-a} d}\right )}{d^2}+\frac {c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-1-a} d}\right )}{d^2}-\frac {c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right )}{d^2}-\frac {c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {1-a} d}\right )}{d^2} \]

[Out]

c*ln((b*x+a-1)/(b*x+a))*ln(c+d*x^(1/2))/d^2-c*ln((b*x+a+1)/(b*x+a))*ln(c+d*x^(1/2))/d^2+c*ln(c+d*x^(1/2))*ln(d
*((-1-a)^(1/2)-b^(1/2)*x^(1/2))/(d*(-1-a)^(1/2)+c*b^(1/2)))/d^2-c*ln(c+d*x^(1/2))*ln(d*((1-a)^(1/2)-b^(1/2)*x^
(1/2))/(d*(1-a)^(1/2)+c*b^(1/2)))/d^2+c*ln(c+d*x^(1/2))*ln(-d*((-1-a)^(1/2)+b^(1/2)*x^(1/2))/(-d*(-1-a)^(1/2)+
c*b^(1/2)))/d^2-c*ln(c+d*x^(1/2))*ln(-d*((1-a)^(1/2)+b^(1/2)*x^(1/2))/(-d*(1-a)^(1/2)+c*b^(1/2)))/d^2+c*polylo
g(2,b^(1/2)*(c+d*x^(1/2))/(-d*(-1-a)^(1/2)+c*b^(1/2)))/d^2+c*polylog(2,b^(1/2)*(c+d*x^(1/2))/(d*(-1-a)^(1/2)+c
*b^(1/2)))/d^2-c*polylog(2,b^(1/2)*(c+d*x^(1/2))/(-d*(1-a)^(1/2)+c*b^(1/2)))/d^2-c*polylog(2,b^(1/2)*(c+d*x^(1
/2))/(d*(1-a)^(1/2)+c*b^(1/2)))/d^2-2*arctanh(b^(1/2)*x^(1/2)/(1-a)^(1/2))*(1-a)^(1/2)/d/b^(1/2)+2*arctan(b^(1
/2)*x^(1/2)/(1+a)^(1/2))*(1+a)^(1/2)/d/b^(1/2)-ln((b*x+a-1)/(b*x+a))*x^(1/2)/d+ln((b*x+a+1)/(b*x+a))*x^(1/2)/d

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.00, number of steps used = 55, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {6251, 196, 45, 2608, 2603, 12, 492, 211, 2604, 2465, 266, 2463, 2441, 2440, 2438, 214} \[ \int \frac {\coth ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=\frac {2 \sqrt {a+1} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+1}}\right )}{\sqrt {b} d}-\frac {2 \sqrt {1-a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} d}+\frac {c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a-1} d}\right )}{d^2}+\frac {c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-a-1} d}\right )}{d^2}-\frac {c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right )}{d^2}-\frac {c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {1-a} d}\right )}{d^2}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (\sqrt {-a-1}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} d+\sqrt {b} c}\right )}{d^2}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} d+\sqrt {b} c}\right )}{d^2}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {-a-1}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a-1} d}\right )}{d^2}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right )}{d^2}+\frac {c \log \left (-\frac {-a-b x+1}{a+b x}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {c \log \left (\frac {a+b x+1}{a+b x}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {\sqrt {x} \log \left (-\frac {-a-b x+1}{a+b x}\right )}{d}+\frac {\sqrt {x} \log \left (\frac {a+b x+1}{a+b x}\right )}{d} \]

[In]

Int[ArcCoth[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)/(a + b*x))])/d + (c*Log[c + d*Sqrt[x]]*Log[-((1 - a - b*x)/(a + b*x))])/d^2 + (Sqrt[x]*Log[(
1 + a + b*x)/(a + b*x)])/d - (c*Log[c + d*Sqrt[x]]*Log[(1 + a + b*x)/(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 + Sqr
t[-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*Sqrt[x]))/(Sqrt[b]*c + Sqrt[1 - a]*d)])/d^2

Rule 12

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 196

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

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 492

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(-a)*(e^n/(b*c -
a*d)), Int[(e*x)^(m - n)/(a + b*x^n), x], x] + Dist[c*(e^n/(b*c - a*d)), Int[(e*x)^(m - n)/(c + d*x^n), x], x
] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1]

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 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 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2603

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[x*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2604

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[Log[d + e*x]*((a + b
*Log[c*RFx^p])^n/e), x] - Dist[b*n*(p/e), Int[Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2608

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 6251

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

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int \frac {\log \left (\frac {-1+a+b x}{a+b x}\right )}{c+d \sqrt {x}} \, dx\right )+\frac {1}{2} \int \frac {\log \left (\frac {1+a+b x}{a+b x}\right )}{c+d \sqrt {x}} \, dx \\ & = -\text {Subst}\left (\int \frac {x \log \left (\frac {-1+a+b x^2}{a+b x^2}\right )}{c+d x} \, dx,x,\sqrt {x}\right )+\text {Subst}\left (\int \frac {x \log \left (\frac {1+a+b x^2}{a+b x^2}\right )}{c+d x} \, dx,x,\sqrt {x}\right ) \\ & = -\text {Subst}\left (\int \left (\frac {\log \left (\frac {-1+a+b x^2}{a+b x^2}\right )}{d}-\frac {c \log \left (\frac {-1+a+b x^2}{a+b x^2}\right )}{d (c+d x)}\right ) \, dx,x,\sqrt {x}\right )+\text {Subst}\left (\int \left (\frac {\log \left (\frac {1+a+b x^2}{a+b x^2}\right )}{d}-\frac {c \log \left (\frac {1+a+b x^2}{a+b x^2}\right )}{d (c+d x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\text {Subst}\left (\int \log \left (\frac {-1+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\text {Subst}\left (\int \log \left (\frac {1+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {c \text {Subst}\left (\int \frac {\log \left (\frac {-1+a+b x^2}{a+b x^2}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}-\frac {c \text {Subst}\left (\int \frac {\log \left (\frac {1+a+b x^2}{a+b x^2}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {\sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{d^2}+\frac {\sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{d^2}-\frac {c \text {Subst}\left (\int \frac {\left (a+b x^2\right ) \left (-\frac {2 b x \left (-1+a+b x^2\right )}{\left (a+b x^2\right )^2}+\frac {2 b x}{a+b x^2}\right ) \log (c+d x)}{-1+a+b x^2} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {c \text {Subst}\left (\int \frac {\left (a+b x^2\right ) \left (\frac {2 b x}{a+b x^2}-\frac {2 b x \left (1+a+b x^2\right )}{\left (a+b x^2\right )^2}\right ) \log (c+d x)}{1+a+b x^2} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {\text {Subst}\left (\int \frac {2 b x^2}{\left (-1+a+b x^2\right ) \left (a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{d}-\frac {\text {Subst}\left (\int \frac {2 b x^2}{\left (-a-b x^2\right ) \left (1+a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {\sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{d^2}+\frac {\sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{d^2}-\frac {c \text {Subst}\left (\int \left (\frac {2 b x \log (c+d x)}{-1+a+b x^2}-\frac {2 b x \log (c+d x)}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {c \text {Subst}\left (\int \left (-\frac {2 b x \log (c+d x)}{a+b x^2}+\frac {2 b x \log (c+d x)}{1+a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(2 b) \text {Subst}\left (\int \frac {x^2}{\left (-1+a+b x^2\right ) \left (a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{d}-\frac {(2 b) \text {Subst}\left (\int \frac {x^2}{\left (-a-b x^2\right ) \left (1+a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {\sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{d^2}+\frac {\sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{d^2}-\frac {(2 b c) \text {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) \text {Subst}\left (\int \frac {x \log (c+d x)}{1+a+b x^2} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(2 (1-a)) \text {Subst}\left (\int \frac {1}{-1+a+b x^2} \, dx,x,\sqrt {x}\right )}{d}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{-a-b x^2} \, dx,x,\sqrt {x}\right )}{d}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{d}+\frac {(2 (1+a)) \text {Subst}\left (\int \frac {1}{1+a+b x^2} \, dx,x,\sqrt {x}\right )}{d} \\ & = \frac {2 \sqrt {1+a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} d}-\frac {2 \sqrt {1-a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} d}-\frac {\sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{d^2}+\frac {\sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{d^2}+\frac {(2 b c) \text {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) \text {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 \sqrt {1+a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} d}-\frac {2 \sqrt {1-a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} d}-\frac {\sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{d^2}+\frac {\sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{d^2}-\frac {\left (\sqrt {b} c\right ) \text {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 ) \text {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 ) \text {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 ) \text {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} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} d}-\frac {2 \sqrt {1-a} \text {arctanh}\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 \left (-\frac {1-a-b x}{a+b x}\right )}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{d^2}+\frac {\sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{d^2}-\frac {c \text {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 \text {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 \text {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 \text {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} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} d}-\frac {2 \sqrt {1-a} \text {arctanh}\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 \left (-\frac {1-a-b x}{a+b x}\right )}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{d^2}+\frac {\sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{d^2}-\frac {c \text {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 \text {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 \text {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 \text {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} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} d}-\frac {2 \sqrt {1-a} \text {arctanh}\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 \left (-\frac {1-a-b x}{a+b x}\right )}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{d^2}+\frac {\sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{d^2}+\frac {c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-1-a} d}\right )}{d^2}+\frac {c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-1-a} d}\right )}{d^2}-\frac {c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right )}{d^2}-\frac {c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {1-a} d}\right )}{d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 575, normalized size of antiderivative = 0.93 \[ \int \frac {\coth ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=\frac {\frac {2 \sqrt {1+a} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b}}-\frac {2 \sqrt {1-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b}}+c \log \left (\frac {d \left (\sqrt {-1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-1-a} d}\right ) \log \left (c+d \sqrt {x}\right )-c \log \left (\frac {d \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {1-a} d}\right ) \log \left (c+d \sqrt {x}\right )+c \log \left (\frac {d \left (\sqrt {-1-a}+\sqrt {b} \sqrt {x}\right )}{-\sqrt {b} c+\sqrt {-1-a} d}\right ) \log \left (c+d \sqrt {x}\right )-c \log \left (\frac {d \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{-\sqrt {b} c+\sqrt {1-a} d}\right ) \log \left (c+d \sqrt {x}\right )-d \sqrt {x} \log \left (\frac {-1+a+b x}{a+b x}\right )+c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {-1+a+b x}{a+b x}\right )+d \sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )-c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )+c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-1-a} d}\right )+c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-1-a} d}\right )-c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right )-c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {1-a} d}\right )}{d^2} \]

[In]

Integrate[ArcCoth[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)/(a + b*x)
] + c*Log[c + d*Sqrt[x]]*Log[(-1 + a + b*x)/(a + b*x)] + d*Sqrt[x]*Log[(1 + a + b*x)/(a + b*x)] - c*Log[c + d*
Sqrt[x]]*Log[(1 + a + b*x)/(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

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 646, normalized size of antiderivative = 1.04

method	result	size
derivativedivides	\(\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) \sqrt {x}}{d}-\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) c \ln \left (c +d \sqrt {x}\right )}{d^{2}}+\frac {4 b \left (d^{2} \left (\frac {\left (1+a \right ) \arctan \left (\frac {-2 b c +2 b \left (c +d \sqrt {x}\right )}{2 \sqrt {a b \,d^{2}+b \,d^{2}}}\right )}{2 b \sqrt {a b \,d^{2}+b \,d^{2}}}+\frac {\left (1-a \right ) \arctan \left (\frac {-2 b c +2 b \left (c +d \sqrt {x}\right )}{2 \sqrt {a b \,d^{2}-b \,d^{2}}}\right )}{2 b \sqrt {a b \,d^{2}-b \,d^{2}}}\right )+c \,d^{2} \left (\frac {\frac {\ln \left (c +d \sqrt {x}\right ) \left (\ln \left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-b \,d^{2}}}{b c +\sqrt {-a b \,d^{2}-b \,d^{2}}}\right )+\ln \left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-b \,d^{2}}}{-b c +\sqrt {-a b \,d^{2}-b \,d^{2}}}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-b \,d^{2}}}{b c +\sqrt {-a b \,d^{2}-b \,d^{2}}}\right )+\operatorname {dilog}\left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-b \,d^{2}}}{-b c +\sqrt {-a b \,d^{2}-b \,d^{2}}}\right )}{2 b}}{2 d^{2}}+\frac {-\frac {\ln \left (c +d \sqrt {x}\right ) \left (\ln \left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+b \,d^{2}}}{b c +\sqrt {-a b \,d^{2}+b \,d^{2}}}\right )+\ln \left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+b \,d^{2}}}{-b c +\sqrt {-a b \,d^{2}+b \,d^{2}}}\right )\right )}{2 b}-\frac {\operatorname {dilog}\left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+b \,d^{2}}}{b c +\sqrt {-a b \,d^{2}+b \,d^{2}}}\right )+\operatorname {dilog}\left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+b \,d^{2}}}{-b c +\sqrt {-a b \,d^{2}+b \,d^{2}}}\right )}{2 b}}{2 d^{2}}\right )\right )}{d^{2}}\)	\(646\)
default	\(\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) \sqrt {x}}{d}-\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) c \ln \left (c +d \sqrt {x}\right )}{d^{2}}+\frac {4 b \left (d^{2} \left (\frac {\left (1+a \right ) \arctan \left (\frac {-2 b c +2 b \left (c +d \sqrt {x}\right )}{2 \sqrt {a b \,d^{2}+b \,d^{2}}}\right )}{2 b \sqrt {a b \,d^{2}+b \,d^{2}}}+\frac {\left (1-a \right ) \arctan \left (\frac {-2 b c +2 b \left (c +d \sqrt {x}\right )}{2 \sqrt {a b \,d^{2}-b \,d^{2}}}\right )}{2 b \sqrt {a b \,d^{2}-b \,d^{2}}}\right )+c \,d^{2} \left (\frac {\frac {\ln \left (c +d \sqrt {x}\right ) \left (\ln \left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-b \,d^{2}}}{b c +\sqrt {-a b \,d^{2}-b \,d^{2}}}\right )+\ln \left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-b \,d^{2}}}{-b c +\sqrt {-a b \,d^{2}-b \,d^{2}}}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-b \,d^{2}}}{b c +\sqrt {-a b \,d^{2}-b \,d^{2}}}\right )+\operatorname {dilog}\left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-b \,d^{2}}}{-b c +\sqrt {-a b \,d^{2}-b \,d^{2}}}\right )}{2 b}}{2 d^{2}}+\frac {-\frac {\ln \left (c +d \sqrt {x}\right ) \left (\ln \left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+b \,d^{2}}}{b c +\sqrt {-a b \,d^{2}+b \,d^{2}}}\right )+\ln \left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+b \,d^{2}}}{-b c +\sqrt {-a b \,d^{2}+b \,d^{2}}}\right )\right )}{2 b}-\frac {\operatorname {dilog}\left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+b \,d^{2}}}{b c +\sqrt {-a b \,d^{2}+b \,d^{2}}}\right )+\operatorname {dilog}\left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+b \,d^{2}}}{-b c +\sqrt {-a b \,d^{2}+b \,d^{2}}}\right )}{2 b}}{2 d^{2}}\right )\right )}{d^{2}}\)	\(646\)

[In]

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

[Out]

2*arccoth(b*x+a)/d*x^(1/2)-2*arccoth(b*x+a)*c/d^2*ln(c+d*x^(1/2))+4*b/d^2*(d^2*(1/2*(1+a)/b/(a*b*d^2+b*d^2)^(1
/2)*arctan(1/2*(-2*b*c+2*b*(c+d*x^(1/2)))/(a*b*d^2+b*d^2)^(1/2))+1/2*(1-a)/b/(a*b*d^2-b*d^2)^(1/2)*arctan(1/2*
(-2*b*c+2*b*(c+d*x^(1/2)))/(a*b*d^2-b*d^2)^(1/2)))+c*d^2*(1/2/d^2*(1/2*ln(c+d*x^(1/2))*(ln((b*c-b*(c+d*x^(1/2)
)+(-a*b*d^2-b*d^2)^(1/2))/(b*c+(-a*b*d^2-b*d^2)^(1/2)))+ln((-b*c+b*(c+d*x^(1/2))+(-a*b*d^2-b*d^2)^(1/2))/(-b*c
+(-a*b*d^2-b*d^2)^(1/2))))/b+1/2*(dilog((b*c-b*(c+d*x^(1/2))+(-a*b*d^2-b*d^2)^(1/2))/(b*c+(-a*b*d^2-b*d^2)^(1/
2)))+dilog((-b*c+b*(c+d*x^(1/2))+(-a*b*d^2-b*d^2)^(1/2))/(-b*c+(-a*b*d^2-b*d^2)^(1/2))))/b)+1/2/d^2*(-1/2*ln(c
+d*x^(1/2))*(ln((b*c-b*(c+d*x^(1/2))+(-a*b*d^2+b*d^2)^(1/2))/(b*c+(-a*b*d^2+b*d^2)^(1/2)))+ln((-b*c+b*(c+d*x^(
1/2))+(-a*b*d^2+b*d^2)^(1/2))/(-b*c+(-a*b*d^2+b*d^2)^(1/2))))/b-1/2*(dilog((b*c-b*(c+d*x^(1/2))+(-a*b*d^2+b*d^
2)^(1/2))/(b*c+(-a*b*d^2+b*d^2)^(1/2)))+dilog((-b*c+b*(c+d*x^(1/2))+(-a*b*d^2+b*d^2)^(1/2))/(-b*c+(-a*b*d^2+b*
d^2)^(1/2))))/b)))

Fricas [F]

\[ \int \frac {\coth ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{d \sqrt {x} + c} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\coth ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{d \sqrt {x} + c} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\coth ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{d \sqrt {x} + c} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=\int \frac {\mathrm {acoth}\left (a+b\,x\right )}{c+d\,\sqrt {x}} \,d x \]

[In]

int(acoth(a + b*x)/(c + d*x^(1/2)),x)

[Out]

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