3.1.59 \(\int \frac {\tanh ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx\) [59]

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

[Out]

c*ln(-b*x-a+1)*ln(c+d*x^(1/2))/d^2-c*ln(b*x+a+1)*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*l
n(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*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*po
lylog(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)*x^(1/2)/d+ln(b*x+a+1)*x^(1/2)/d

________________________________________________________________________________________

Rubi [A]
time = 0.87, antiderivative size = 585, normalized size of antiderivative = 1.00, 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 = {6250, 2455, 2516, 2498, 327, 211, 2512, 266, 2463, 2441, 2440, 2438, 214} \begin {gather*} \frac {2 \sqrt {a+1} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+1}}\right )}{\sqrt {b} d}+\frac {c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a-1} d}\right )}{d^2}+\frac {c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-a-1} 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 \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 {1-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} d} \end {gather*}

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 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 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 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 2516

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

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Mathematica [A]
time = 0.38, size = 549, normalized size = 0.94 \begin {gather*} \frac {\frac {2 \sqrt {1+a} d \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b}}-\frac {2 \sqrt {1-a} d \tanh ^{-1}\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 (1-a-b x)+c \log \left (c+d \sqrt {x}\right ) \log (1-a-b x)+d \sqrt {x} \log (1+a+b x)-c \log \left (c+d \sqrt {x}\right ) \log (1+a+b x)+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 {b} c+\sqrt {-1-a} d}\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 {b} c+\sqrt {1-a} d}\right )}{d^2} \end {gather*}

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 = 2.07, size = 773, normalized size = 1.32

method	result	size
derivativedivides	\(\frac {2 \arctanh \left (b x +a \right ) \sqrt {x}}{d}-\frac {2 \arctanh \left (b x +a \right ) c \ln \left (c +d \sqrt {x}\right )}{d^{2}}-\frac {4 b \left (-\frac {d^{2} \arctan \left (\frac {-2 b c +2 b \left (c +d \sqrt {x}\right )}{2 \sqrt {a b \,d^{2}+d^{2} b}}\right ) a}{2 b \sqrt {a b \,d^{2}+d^{2} b}}-\frac {d^{2} \arctan \left (\frac {-2 b c +2 b \left (c +d \sqrt {x}\right )}{2 \sqrt {a b \,d^{2}+d^{2} b}}\right )}{2 b \sqrt {a b \,d^{2}+d^{2} b}}+\frac {d^{2} \arctan \left (\frac {-2 b c +2 b \left (c +d \sqrt {x}\right )}{2 \sqrt {a b \,d^{2}-d^{2} b}}\right ) a}{2 b \sqrt {a b \,d^{2}-d^{2} b}}-\frac {d^{2} \arctan \left (\frac {-2 b c +2 b \left (c +d \sqrt {x}\right )}{2 \sqrt {a b \,d^{2}-d^{2} b}}\right )}{2 b \sqrt {a b \,d^{2}-d^{2} b}}-\frac {c \ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-d^{2} b}}{b c +\sqrt {-a b \,d^{2}-d^{2} b}}\right )}{4 b}-\frac {c \ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-d^{2} b}}{-b c +\sqrt {-a b \,d^{2}-d^{2} b}}\right )}{4 b}-\frac {c \dilog \left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-d^{2} b}}{b c +\sqrt {-a b \,d^{2}-d^{2} b}}\right )}{4 b}-\frac {c \dilog \left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-d^{2} b}}{-b c +\sqrt {-a b \,d^{2}-d^{2} b}}\right )}{4 b}+\frac {c \ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+d^{2} b}}{b c +\sqrt {-a b \,d^{2}+d^{2} b}}\right )}{4 b}+\frac {c \ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+d^{2} b}}{-b c +\sqrt {-a b \,d^{2}+d^{2} b}}\right )}{4 b}+\frac {c \dilog \left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+d^{2} b}}{b c +\sqrt {-a b \,d^{2}+d^{2} b}}\right )}{4 b}+\frac {c \dilog \left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+d^{2} b}}{-b c +\sqrt {-a b \,d^{2}+d^{2} b}}\right )}{4 b}\right )}{d^{2}}\)	\(773\)
default	\(\frac {2 \arctanh \left (b x +a \right ) \sqrt {x}}{d}-\frac {2 \arctanh \left (b x +a \right ) c \ln \left (c +d \sqrt {x}\right )}{d^{2}}-\frac {4 b \left (-\frac {d^{2} \arctan \left (\frac {-2 b c +2 b \left (c +d \sqrt {x}\right )}{2 \sqrt {a b \,d^{2}+d^{2} b}}\right ) a}{2 b \sqrt {a b \,d^{2}+d^{2} b}}-\frac {d^{2} \arctan \left (\frac {-2 b c +2 b \left (c +d \sqrt {x}\right )}{2 \sqrt {a b \,d^{2}+d^{2} b}}\right )}{2 b \sqrt {a b \,d^{2}+d^{2} b}}+\frac {d^{2} \arctan \left (\frac {-2 b c +2 b \left (c +d \sqrt {x}\right )}{2 \sqrt {a b \,d^{2}-d^{2} b}}\right ) a}{2 b \sqrt {a b \,d^{2}-d^{2} b}}-\frac {d^{2} \arctan \left (\frac {-2 b c +2 b \left (c +d \sqrt {x}\right )}{2 \sqrt {a b \,d^{2}-d^{2} b}}\right )}{2 b \sqrt {a b \,d^{2}-d^{2} b}}-\frac {c \ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-d^{2} b}}{b c +\sqrt {-a b \,d^{2}-d^{2} b}}\right )}{4 b}-\frac {c \ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-d^{2} b}}{-b c +\sqrt {-a b \,d^{2}-d^{2} b}}\right )}{4 b}-\frac {c \dilog \left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-d^{2} b}}{b c +\sqrt {-a b \,d^{2}-d^{2} b}}\right )}{4 b}-\frac {c \dilog \left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}-d^{2} b}}{-b c +\sqrt {-a b \,d^{2}-d^{2} b}}\right )}{4 b}+\frac {c \ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+d^{2} b}}{b c +\sqrt {-a b \,d^{2}+d^{2} b}}\right )}{4 b}+\frac {c \ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+d^{2} b}}{-b c +\sqrt {-a b \,d^{2}+d^{2} b}}\right )}{4 b}+\frac {c \dilog \left (\frac {b c -b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+d^{2} b}}{b c +\sqrt {-a b \,d^{2}+d^{2} b}}\right )}{4 b}+\frac {c \dilog \left (\frac {-b c +b \left (c +d \sqrt {x}\right )+\sqrt {-a b \,d^{2}+d^{2} b}}{-b c +\sqrt {-a b \,d^{2}+d^{2} b}}\right )}{4 b}\right )}{d^{2}}\)	\(773\)

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]

2*arctanh(b*x+a)/d*x^(1/2)-2*arctanh(b*x+a)*c/d^2*ln(c+d*x^(1/2))-4*b/d^2*(-1/2*d^2/b/(a*b*d^2+b*d^2)^(1/2)*ar
ctan(1/2*(-2*b*c+2*b*(c+d*x^(1/2)))/(a*b*d^2+b*d^2)^(1/2))*a-1/2*d^2/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*d^2/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))*a-1/2*d^2/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))-1/4*c/b*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
)))-1/4*c/b*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)))-1/
4*c/b*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)))-1/4*c/b*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)))+1/4*c/b*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)))+1/4*c/b*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)))+1/4*c/b*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)))+1/4*c/b*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))))

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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)/(d*sqrt(x) + c), 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((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)] 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)/(d*sqrt(x) + c), 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+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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