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

Optimal. Leaf size=358 \[ \frac{b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{2 d}+\frac{b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{2 d}-\frac{b \text{PolyLog}\left (2,1-\frac{2}{c \sqrt{x}+1}\right )}{d}-\frac{b \text{PolyLog}\left (2,-c \sqrt{x}\right )}{d}+\frac{b \text{PolyLog}\left (2,c \sqrt{x}\right )}{d}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{d}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{d}+\frac{2 \log \left (\frac{2}{c \sqrt{x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{d}+\frac{a \log (x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.58681, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {36, 29, 31, 1593, 5992, 5912, 6044, 5920, 2402, 2315, 2447} \[ \frac{b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{2 d}+\frac{b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{2 d}-\frac{b \text{PolyLog}\left (2,1-\frac{2}{c \sqrt{x}+1}\right )}{d}-\frac{b \text{PolyLog}\left (2,-c \sqrt{x}\right )}{d}+\frac{b \text{PolyLog}\left (2,c \sqrt{x}\right )}{d}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{d}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{d}+\frac{2 \log \left (\frac{2}{c \sqrt{x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{d}+\frac{a \log (x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 5992

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTanh[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[
a, 0])

Rule 5912

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Simp[(b*PolyLog[2, -(c*x)])/2
, x] + Simp[(b*PolyLog[2, c*x])/2, x]) /; FreeQ[{a, b, c}, x]

Rule 6044

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a,
b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (GtQ[q, 0] || IntegerQ[m])

Rule 5920

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

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

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

Mathematica [A]  time = 1.12444, size = 302, normalized size = 0.84 \[ -\frac{b \text{PolyLog}\left (2,-\frac{\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d-2 c \sqrt{-d} \sqrt{e}-e}\right )+b \text{PolyLog}\left (2,-\frac{\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d+2 c \sqrt{-d} \sqrt{e}-e}\right )+2 b \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )+2 a \log (d+e x)-2 a \log (x)+2 b \tanh ^{-1}\left (c \sqrt{x}\right ) \log \left (\frac{\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d-2 c \sqrt{-d} \sqrt{e}-e}+1\right )+2 b \tanh ^{-1}\left (c \sqrt{x}\right ) \log \left (\frac{\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d+2 c \sqrt{-d} \sqrt{e}-e}+1\right )-4 b \tanh ^{-1}\left (c \sqrt{x}\right )^2-4 b \tanh ^{-1}\left (c \sqrt{x}\right ) \log \left (1-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )}{2 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(-4*b*ArcTanh[c*Sqrt[x]]^2 - 4*b*ArcTanh[c*Sqrt[x]]*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])] + 2*b*ArcTanh[c*Sqrt[x
]]*Log[1 + ((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d - 2*c*Sqrt[-d]*Sqrt[e] - e)] + 2*b*ArcTanh[c*Sqrt[x]]
*Log[1 + ((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d + 2*c*Sqrt[-d]*Sqrt[e] - e)] - 2*a*Log[x] + 2*a*Log[d +
e*x] + 2*b*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])] + b*PolyLog[2, -(((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2
*d - 2*c*Sqrt[-d]*Sqrt[e] - e))] + b*PolyLog[2, -(((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d + 2*c*Sqrt[-d]
*Sqrt[e] - e))])/(2*d)

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Maple [A]  time = 0.062, size = 540, normalized size = 1.5 \begin{align*} -{\frac{a\ln \left ({c}^{2}ex+{c}^{2}d \right ) }{d}}+2\,{\frac{a\ln \left ( c\sqrt{x} \right ) }{d}}-{\frac{b\ln \left ({c}^{2}ex+{c}^{2}d \right ) }{d}{\it Artanh} \left ( c\sqrt{x} \right ) }+2\,{\frac{b{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( c\sqrt{x} \right ) }{d}}-{\frac{b\ln \left ({c}^{2}ex+{c}^{2}d \right ) }{2\,d}\ln \left ( c\sqrt{x}-1 \right ) }+{\frac{b}{2\,d}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({ \left ( c\sqrt{-de}-e \left ( c\sqrt{x}-1 \right ) -e \right ) \left ( c\sqrt{-de}-e \right ) ^{-1}} \right ) }+{\frac{b}{2\,d}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({ \left ( c\sqrt{-de}+e \left ( c\sqrt{x}-1 \right ) +e \right ) \left ( c\sqrt{-de}+e \right ) ^{-1}} \right ) }+{\frac{b}{2\,d}{\it dilog} \left ({ \left ( c\sqrt{-de}-e \left ( c\sqrt{x}-1 \right ) -e \right ) \left ( c\sqrt{-de}-e \right ) ^{-1}} \right ) }+{\frac{b}{2\,d}{\it dilog} \left ({ \left ( c\sqrt{-de}+e \left ( c\sqrt{x}-1 \right ) +e \right ) \left ( c\sqrt{-de}+e \right ) ^{-1}} \right ) }+{\frac{b\ln \left ({c}^{2}ex+{c}^{2}d \right ) }{2\,d}\ln \left ( 1+c\sqrt{x} \right ) }-{\frac{b}{2\,d}\ln \left ( 1+c\sqrt{x} \right ) \ln \left ({ \left ( c\sqrt{-de}-e \left ( 1+c\sqrt{x} \right ) +e \right ) \left ( c\sqrt{-de}+e \right ) ^{-1}} \right ) }-{\frac{b}{2\,d}\ln \left ( 1+c\sqrt{x} \right ) \ln \left ({ \left ( c\sqrt{-de}+e \left ( 1+c\sqrt{x} \right ) -e \right ) \left ( c\sqrt{-de}-e \right ) ^{-1}} \right ) }-{\frac{b}{2\,d}{\it dilog} \left ({ \left ( c\sqrt{-de}-e \left ( 1+c\sqrt{x} \right ) +e \right ) \left ( c\sqrt{-de}+e \right ) ^{-1}} \right ) }-{\frac{b}{2\,d}{\it dilog} \left ({ \left ( c\sqrt{-de}+e \left ( 1+c\sqrt{x} \right ) -e \right ) \left ( c\sqrt{-de}-e \right ) ^{-1}} \right ) }-{\frac{b}{d}{\it dilog} \left ( c\sqrt{x} \right ) }-{\frac{b}{d}{\it dilog} \left ( 1+c\sqrt{x} \right ) }-{\frac{b}{d}\ln \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -a{\left (\frac{\log \left (e x + d\right )}{d} - \frac{\log \left (x\right )}{d}\right )} + b \int \frac{\log \left (c \sqrt{x} + 1\right )}{2 \,{\left (e x^{\frac{3}{2}} + d \sqrt{x}\right )} \sqrt{x}}\,{d x} - b \int \frac{\log \left (-c \sqrt{x} + 1\right )}{2 \,{\left (e x^{\frac{3}{2}} + d \sqrt{x}\right )} \sqrt{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*(log(e*x + d)/d - log(x)/d) + b*integrate(1/2*log(c*sqrt(x) + 1)/((e*x^(3/2) + d*sqrt(x))*sqrt(x)), x) - b*
integrate(1/2*log(-c*sqrt(x) + 1)/((e*x^(3/2) + d*sqrt(x))*sqrt(x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arctanh(c*sqrt(x)) + a)/(e*x^2 + d*x), 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((a+b*atanh(c*x**(1/2)))/x/(e*x+d),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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