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

Optimal. Leaf size=318 \[ -\frac{b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{2 e}-\frac{b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{2 e}+\frac{b \text{PolyLog}\left (2,1-\frac{2}{c \sqrt{x}+1}\right )}{e}+\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{e}+\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{e}-\frac{2 \log \left (\frac{2}{c \sqrt{x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.321541, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6044, 5920, 2402, 2315, 2447} \[ -\frac{b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{2 e}-\frac{b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{2 e}+\frac{b \text{PolyLog}\left (2,1-\frac{2}{c \sqrt{x}+1}\right )}{e}+\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{e}+\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{e}-\frac{2 \log \left (\frac{2}{c \sqrt{x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 1.53407, size = 551, normalized size = 1.73 \[ \frac{a \log (d+e x)}{e}-\frac{b \left (\text{PolyLog}\left (2,\frac{\left (-2 \sqrt{-c^2 d e}+c^2 (-d)+e\right ) e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d+e}\right )+\text{PolyLog}\left (2,\frac{\left (2 \sqrt{-c^2 d e}+c^2 (-d)+e\right ) e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d+e}\right )-2 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )-2 \tanh ^{-1}\left (c \sqrt{x}\right ) \log \left (\frac{e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )} \left (-2 \sqrt{-c^2 d e}+c^2 d \left (e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )+e \left (e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}-1\right )\right )}{c^2 d+e}\right )-2 \tanh ^{-1}\left (c \sqrt{x}\right ) \log \left (\frac{e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )} \left (2 \sqrt{-c^2 d e}+c^2 d \left (e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )+e \left (e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}-1\right )\right )}{c^2 d+e}\right )+4 i \sin ^{-1}\left (\sqrt{\frac{c^2 d}{c^2 d+e}}\right ) \tanh ^{-1}\left (\frac{c e \sqrt{x}}{\sqrt{-c^2 d e}}\right )+2 i \sin ^{-1}\left (\sqrt{\frac{c^2 d}{c^2 d+e}}\right ) \log \left (\frac{e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )} \left (-2 \sqrt{-c^2 d e}+c^2 d \left (e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )+e \left (e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}-1\right )\right )}{c^2 d+e}\right )-2 i \sin ^{-1}\left (\sqrt{\frac{c^2 d}{c^2 d+e}}\right ) \log \left (\frac{e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )} \left (2 \sqrt{-c^2 d e}+c^2 d \left (e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )+e \left (e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}-1\right )\right )}{c^2 d+e}\right )+4 \tanh ^{-1}\left (c \sqrt{x}\right ) \log \left (e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )\right )}{2 e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*Log[d + e*x])/e - (b*((4*I)*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]]*ArcTanh[(c*e*Sqrt[x])/Sqrt[-(c^2*d*e)]] + 4*A
rcTanh[c*Sqrt[x]]*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])] + (2*I)*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]]*Log[(-2*Sqrt[-(
c^2*d*e)] + e*(-1 + E^(2*ArcTanh[c*Sqrt[x]])) + c^2*d*(1 + E^(2*ArcTanh[c*Sqrt[x]])))/((c^2*d + e)*E^(2*ArcTan
h[c*Sqrt[x]]))] - 2*ArcTanh[c*Sqrt[x]]*Log[(-2*Sqrt[-(c^2*d*e)] + e*(-1 + E^(2*ArcTanh[c*Sqrt[x]])) + c^2*d*(1
+ E^(2*ArcTanh[c*Sqrt[x]])))/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] - (2*I)*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]
]*Log[(2*Sqrt[-(c^2*d*e)] + e*(-1 + E^(2*ArcTanh[c*Sqrt[x]])) + c^2*d*(1 + E^(2*ArcTanh[c*Sqrt[x]])))/((c^2*d
+ e)*E^(2*ArcTanh[c*Sqrt[x]]))] - 2*ArcTanh[c*Sqrt[x]]*Log[(2*Sqrt[-(c^2*d*e)] + e*(-1 + E^(2*ArcTanh[c*Sqrt[x
]])) + c^2*d*(1 + E^(2*ArcTanh[c*Sqrt[x]])))/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] - 2*PolyLog[2, -E^(-2*Arc
Tanh[c*Sqrt[x]])] + PolyLog[2, (-(c^2*d) + e - 2*Sqrt[-(c^2*d*e)])/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] + P
olyLog[2, (-(c^2*d) + e + 2*Sqrt[-(c^2*d*e)])/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))]))/(2*e)

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Maple [A]  time = 0.053, size = 462, normalized size = 1.5 \begin{align*}{\frac{a\ln \left ({c}^{2}ex+{c}^{2}d \right ) }{e}}+{\frac{b\ln \left ({c}^{2}ex+{c}^{2}d \right ) }{e}{\it Artanh} \left ( c\sqrt{x} \right ) }+{\frac{b\ln \left ({c}^{2}ex+{c}^{2}d \right ) }{2\,e}\ln \left ( c\sqrt{x}-1 \right ) }-{\frac{b}{2\,e}\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\,e}\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\,e}{\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\,e}{\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\,e}\ln \left ( 1+c\sqrt{x} \right ) }+{\frac{b}{2\,e}\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\,e}\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\,e}{\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\,e}{\it dilog} \left ({ \left ( c\sqrt{-de}+e \left ( 1+c\sqrt{x} \right ) -e \right ) \left ( c\sqrt{-de}-e \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arctanh(c*sqrt(x)) + a)/(e*x + d), 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)))/(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}{e x + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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