Integrand size = 23, antiderivative size = 358 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x (d+e x)} \, dx=\frac {2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{d}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{d}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{d}+\frac {a \log (x)}{d}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c \sqrt {x}}\right )}{d}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 d}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-c \sqrt {x}\right )}{d}+\frac {b \operatorname {PolyLog}\left (2,c \sqrt {x}\right )}{d} \] Output:
2*(a+b*arctanh(c*x^(1/2)))*ln(2/(1+c*x^(1/2)))/d-(a+b*arctanh(c*x^(1/2)))* ln(2*c*((-d)^(1/2)-e^(1/2)*x^(1/2))/(c*(-d)^(1/2)-e^(1/2))/(1+c*x^(1/2)))/ d-(a+b*arctanh(c*x^(1/2)))*ln(2*c*((-d)^(1/2)+e^(1/2)*x^(1/2))/(c*(-d)^(1/ 2)+e^(1/2))/(1+c*x^(1/2)))/d+a*ln(x)/d-b*polylog(2,1-2/(1+c*x^(1/2)))/d+1/ 2*b*polylog(2,1-2*c*((-d)^(1/2)-e^(1/2)*x^(1/2))/(c*(-d)^(1/2)-e^(1/2))/(1 +c*x^(1/2)))/d+1/2*b*polylog(2,1-2*c*((-d)^(1/2)+e^(1/2)*x^(1/2))/(c*(-d)^ (1/2)+e^(1/2))/(1+c*x^(1/2)))/d-b*polylog(2,-c*x^(1/2))/d+b*polylog(2,c*x^ (1/2))/d
Result contains complex when optimal does not.
Time = 1.50 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.24 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x (d+e x)} \, dx=\frac {-2 b \text {arctanh}\left (c \sqrt {x}\right )^2+4 i b \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right ) \text {arctanh}\left (\frac {c e \sqrt {x}}{\sqrt {-c^2 d e}}\right )+2 b \text {arctanh}\left (c \sqrt {x}\right ) \left (\text {arctanh}\left (c \sqrt {x}\right )+2 \log \left (1-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )+2 i b \left (\arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )+i \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {e^{-2 \text {arctanh}\left (c \sqrt {x}\right )} \left (-2 \sqrt {-c^2 d e}+e \left (-1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+c^2 d \left (1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )}{c^2 d+e}\right )-2 b \left (i \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )+\text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {e^{-2 \text {arctanh}\left (c \sqrt {x}\right )} \left (2 \sqrt {-c^2 d e}+e \left (-1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+c^2 d \left (1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )}{c^2 d+e}\right )+2 a \log (x)-2 a \log (d+e x)-2 b \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+b \operatorname {PolyLog}\left (2,\frac {\left (-c^2 d+e-2 \sqrt {-c^2 d e}\right ) e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}}{c^2 d+e}\right )+b \operatorname {PolyLog}\left (2,\frac {\left (-c^2 d+e+2 \sqrt {-c^2 d e}\right ) e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}}{c^2 d+e}\right )}{2 d} \] Input:
Integrate[(a + b*ArcTanh[c*Sqrt[x]])/(x*(d + e*x)),x]
Output:
(-2*b*ArcTanh[c*Sqrt[x]]^2 + (4*I)*b*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]]*Arc Tanh[(c*e*Sqrt[x])/Sqrt[-(c^2*d*e)]] + 2*b*ArcTanh[c*Sqrt[x]]*(ArcTanh[c*S qrt[x]] + 2*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])]) + (2*I)*b*(ArcSin[Sqrt[(c^ 2*d)/(c^2*d + e)]] + I*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*b*(I*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]] + 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*Arc Tanh[c*Sqrt[x]]))] + 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 - 2*Sqrt[-(c^2*d*e)])/(( c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] + b*PolyLog[2, (-(c^2*d) + e + 2*Sqr t[-(c^2*d*e)])/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))])/(2*d)
Time = 1.36 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {7267, 2026, 6554, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x (d+e x)} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{e x^{3/2}+d \sqrt {x}}d\sqrt {x}\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle 2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} (d+e x)}d\sqrt {x}\) |
| \(\Big \downarrow \) 6554 |
| \(\displaystyle 2 \int \left (\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{d \sqrt {x}}-\frac {e \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d (d+e x)}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{2 d}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{2 d}+\frac {\log \left (\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d}+\frac {a \log \left (\sqrt {x}\right )}{d}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{4 d}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (\sqrt {-d} c+\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{4 d}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt {x} c+1}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-c \sqrt {x}\right )}{2 d}+\frac {b \operatorname {PolyLog}\left (2,c \sqrt {x}\right )}{2 d}\right )\) |
Input:
Int[(a + b*ArcTanh[c*Sqrt[x]])/(x*(d + e*x)),x]
Output:
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] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e] )*(1 + c*Sqrt[x]))])/(2*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]))])/(2*d) + ( a*Log[Sqrt[x]])/d - (b*PolyLog[2, 1 - 2/(1 + c*Sqrt[x])])/(2*d) + (b*PolyL og[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/(4*d) + (b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x])) /((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/(4*d) - (b*PolyLog[2, -(c*Sqrt [x])])/(2*d) + (b*PolyLog[2, c*Sqrt[x]])/(2*d))
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
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] )
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.09 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.39
| method | result | size |
| parts | \(\frac {a \ln \left (x \right )}{d}-\frac {a \ln \left (e x +d \right )}{d}+b \left (\frac {2 \,\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}\right )}{d}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{d}-c^{2} \left (\frac {\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )-\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )}{d \,c^{2}}-\frac {2 \left (-\frac {\operatorname {dilog}\left (c \sqrt {x}\right )}{2}-\frac {\operatorname {dilog}\left (1+c \sqrt {x}\right )}{2}-\frac {\ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}\right )}{d \,c^{2}}\right )\right )\) | \(499\) |
| derivativedivides | \(-\frac {a \ln \left (c^{2} e x +c^{2} d \right )}{d}+\frac {2 a \ln \left (c \sqrt {x}\right )}{d}+2 b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 d \,c^{2}}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}\right )}{d \,c^{2}}-\frac {\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )-\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )}{2 d \,c^{2}}+\frac {-\frac {\operatorname {dilog}\left (c \sqrt {x}\right )}{2}-\frac {\operatorname {dilog}\left (1+c \sqrt {x}\right )}{2}-\frac {\ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}}{d \,c^{2}}\right )\) | \(514\) |
| default | \(-\frac {a \ln \left (c^{2} e x +c^{2} d \right )}{d}+\frac {2 a \ln \left (c \sqrt {x}\right )}{d}+2 b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 d \,c^{2}}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}\right )}{d \,c^{2}}-\frac {\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )-\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )}{2 d \,c^{2}}+\frac {-\frac {\operatorname {dilog}\left (c \sqrt {x}\right )}{2}-\frac {\operatorname {dilog}\left (1+c \sqrt {x}\right )}{2}-\frac {\ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}}{d \,c^{2}}\right )\) | \(514\) |
Input:
int((a+b*arctanh(c*x^(1/2)))/x/(e*x+d),x,method=_RETURNVERBOSE)
Output:
a*ln(x)/d-a/d*ln(e*x+d)+b*(2*arctanh(c*x^(1/2))/d*ln(c*x^(1/2))-arctanh(c* x^(1/2))/d*ln(c^2*e*x+c^2*d)-c^2*(1/d/c^2*(1/2*ln(c*x^(1/2)-1)*ln(c^2*e*x+ c^2*d)-e*(1/2*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))+ln((c*(-d*e)^(1/2)+e*(c*x^(1/2)-1)+e)/(c*(-d*e)^(1/2)+e)))/ e+1/2*(dilog((c*(-d*e)^(1/2)-e*(c*x^(1/2)-1)-e)/(c*(-d*e)^(1/2)-e))+dilog( (c*(-d*e)^(1/2)+e*(c*x^(1/2)-1)+e)/(c*(-d*e)^(1/2)+e)))/e)-1/2*ln(1+c*x^(1 /2))*ln(c^2*e*x+c^2*d)+e*(1/2*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))+ln((c*(-d*e)^(1/2)+e*(1+c*x^(1/2))-e)/(c*(- d*e)^(1/2)-e)))/e+1/2*(dilog((c*(-d*e)^(1/2)-e*(1+c*x^(1/2))+e)/(c*(-d*e)^ (1/2)+e))+dilog((c*(-d*e)^(1/2)+e*(1+c*x^(1/2))-e)/(c*(-d*e)^(1/2)-e)))/e) )-2/d/c^2*(-1/2*dilog(c*x^(1/2))-1/2*dilog(1+c*x^(1/2))-1/2*ln(c*x^(1/2))* ln(1+c*x^(1/2)))))
\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{{\left (e x + d\right )} x} \,d x } \] Input:
integrate((a+b*arctanh(c*x^(1/2)))/x/(e*x+d),x, algorithm="fricas")
Output:
integral((b*arctanh(c*sqrt(x)) + a)/(e*x^2 + d*x), x)
Timed out. \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x (d+e x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*atanh(c*x**(1/2)))/x/(e*x+d),x)
Output:
Timed out
\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{{\left (e x + d\right )} x} \,d x } \] Input:
integrate((a+b*arctanh(c*x^(1/2)))/x/(e*x+d),x, algorithm="maxima")
Output:
-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)
\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{{\left (e x + d\right )} x} \,d x } \] Input:
integrate((a+b*arctanh(c*x^(1/2)))/x/(e*x+d),x, algorithm="giac")
Output:
integrate((b*arctanh(c*sqrt(x)) + a)/((e*x + d)*x), x)
Timed out. \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x (d+e x)} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{x\,\left (d+e\,x\right )} \,d x \] Input:
int((a + b*atanh(c*x^(1/2)))/(x*(d + e*x)),x)
Output:
int((a + b*atanh(c*x^(1/2)))/(x*(d + e*x)), x)
\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x (d+e x)} \, dx=\frac {\left (\int \frac {\mathit {atanh} \left (\sqrt {x}\, c \right )}{e \,x^{2}+d x}d x \right ) b d -\mathrm {log}\left (e x +d \right ) a +\mathrm {log}\left (x \right ) a}{d} \] Input:
int((a+b*atanh(c*x^(1/2)))/x/(e*x+d),x)
Output:
(int(atanh(sqrt(x)*c)/(d*x + e*x**2),x)*b*d - log(d + e*x)*a + log(x)*a)/d