Integrand size = 23, antiderivative size = 413 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2 (d+e x)} \, dx=-\frac {b c}{d \sqrt {x}}+\frac {b c^2 \text {arctanh}\left (c \sqrt {x}\right )}{d}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{d x}-\frac {2 e \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{d^2}+\frac {e \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{d^2}+\frac {e \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{d^2}-\frac {a e \log (x)}{d^2}+\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2}{1+c \sqrt {x}}\right )}{d^2}-\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 d^2}-\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 d^2}+\frac {b e \operatorname {PolyLog}\left (2,-c \sqrt {x}\right )}{d^2}-\frac {b e \operatorname {PolyLog}\left (2,c \sqrt {x}\right )}{d^2} \]
b*c^2*arctanh(c*x^(1/2))/d+(-a-b*arctanh(c*x^(1/2)))/d/x-a*e*ln(x)/d^2-2*e *(a+b*arctanh(c*x^(1/2)))*ln(2/(1+c*x^(1/2)))/d^2+e*(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^2+e*(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^2+b*e*polylog(2,-c*x^(1/2))/d^2-b*e*po lylog(2,c*x^(1/2))/d^2+b*e*polylog(2,1-2/(1+c*x^(1/2)))/d^2-1/2*b*e*polylo g(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^2-1/2*b*e*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^2-b*c/d/x^(1/2)
Result contains complex when optimal does not.
Time = 2.11 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2 (d+e x)} \, dx=\frac {-2 a d-2 a e x \log (x)+2 a e x \log (d+e x)-2 b \left (c d \sqrt {x}+\text {arctanh}\left (c \sqrt {x}\right ) \left (d-c^2 d x+e x \text {arctanh}\left (c \sqrt {x}\right )+2 e x \log \left (1-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )-e x \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )+b e x \left (2 \text {arctanh}\left (c \sqrt {x}\right )^2-4 i \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right ) \text {arctanh}\left (\frac {c e \sqrt {x}}{\sqrt {-c^2 d e}}\right )+2 \left (-i \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )+\text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {e^{-2 \text {arctanh}\left (c \sqrt {x}\right )} \left (-2 \sqrt {-c^2 d e}+e \left (-1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+c^2 d \left (1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )}{c^2 d+e}\right )+2 \left (i \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )+\text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {e^{-2 \text {arctanh}\left (c \sqrt {x}\right )} \left (2 \sqrt {-c^2 d e}+e \left (-1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+c^2 d \left (1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )}{c^2 d+e}\right )-\operatorname {PolyLog}\left (2,\frac {\left (-c^2 d+e-2 \sqrt {-c^2 d e}\right ) e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}}{c^2 d+e}\right )-\operatorname {PolyLog}\left (2,\frac {\left (-c^2 d+e+2 \sqrt {-c^2 d e}\right ) e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}}{c^2 d+e}\right )\right )}{2 d^2 x} \]
(-2*a*d - 2*a*e*x*Log[x] + 2*a*e*x*Log[d + e*x] - 2*b*(c*d*Sqrt[x] + ArcTa nh[c*Sqrt[x]]*(d - c^2*d*x + e*x*ArcTanh[c*Sqrt[x]] + 2*e*x*Log[1 - E^(-2* ArcTanh[c*Sqrt[x]])]) - e*x*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])]) + b*e*x *(2*ArcTanh[c*Sqrt[x]]^2 - (4*I)*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]]*ArcTanh [(c*e*Sqrt[x])/Sqrt[-(c^2*d*e)]] + 2*((-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*ArcT anh[c*Sqrt[x]]))] + 2*(I*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]] + ArcTanh[c*Sqr t[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]]))] - PolyLog[2, (-(c^2*d) + e - 2*Sqrt[-(c^2*d*e)])/((c^2*d + e)*E^(2*ArcTanh[ 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]]))]))/(2*d^2*x)
Time = 1.13 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {7267, 2026, 6544, 6452, 264, 219, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{e x^{5/2}+d x^{3/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle 2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^{3/2} (d+e x)}d\sqrt {x}\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle 2 \left (\frac {\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^{3/2}}d\sqrt {x}}{d}-\frac {e \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} (d+e x)}d\sqrt {x}}{d}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} b c \int \frac {1}{x \left (1-c^2 x\right )}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}}{d}-\frac {e \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} (d+e x)}d\sqrt {x}}{d}\right )\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x}d\sqrt {x}-\frac {1}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}}{d}-\frac {e \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} (d+e x)}d\sqrt {x}}{d}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} b c \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}}{d}-\frac {e \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} (d+e x)}d\sqrt {x}}{d}\right )\) |
| \(\Big \downarrow \) 6554 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} b c \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}}{d}-\frac {e \int \left (\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{d \sqrt {x}}-\frac {e \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d (d+e x)}\right )d\sqrt {x}}{d}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} b c \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}}{d}-\frac {e \left (-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{2 d}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{2 d}+\frac {\log \left (\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d}+\frac {a \log \left (\sqrt {x}\right )}{d}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{4 d}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (\sqrt {-d} c+\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{4 d}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt {x} c+1}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-c \sqrt {x}\right )}{2 d}+\frac {b \operatorname {PolyLog}\left (2,c \sqrt {x}\right )}{2 d}\right )}{d}\right )\) |
2*((-1/2*(a + b*ArcTanh[c*Sqrt[x]])/x + (b*c*(-(1/Sqrt[x]) + c*ArcTanh[c*S qrt[x]]))/2)/d - (e*(((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*Sqr t[x]))])/(2*d) + (a*Log[Sqrt[x]])/d - (b*PolyLog[2, 1 - 2/(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]))])/(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*P olyLog[2, -(c*Sqrt[x])])/(2*d) + (b*PolyLog[2, c*Sqrt[x]])/(2*d)))/d)
3.1.48.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x ], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/(d + e*x ^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
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 = 0.99 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.40
| method | result | size |
| parts | \(a \left (\frac {e \ln \left (e x +d \right )}{d^{2}}-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}\right )+2 b \,c^{2} \left (\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) e \ln \left (c^{2} e x +c^{2} d \right )}{2 c^{2} d^{2}}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 d \,c^{2} x}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) e \ln \left (c \sqrt {x}\right )}{c^{2} d^{2}}-\frac {c^{2} \left (\frac {\frac {1}{c \sqrt {x}}-\frac {\ln \left (1+c \sqrt {x}\right )}{2}+\frac {\ln \left (c \sqrt {x}-1\right )}{2}}{d \,c^{2}}-\frac {e \left (-\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 )+\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 )\right )}{d^{2} c^{4}}+\frac {2 e \left (-\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}-\frac {\operatorname {dilog}\left (c \sqrt {x}\right )}{2}\right )}{d^{2} c^{4}}\right )}{2}\right )\) | \(577\) |
| derivativedivides | \(2 c^{2} \left (-\frac {a}{2 d \,c^{2} x}-\frac {a e \ln \left (c \sqrt {x}\right )}{c^{2} d^{2}}+\frac {a e \ln \left (c^{2} e x +c^{2} d \right )}{2 c^{2} d^{2}}+b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 d \,c^{4} x}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) e \ln \left (c \sqrt {x}\right )}{d^{2} c^{4}}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) e \ln \left (c^{2} e x +c^{2} d \right )}{2 d^{2} c^{4}}-\frac {\frac {1}{c \sqrt {x}}-\frac {\ln \left (1+c \sqrt {x}\right )}{2}+\frac {\ln \left (c \sqrt {x}-1\right )}{2}}{2 d \,c^{2}}-\frac {e \left (-\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}-\frac {\operatorname {dilog}\left (c \sqrt {x}\right )}{2}\right )}{d^{2} c^{4}}+\frac {e \left (-\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 )+\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 )\right )}{2 d^{2} c^{4}}\right )\right )\) | \(597\) |
| default | \(2 c^{2} \left (-\frac {a}{2 d \,c^{2} x}-\frac {a e \ln \left (c \sqrt {x}\right )}{c^{2} d^{2}}+\frac {a e \ln \left (c^{2} e x +c^{2} d \right )}{2 c^{2} d^{2}}+b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 d \,c^{4} x}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) e \ln \left (c \sqrt {x}\right )}{d^{2} c^{4}}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) e \ln \left (c^{2} e x +c^{2} d \right )}{2 d^{2} c^{4}}-\frac {\frac {1}{c \sqrt {x}}-\frac {\ln \left (1+c \sqrt {x}\right )}{2}+\frac {\ln \left (c \sqrt {x}-1\right )}{2}}{2 d \,c^{2}}-\frac {e \left (-\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}-\frac {\operatorname {dilog}\left (c \sqrt {x}\right )}{2}\right )}{d^{2} c^{4}}+\frac {e \left (-\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 )+\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 )\right )}{2 d^{2} c^{4}}\right )\right )\) | \(597\) |
a*(e/d^2*ln(e*x+d)-1/d/x-e/d^2*ln(x))+2*b*c^2*(1/2/c^2*arctanh(c*x^(1/2))* e/d^2*ln(c^2*e*x+c^2*d)-1/2*arctanh(c*x^(1/2))/d/c^2/x-1/c^2*arctanh(c*x^( 1/2))*e/d^2*ln(c*x^(1/2))-1/2*c^2*(1/d/c^2*(1/c/x^(1/2)-1/2*ln(1+c*x^(1/2) )+1/2*ln(c*x^(1/2)-1))-e/d^2/c^4*(-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*(d ilog((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)+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))+2*e/d^ 2/c^4*(-1/2*dilog(1+c*x^(1/2))-1/2*ln(c*x^(1/2))*ln(1+c*x^(1/2))-1/2*dilog (c*x^(1/2)))))
\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2 (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2 (d+e x)} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2 (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \]
a*(e*log(e*x + d)/d^2 - e*log(x)/d^2 - 1/(d*x)) + b*integrate(1/2*log(c*sq rt(x) + 1)/((e*x^(5/2) + d*x^(3/2))*sqrt(x)), x) - b*integrate(1/2*log(-c* sqrt(x) + 1)/((e*x^(5/2) + d*x^(3/2))*sqrt(x)), x)
\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2 (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2 (d+e x)} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{x^2\,\left (d+e\,x\right )} \,d x \]