Integrand size = 21, antiderivative size = 374 \[ \int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\frac {b \sqrt {x}}{c e}-\frac {b \text {arctanh}\left (c \sqrt {x}\right )}{c^2 e}+\frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{e}+\frac {2 d \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{e^2}-\frac {d \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e^2}-\frac {d \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e^2}-\frac {b d \operatorname {PolyLog}\left (2,1-\frac {2}{1+c \sqrt {x}}\right )}{e^2}+\frac {b d \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 e^2}+\frac {b d \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 e^2} \] Output:
b*x^(1/2)/c/e-b*arctanh(c*x^(1/2))/c^2/e+x*(a+b*arctanh(c*x^(1/2)))/e+2*d* (a+b*arctanh(c*x^(1/2)))*ln(2/(1+c*x^(1/2)))/e^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)) )/e^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)))/e^2-b*d*polylog(2,1-2/(1+c*x^(1/2)))/e^2+ 1/2*b*d*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)))/e^2+1/2*b*d*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)))/e^2
Result contains complex when optimal does not.
Time = 1.65 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.24 \[ \int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\frac {\frac {2 b e \sqrt {x}}{c}+2 a e x+\frac {2 b e \left (-1+c^2 x\right ) \text {arctanh}\left (c \sqrt {x}\right )}{c^2}+4 i b d \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right ) \text {arctanh}\left (\frac {c e \sqrt {x}}{\sqrt {-c^2 d e}}\right )+4 b d \text {arctanh}\left (c \sqrt {x}\right ) \log \left (1+e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+2 i b d \left (\arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )+i \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {e^{-2 \text {arctanh}\left (c \sqrt {x}\right )} \left (-2 \sqrt {-c^2 d e}+e \left (-1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+c^2 d \left (1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )}{c^2 d+e}\right )-2 b d \left (i \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )+\text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {e^{-2 \text {arctanh}\left (c \sqrt {x}\right )} \left (2 \sqrt {-c^2 d e}+e \left (-1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+c^2 d \left (1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )}{c^2 d+e}\right )-2 a d \log (d+e x)-2 b d \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+b d \operatorname {PolyLog}\left (2,\frac {\left (-c^2 d+e-2 \sqrt {-c^2 d e}\right ) e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}}{c^2 d+e}\right )+b d \operatorname {PolyLog}\left (2,\frac {\left (-c^2 d+e+2 \sqrt {-c^2 d e}\right ) e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}}{c^2 d+e}\right )}{2 e^2} \] Input:
Integrate[(x*(a + b*ArcTanh[c*Sqrt[x]]))/(d + e*x),x]
Output:
((2*b*e*Sqrt[x])/c + 2*a*e*x + (2*b*e*(-1 + c^2*x)*ArcTanh[c*Sqrt[x]])/c^2 + (4*I)*b*d*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]]*ArcTanh[(c*e*Sqrt[x])/Sqrt[ -(c^2*d*e)]] + 4*b*d*ArcTanh[c*Sqrt[x]]*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])] + (2*I)*b*d*(ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]] + I*ArcTanh[c*Sqrt[x]])*Lo g[(-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*d*( 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*Sq rt[x]])))/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] - 2*a*d*Log[d + e*x] - 2 *b*d*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])] + b*d*PolyLog[2, (-(c^2*d) + e - 2*Sqrt[-(c^2*d*e)])/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] + b*d*PolyL og[2, (-(c^2*d) + e + 2*Sqrt[-(c^2*d*e)])/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt [x]]))])/(2*e^2)
Time = 0.92 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {7267, 6542, 6452, 262, 219, 6606, 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.
| \(\displaystyle \int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle 2 \left (\frac {\int \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{e}-\frac {d \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}}{e}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \int \frac {x}{1-c^2 x}d\sqrt {x}}{e}-\frac {d \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}}{e}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\int \frac {1}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\sqrt {x}}{c^2}\right )}{e}-\frac {d \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}}{e}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{e}-\frac {d \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}}{e}\right )\) |
| \(\Big \downarrow \) 6606 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{e}-\frac {d \int \left (\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}\right )d\sqrt {x}}{e}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{e}-\frac {d \left (\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{2 e}+\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{2 e}-\frac {\log \left (\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{e}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{4 e}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (\sqrt {-d} c+\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{4 e}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt {x} c+1}\right )}{2 e}\right )}{e}\right )\) |
Input:
Int[(x*(a + b*ArcTanh[c*Sqrt[x]]))/(d + e*x),x]
Output:
2*(((x*(a + b*ArcTanh[c*Sqrt[x]]))/2 - (b*c*(-(Sqrt[x]/c^2) + ArcTanh[c*Sq rt[x]]/c^3))/2)/e - (d*(-(((a + b*ArcTanh[c*Sqrt[x]])*Log[2/(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]))])/(2*e) + ((a + b*ArcTanh[c*Sqr t[x]])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/(2*e) + (b*PolyLog[2, 1 - 2/(1 + c*Sqrt[x])])/(2*e) - (b*Po lyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/(4*e) - (b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x ]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/(4*e)))/e)
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*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, 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[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTanh[c* x])^p, x], x] - Simp[d*(f^2/e) 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] && GtQ[m, 1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_)^2)^(q_.), x_Symbol] :> With[{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])
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 = 0.07 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.32
| method | result | size |
| parts | \(\frac {a x}{e}-\frac {a d \ln \left (e x +d \right )}{e^{2}}+\frac {2 b \left (\frac {c^{4} \operatorname {arctanh}\left (c \sqrt {x}\right ) x}{2 e}-\frac {c^{4} \operatorname {arctanh}\left (c \sqrt {x}\right ) d \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{2}}-\frac {c^{2} \left (\frac {d \,c^{2} \left (\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )-\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )\right )}{e^{2}}-\frac {c \sqrt {x}+\frac {\ln \left (c \sqrt {x}-1\right )}{2}-\frac {\ln \left (1+c \sqrt {x}\right )}{2}}{e}\right )}{2}\right )}{c^{4}}\) | \(494\) |
| derivativedivides | \(\frac {\frac {a \,c^{4} x}{e}-\frac {a \,c^{4} d \ln \left (c^{2} e x +c^{2} d \right )}{e^{2}}+2 b \,c^{2} \left (\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{2} x}{2 e}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) d \,c^{2} \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{2}}-\frac {d \,c^{2} \left (\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )-\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )\right )}{2 e^{2}}+\frac {c \sqrt {x}+\frac {\ln \left (c \sqrt {x}-1\right )}{2}-\frac {\ln \left (1+c \sqrt {x}\right )}{2}}{2 e}\right )}{c^{4}}\) | \(507\) |
| default | \(\frac {\frac {a \,c^{4} x}{e}-\frac {a \,c^{4} d \ln \left (c^{2} e x +c^{2} d \right )}{e^{2}}+2 b \,c^{2} \left (\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{2} x}{2 e}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) d \,c^{2} \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{2}}-\frac {d \,c^{2} \left (\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )-\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )\right )}{2 e^{2}}+\frac {c \sqrt {x}+\frac {\ln \left (c \sqrt {x}-1\right )}{2}-\frac {\ln \left (1+c \sqrt {x}\right )}{2}}{2 e}\right )}{c^{4}}\) | \(507\) |
Input:
int(x*(a+b*arctanh(c*x^(1/2)))/(e*x+d),x,method=_RETURNVERBOSE)
Output:
a/e*x-a*d/e^2*ln(e*x+d)+2*b/c^4*(1/2*c^4*arctanh(c*x^(1/2))/e*x-1/2*c^4*ar ctanh(c*x^(1/2))*d/e^2*ln(c^2*e*x+c^2*d)-1/2*c^2*(d*c^2/e^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))-1/e*(c*x^(1/2)+1/2*ln(c*x^(1/2)-1)-1/2*ln(1+c*x^(1/2)) )))
\[ \int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )} x}{e x + d} \,d x } \] Input:
integrate(x*(a+b*arctanh(c*x^(1/2)))/(e*x+d),x, algorithm="fricas")
Output:
integral((b*x*arctanh(c*sqrt(x)) + a*x)/(e*x + d), x)
\[ \int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\int \frac {x \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )}{d + e x}\, dx \] Input:
integrate(x*(a+b*atanh(c*x**(1/2)))/(e*x+d),x)
Output:
Integral(x*(a + b*atanh(c*sqrt(x)))/(d + e*x), x)
\[ \int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )} x}{e x + d} \,d x } \] Input:
integrate(x*(a+b*arctanh(c*x^(1/2)))/(e*x+d),x, algorithm="maxima")
Output:
a*(x/e - d*log(e*x + d)/e^2) + b*integrate(1/2*x*log(c*sqrt(x) + 1)/(e*x + d), x) - b*integrate(1/2*x*log(-c*sqrt(x) + 1)/(e*x + d), x)
\[ \int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )} x}{e x + d} \,d x } \] Input:
integrate(x*(a+b*arctanh(c*x^(1/2)))/(e*x+d),x, algorithm="giac")
Output:
integrate((b*arctanh(c*sqrt(x)) + a)*x/(e*x + d), x)
Timed out. \[ \int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\int \frac {x\,\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}{d+e\,x} \,d x \] Input:
int((x*(a + b*atanh(c*x^(1/2))))/(d + e*x),x)
Output:
int((x*(a + b*atanh(c*x^(1/2))))/(d + e*x), x)
\[ \int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\frac {\left (\int \frac {\mathit {atanh} \left (\sqrt {x}\, c \right ) x}{e x +d}d x \right ) b \,e^{2}-\mathrm {log}\left (e x +d \right ) a d +a e x}{e^{2}} \] Input:
int(x*(a+b*atanh(c*x^(1/2)))/(e*x+d),x)
Output:
(int((atanh(sqrt(x)*c)*x)/(d + e*x),x)*b*e**2 - log(d + e*x)*a*d + a*e*x)/ e**2