Integrand size = 18, antiderivative size = 661 \[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=-\frac {2 \sqrt {1+a} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} c^2}+\frac {2 \sqrt {1-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} c^2}-\frac {d^2 \log \left (\frac {c \left (\sqrt {-1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {d^2 \log \left (\frac {c \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {d^2 \log \left (\frac {c \left (\sqrt {-1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-1-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {d^2 \log \left (\frac {c \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {d \sqrt {x} \log (1-a-b x)}{c^2}+\frac {(1-a-b x) \log (1-a-b x)}{2 b c}-\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log (1-a-b x)}{c^3}-\frac {d \sqrt {x} \log (1+a+b x)}{c^2}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log (1+a+b x)}{c^3}-\frac {d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-1-a} c-\sqrt {b} d}\right )}{c^3}+\frac {d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )}{c^3}-\frac {d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right )}{c^3}+\frac {d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{c^3} \] Output:
-2*(1+a)^(1/2)*d*arctan(b^(1/2)*x^(1/2)/(1+a)^(1/2))/b^(1/2)/c^2+2*(1-a)^( 1/2)*d*arctanh(b^(1/2)*x^(1/2)/(1-a)^(1/2))/b^(1/2)/c^2-d^2*ln(c*((-1-a)^( 1/2)-b^(1/2)*x^(1/2))/((-1-a)^(1/2)*c+b^(1/2)*d))*ln(d+c*x^(1/2))/c^3+d^2* ln(c*((1-a)^(1/2)-b^(1/2)*x^(1/2))/((1-a)^(1/2)*c+b^(1/2)*d))*ln(d+c*x^(1/ 2))/c^3-d^2*ln(c*((-1-a)^(1/2)+b^(1/2)*x^(1/2))/((-1-a)^(1/2)*c-b^(1/2)*d) )*ln(d+c*x^(1/2))/c^3+d^2*ln(c*((1-a)^(1/2)+b^(1/2)*x^(1/2))/((1-a)^(1/2)* c-b^(1/2)*d))*ln(d+c*x^(1/2))/c^3+d*x^(1/2)*ln(-b*x-a+1)/c^2+1/2*(-b*x-a+1 )*ln(-b*x-a+1)/b/c-d^2*ln(d+c*x^(1/2))*ln(-b*x-a+1)/c^3-d*x^(1/2)*ln(b*x+a +1)/c^2+1/2*(b*x+a+1)*ln(b*x+a+1)/b/c+d^2*ln(d+c*x^(1/2))*ln(b*x+a+1)/c^3- d^2*polylog(2,-b^(1/2)*(d+c*x^(1/2))/((-1-a)^(1/2)*c-b^(1/2)*d))/c^3+d^2*p olylog(2,-b^(1/2)*(d+c*x^(1/2))/((1-a)^(1/2)*c-b^(1/2)*d))/c^3-d^2*polylog (2,b^(1/2)*(d+c*x^(1/2))/((-1-a)^(1/2)*c+b^(1/2)*d))/c^3+d^2*polylog(2,b^( 1/2)*(d+c*x^(1/2))/((1-a)^(1/2)*c+b^(1/2)*d))/c^3
Time = 0.31 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.01 \[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=-\frac {2 \sqrt {1+a} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} c^2}+\frac {2 \sqrt {1-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} c^2}-\frac {d^2 \log \left (\frac {c \left (\sqrt {-1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {d^2 \log \left (\frac {c \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {d^2 \log \left (\frac {c \left (\sqrt {-1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-1-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {d^2 \log \left (\frac {c \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {d \sqrt {x} \log (1-a-b x)}{c^2}-\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log (1-a-b x)}{c^3}+\frac {x+\frac {(1-a-b x) \log (1-a-b x)}{b}}{2 c}-\frac {d \sqrt {x} \log (1+a+b x)}{c^2}+\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log (1+a+b x)}{c^3}-\frac {x-\frac {(1+a+b x) \log (1+a+b x)}{b}}{2 c}-\frac {d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-1-a} c-\sqrt {b} d}\right )}{c^3}+\frac {d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )}{c^3}-\frac {d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right )}{c^3}+\frac {d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{c^3} \] Input:
Integrate[ArcTanh[a + b*x]/(c + d/Sqrt[x]),x]
Output:
(-2*Sqrt[1 + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]])/(Sqrt[b]*c^2) + ( 2*Sqrt[1 - a]*d*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[1 - a]])/(Sqrt[b]*c^2) - (d ^2*Log[(c*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)]* Log[d + c*Sqrt[x]])/c^3 + (d^2*Log[(c*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sq rt[1 - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 - (d^2*Log[(c*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/ c^3 + (d^2*Log[(c*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b ]*d)]*Log[d + c*Sqrt[x]])/c^3 + (d*Sqrt[x]*Log[1 - a - b*x])/c^2 - (d^2*Lo g[d + c*Sqrt[x]]*Log[1 - a - b*x])/c^3 + (x + ((1 - a - b*x)*Log[1 - a - b *x])/b)/(2*c) - (d*Sqrt[x]*Log[1 + a + b*x])/c^2 + (d^2*Log[d + c*Sqrt[x]] *Log[1 + a + b*x])/c^3 - (x - ((1 + a + b*x)*Log[1 + a + b*x])/b)/(2*c) - (d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d))] )/c^3 + (d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[ b]*d))])/c^3 - (d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)])/c^3 + (d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a] *c + Sqrt[b]*d)])/c^3
Time = 1.38 (sec) , antiderivative size = 661, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6665, 2855, 2005, 2916, 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.
| \(\displaystyle \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx\) |
| \(\Big \downarrow \) 6665 |
| \(\displaystyle \frac {1}{2} \int \frac {\log (a+b x+1)}{c+\frac {d}{\sqrt {x}}}dx-\frac {1}{2} \int \frac {\log (-a-b x+1)}{c+\frac {d}{\sqrt {x}}}dx\) |
| \(\Big \downarrow \) 2855 |
| \(\displaystyle \int \frac {\sqrt {x} \log (a+b x+1)}{c+\frac {d}{\sqrt {x}}}d\sqrt {x}-\int \frac {\sqrt {x} \log (-a-b x+1)}{c+\frac {d}{\sqrt {x}}}d\sqrt {x}\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle \int \frac {x \log (a+b x+1)}{\sqrt {x} c+d}d\sqrt {x}-\int \frac {x \log (-a-b x+1)}{\sqrt {x} c+d}d\sqrt {x}\) |
| \(\Big \downarrow \) 2916 |
| \(\displaystyle \int \left (\frac {\log (a+b x+1) d^2}{c^2 \left (\sqrt {x} c+d\right )}-\frac {\log (a+b x+1) d}{c^2}+\frac {\sqrt {x} \log (a+b x+1)}{c}\right )d\sqrt {x}-\int \left (\frac {\log (-a-b x+1) d^2}{c^2 \left (\sqrt {x} c+d\right )}-\frac {\log (-a-b x+1) d}{c^2}+\frac {\sqrt {x} \log (-a-b x+1)}{c}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {a+1} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+1}}\right )}{\sqrt {b} c^2}+\frac {2 \sqrt {1-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} c^2}-\frac {d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-1} c-\sqrt {b} d}\right )}{c^3}+\frac {d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )}{c^3}-\frac {d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-1} c+\sqrt {b} d}\right )}{c^3}+\frac {d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{c^3}-\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {-a-1}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} c+\sqrt {b} d}\right )}{c^3}+\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{c^3}-\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {-a-1}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} c-\sqrt {b} d}\right )}{c^3}+\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )}{c^3}-\frac {d^2 \log (-a-b x+1) \log \left (c \sqrt {x}+d\right )}{c^3}+\frac {d^2 \log (a+b x+1) \log \left (c \sqrt {x}+d\right )}{c^3}+\frac {d \sqrt {x} \log (-a-b x+1)}{c^2}-\frac {d \sqrt {x} \log (a+b x+1)}{c^2}+\frac {(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac {(a+b x+1) \log (a+b x+1)}{2 b c}\) |
Input:
Int[ArcTanh[a + b*x]/(c + d/Sqrt[x]),x]
Output:
(-2*Sqrt[1 + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]])/(Sqrt[b]*c^2) + ( 2*Sqrt[1 - a]*d*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[1 - a]])/(Sqrt[b]*c^2) - (d ^2*Log[(c*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)]* Log[d + c*Sqrt[x]])/c^3 + (d^2*Log[(c*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sq rt[1 - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 - (d^2*Log[(c*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/ c^3 + (d^2*Log[(c*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b ]*d)]*Log[d + c*Sqrt[x]])/c^3 + (d*Sqrt[x]*Log[1 - a - b*x])/c^2 + ((1 - a - b*x)*Log[1 - a - b*x])/(2*b*c) - (d^2*Log[d + c*Sqrt[x]]*Log[1 - a - b* x])/c^3 - (d*Sqrt[x]*Log[1 + a + b*x])/c^2 + ((1 + a + b*x)*Log[1 + a + b* x])/(2*b*c) + (d^2*Log[d + c*Sqrt[x]]*Log[1 + a + b*x])/c^3 - (d^2*PolyLog [2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d))])/c^3 + (d^2 *PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b]*d))])/c^3 - (d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)] )/c^3 + (d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a]*c + Sqrt[b] *d)])/c^3
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_)^(r_))^(q_.), x_Symbol] :> With[{k = Denominator[r]}, Simp[k Subst [Int[x^(k - 1)*(f + g*x^(k*r))^q*(a + b*Log[c*(d + e*x^k)^n])^p, x], x, x^( 1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && FractionQ[r] && I GtQ[p, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log [c*(d + e*x^n)^p])^q, x^m*(f + g*x)^r, x], x] /; FreeQ[{a, b, c, d, e, f, g , n, p, q}, x] && IntegerQ[m] && IntegerQ[r]
Int[ArcTanh[(c_) + (d_.)*(x_)]/((e_) + (f_.)*(x_)^(n_.)), x_Symbol] :> Simp [1/2 Int[Log[1 + c + d*x]/(e + f*x^n), x], x] - Simp[1/2 Int[Log[1 - c - d*x]/(e + f*x^n), x], x] /; FreeQ[{c, d, e, f}, x] && RationalQ[n]
Time = 0.10 (sec) , antiderivative size = 751, normalized size of antiderivative = 1.14
| method | result | size |
| derivativedivides | \(\frac {\operatorname {arctanh}\left (b x +a \right ) x}{c}-\frac {2 \,\operatorname {arctanh}\left (b x +a \right ) d \sqrt {x}}{c^{2}}+\frac {2 \,\operatorname {arctanh}\left (b x +a \right ) d^{2} \ln \left (d +c \sqrt {x}\right )}{c^{3}}-\frac {4 b \left (c \,d^{2} \left (\frac {\frac {\ln \left (d +c \sqrt {x}\right ) \left (\ln \left (\frac {b d -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )+\ln \left (\frac {-b d +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {b d -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )+\operatorname {dilog}\left (\frac {-b d +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )}{2 b}}{2 c^{2}}+\frac {-\frac {\ln \left (d +c \sqrt {x}\right ) \left (\ln \left (\frac {b d -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )+\ln \left (\frac {-b d +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )\right )}{2 b}-\frac {\operatorname {dilog}\left (\frac {b d -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )+\operatorname {dilog}\left (\frac {-b d +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )}{2 b}}{2 c^{2}}\right )+\frac {c \left (\frac {\left (1+a \right ) \left (-\frac {\ln \left (a \,c^{2}+b \,d^{2}-2 b \left (d +c \sqrt {x}\right ) d +b \left (d +c \sqrt {x}\right )^{2}+c^{2}\right )}{2 b}+\frac {2 d \arctan \left (\frac {-2 b d +2 b \left (d +c \sqrt {x}\right )}{2 \sqrt {a b \,c^{2}+b \,c^{2}}}\right )}{\sqrt {a b \,c^{2}+b \,c^{2}}}\right )}{2 b}-\frac {\left (a -1\right ) \left (-\frac {\ln \left (a \,c^{2}+b \,d^{2}-2 b \left (d +c \sqrt {x}\right ) d +b \left (d +c \sqrt {x}\right )^{2}-c^{2}\right )}{2 b}+\frac {2 d \arctan \left (\frac {-2 b d +2 b \left (d +c \sqrt {x}\right )}{2 \sqrt {a b \,c^{2}-b \,c^{2}}}\right )}{\sqrt {a b \,c^{2}-b \,c^{2}}}\right )}{2 b}\right )}{2}\right )}{c^{2}}\) | \(751\) |
| default | \(\frac {\operatorname {arctanh}\left (b x +a \right ) x}{c}-\frac {2 \,\operatorname {arctanh}\left (b x +a \right ) d \sqrt {x}}{c^{2}}+\frac {2 \,\operatorname {arctanh}\left (b x +a \right ) d^{2} \ln \left (d +c \sqrt {x}\right )}{c^{3}}-\frac {4 b \left (c \,d^{2} \left (\frac {\frac {\ln \left (d +c \sqrt {x}\right ) \left (\ln \left (\frac {b d -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )+\ln \left (\frac {-b d +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {b d -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )+\operatorname {dilog}\left (\frac {-b d +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )}{2 b}}{2 c^{2}}+\frac {-\frac {\ln \left (d +c \sqrt {x}\right ) \left (\ln \left (\frac {b d -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )+\ln \left (\frac {-b d +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )\right )}{2 b}-\frac {\operatorname {dilog}\left (\frac {b d -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )+\operatorname {dilog}\left (\frac {-b d +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )}{2 b}}{2 c^{2}}\right )+\frac {c \left (\frac {\left (1+a \right ) \left (-\frac {\ln \left (a \,c^{2}+b \,d^{2}-2 b \left (d +c \sqrt {x}\right ) d +b \left (d +c \sqrt {x}\right )^{2}+c^{2}\right )}{2 b}+\frac {2 d \arctan \left (\frac {-2 b d +2 b \left (d +c \sqrt {x}\right )}{2 \sqrt {a b \,c^{2}+b \,c^{2}}}\right )}{\sqrt {a b \,c^{2}+b \,c^{2}}}\right )}{2 b}-\frac {\left (a -1\right ) \left (-\frac {\ln \left (a \,c^{2}+b \,d^{2}-2 b \left (d +c \sqrt {x}\right ) d +b \left (d +c \sqrt {x}\right )^{2}-c^{2}\right )}{2 b}+\frac {2 d \arctan \left (\frac {-2 b d +2 b \left (d +c \sqrt {x}\right )}{2 \sqrt {a b \,c^{2}-b \,c^{2}}}\right )}{\sqrt {a b \,c^{2}-b \,c^{2}}}\right )}{2 b}\right )}{2}\right )}{c^{2}}\) | \(751\) |
Input:
int(arctanh(b*x+a)/(c+d/x^(1/2)),x,method=_RETURNVERBOSE)
Output:
arctanh(b*x+a)*x/c-2*arctanh(b*x+a)/c^2*d*x^(1/2)+2*arctanh(b*x+a)*d^2/c^3 *ln(d+c*x^(1/2))-4*b/c^2*(c*d^2*(1/2/c^2*(1/2*ln(d+c*x^(1/2))*(ln((b*d-b*( d+c*x^(1/2))+(-a*b*c^2-b*c^2)^(1/2))/(b*d+(-a*b*c^2-b*c^2)^(1/2)))+ln((-b* d+b*(d+c*x^(1/2))+(-a*b*c^2-b*c^2)^(1/2))/(-b*d+(-a*b*c^2-b*c^2)^(1/2))))/ b+1/2*(dilog((b*d-b*(d+c*x^(1/2))+(-a*b*c^2-b*c^2)^(1/2))/(b*d+(-a*b*c^2-b *c^2)^(1/2)))+dilog((-b*d+b*(d+c*x^(1/2))+(-a*b*c^2-b*c^2)^(1/2))/(-b*d+(- a*b*c^2-b*c^2)^(1/2))))/b)+1/2/c^2*(-1/2*ln(d+c*x^(1/2))*(ln((b*d-b*(d+c*x ^(1/2))+(-a*b*c^2+b*c^2)^(1/2))/(b*d+(-a*b*c^2+b*c^2)^(1/2)))+ln((-b*d+b*( d+c*x^(1/2))+(-a*b*c^2+b*c^2)^(1/2))/(-b*d+(-a*b*c^2+b*c^2)^(1/2))))/b-1/2 *(dilog((b*d-b*(d+c*x^(1/2))+(-a*b*c^2+b*c^2)^(1/2))/(b*d+(-a*b*c^2+b*c^2) ^(1/2)))+dilog((-b*d+b*(d+c*x^(1/2))+(-a*b*c^2+b*c^2)^(1/2))/(-b*d+(-a*b*c ^2+b*c^2)^(1/2))))/b))+1/2*c*(1/2*(1+a)/b*(-1/2/b*ln(a*c^2+b*d^2-2*b*(d+c* x^(1/2))*d+b*(d+c*x^(1/2))^2+c^2)+2*d/(a*b*c^2+b*c^2)^(1/2)*arctan(1/2*(-2 *b*d+2*b*(d+c*x^(1/2)))/(a*b*c^2+b*c^2)^(1/2)))-1/2*(a-1)/b*(-1/2/b*ln(a*c ^2+b*d^2-2*b*(d+c*x^(1/2))*d+b*(d+c*x^(1/2))^2-c^2)+2*d/(a*b*c^2-b*c^2)^(1 /2)*arctan(1/2*(-2*b*d+2*b*(d+c*x^(1/2)))/(a*b*c^2-b*c^2)^(1/2)))))
\[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )}{c + \frac {d}{\sqrt {x}}} \,d x } \] Input:
integrate(arctanh(b*x+a)/(c+d/x^(1/2)),x, algorithm="fricas")
Output:
integral((c*x*arctanh(b*x + a) - d*sqrt(x)*arctanh(b*x + a))/(c^2*x - d^2) , x)
Timed out. \[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\text {Timed out} \] Input:
integrate(atanh(b*x+a)/(c+d/x**(1/2)),x)
Output:
Timed out
\[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )}{c + \frac {d}{\sqrt {x}}} \,d x } \] Input:
integrate(arctanh(b*x+a)/(c+d/x^(1/2)),x, algorithm="maxima")
Output:
integrate(arctanh(b*x + a)/(c + d/sqrt(x)), x)
\[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )}{c + \frac {d}{\sqrt {x}}} \,d x } \] Input:
integrate(arctanh(b*x+a)/(c+d/x^(1/2)),x, algorithm="giac")
Output:
integrate(arctanh(b*x + a)/(c + d/sqrt(x)), x)
Timed out. \[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int \frac {\mathrm {atanh}\left (a+b\,x\right )}{c+\frac {d}{\sqrt {x}}} \,d x \] Input:
int(atanh(a + b*x)/(c + d/x^(1/2)),x)
Output:
int(atanh(a + b*x)/(c + d/x^(1/2)), x)
\[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int \frac {\sqrt {x}\, \mathit {atanh} \left (b x +a \right )}{\sqrt {x}\, c +d}d x \] Input:
int(atanh(b*x+a)/(c+d/x^(1/2)),x)
Output:
int((sqrt(x)*atanh(a + b*x))/(sqrt(x)*c + d),x)