Integrand size = 21, antiderivative size = 758 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\frac {b c \sqrt {1+\frac {1}{c^2 x^2}}}{d^2}-\frac {a}{d^2 x}-\frac {b \text {csch}^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d-e}}-\frac {b e \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d-e}}-\frac {3 \sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{5/2}} \]
-a/d^2/x-b*arccsch(c*x)/d^2/x-1/4*b*e*arctanh((c^2*d-(-d)^(1/2)*e^(1/2)/x) /c/d^(1/2)/(c^2*d-e)^(1/2)/(1+1/c^2/x^2)^(1/2))/d^(5/2)/(c^2*d-e)^(1/2)-1/ 4*b*e*arctanh((c^2*d+(-d)^(1/2)*e^(1/2)/x)/c/d^(1/2)/(c^2*d-e)^(1/2)/(1+1/ c^2/x^2)^(1/2))/d^(5/2)/(c^2*d-e)^(1/2)-3/4*(a+b*arccsch(c*x))*ln(1-c*(1/c /x+(1+1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(-c^2*d+e)^(1/2)))*e^(1/2)/(-d )^(5/2)+3/4*(a+b*arccsch(c*x))*ln(1+c*(1/c/x+(1+1/c^2/x^2)^(1/2))*(-d)^(1/ 2)/(e^(1/2)-(-c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(5/2)-3/4*(a+b*arccsch(c*x))*l n(1-c*(1/c/x+(1+1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)+(-c^2*d+e)^(1/2)))*e ^(1/2)/(-d)^(5/2)+3/4*(a+b*arccsch(c*x))*ln(1+c*(1/c/x+(1+1/c^2/x^2)^(1/2) )*(-d)^(1/2)/(e^(1/2)+(-c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(5/2)+3/4*b*polylog( 2,-c*(1/c/x+(1+1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(-c^2*d+e)^(1/2)))*e^ (1/2)/(-d)^(5/2)-3/4*b*polylog(2,c*(1/c/x+(1+1/c^2/x^2)^(1/2))*(-d)^(1/2)/ (e^(1/2)-(-c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(5/2)+3/4*b*polylog(2,-c*(1/c/x+( 1+1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)+(-c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(5 /2)-3/4*b*polylog(2,c*(1/c/x+(1+1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)+(-c^ 2*d+e)^(1/2)))*e^(1/2)/(-d)^(5/2)+1/4*e*(a+b*arccsch(c*x))/d^2/(-d/x+(-d)^ (1/2)*e^(1/2))-1/4*e*(a+b*arccsch(c*x))/d^2/(d/x+(-d)^(1/2)*e^(1/2))+b*c*( 1+1/c^2/x^2)^(1/2)/d^2
Result contains complex when optimal does not.
Time = 2.83 (sec) , antiderivative size = 1487, normalized size of antiderivative = 1.96 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx =\text {Too large to display} \]
((-8*a*Sqrt[d])/x - (4*a*Sqrt[d]*e*x)/(d + e*x^2) - 12*a*Sqrt[e]*ArcTan[(S qrt[e]*x)/Sqrt[d]] + b*(8*c*Sqrt[d]*Sqrt[1 + 1/(c^2*x^2)] - (8*Sqrt[d]*Arc Csch[c*x])/x - (2*Sqrt[d]*e*ArcCsch[c*x])/((-I)*Sqrt[d]*Sqrt[e] + e*x) - ( 2*Sqrt[d]*e*ArcCsch[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) - (24*I)*Sqrt[e]*ArcSi n[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot [(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - (24*I)*Sqrt[e]*ArcSin [Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[ (Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] + 3*Sqrt[e]*Pi*Log[1 - ( I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (6*I)*Sqr t[e]*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x ])/(c*Sqrt[d])] + 12*Sqrt[e]*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]] *Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - 3*Sqrt[e]*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/( c*Sqrt[d])] + (6*I)*Sqrt[e]*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2 *d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - 12*Sqrt[e]*ArcSin[Sqrt[1 - Sqrt[e ]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCs ch[c*x])/(c*Sqrt[d])] - 3*Sqrt[e]*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (6*I)*Sqrt[e]*ArcCsch[c*x]*Log[1 - (I* (Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 12*Sqrt[e]*A rcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt...
Time = 2.56 (sec) , antiderivative size = 818, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6858, 6238, 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.
| \(\displaystyle \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6858 |
| \(\displaystyle -\int \frac {a+b \text {arcsinh}\left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right )^2 x^4}d\frac {1}{x}\) |
| \(\Big \downarrow \) 6238 |
| \(\displaystyle -\int \left (\frac {\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) e^2}{d^2 \left (\frac {d}{x^2}+e\right )^2}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) e}{d^2 \left (\frac {d}{x^2}+e\right )}+\frac {a+b \text {arcsinh}\left (\frac {1}{c x}\right )}{d^2}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a}{d^2 x}-\frac {b \text {arcsinh}\left (\frac {1}{c x}\right )}{d^2 x}+\frac {e \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{4 d^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}-\frac {b e \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d-e}}-\frac {b e \text {arctanh}\left (\frac {d c^2+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d-e}}-\frac {3 \sqrt {e} \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {\sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )} c}{\sqrt {e}-\sqrt {e-c^2 d}}+1\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {\sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )} c}{\sqrt {e}+\sqrt {e-c^2 d}}+1\right )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{4 (-d)^{5/2}}+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}}}{d^2}\) |
(b*c*Sqrt[1 + 1/(c^2*x^2)])/d^2 - a/(d^2*x) - (b*ArcSinh[1/(c*x)])/(d^2*x) + (e*(a + b*ArcSinh[1/(c*x)]))/(4*d^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (e*(a + b*ArcSinh[1/(c*x)]))/(4*d^2*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*e*ArcTanh[(c^2 *d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2) ])])/(4*d^(5/2)*Sqrt[c^2*d - e]) - (b*e*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e] )/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*d^(5/2)*Sqrt[c ^2*d - e]) - (3*Sqrt[e]*(a + b*ArcSinh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^Arc Sinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(4*(-d)^(5/2)) + (3*Sqrt[e ]*(a + b*ArcSinh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e ] - Sqrt[-(c^2*d) + e])])/(4*(-d)^(5/2)) - (3*Sqrt[e]*(a + b*ArcSinh[1/(c* x)])*Log[1 - (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e] )])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcSinh[1/(c*x)])*Log[1 + (c*Sqrt[- d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(4*(-d)^(5/2)) + ( 3*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[- (c^2*d) + e]))])/(4*(-d)^(5/2)) - (3*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^Ar cSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(4*(-d)^(5/2)) + (3*b*Sqr t[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e]))])/(4*(-d)^(5/2)) - (3*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcSinh[1 /(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(4*(-d)^(5/2))
3.2.10.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[e, c^ 2*d] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcSinh[x/c])^n/x ^(m + 2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0 ] && IntegersQ[m, p]
\[\int \frac {a +b \,\operatorname {arccsch}\left (c x \right )}{x^{2} \left (e \,x^{2}+d \right )^{2}}d x\]
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^2} \,d x \]