Integrand size = 18, antiderivative size = 1096 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=-\frac {b c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d-e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d-e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {5 \left (a+b \text {csch}^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}+\frac {5 \left (a+b \text {csch}^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {5 b \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 )}{16 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 )}{16 d^{5/2} \left (c^2 d-e\right )^{3/2}}+\frac {5 b \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 )}{16 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 )}{16 d^{5/2} \left (c^2 d-e\right )^{3/2}}+\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}} \]
1/16*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)^(3/2)+1/16*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)^(3/2)+5/16*b*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)+5/16*b*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)+3/16*(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)))/(-d)^(5/2)/e^(1/2)-3/16*(a+b*a rccsch(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)))/(-d)^(5/2)/e^(1/2)+3/16*(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)))/(-d)^(5/2)/e^(1/2) -3/16*(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)))/(-d)^(5/2)/e^(1/2)-3/16*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)))/(-d)^(5/2)/e^(1/2 )+3/16*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)))/(-d)^(5/2)/e^(1/2)-3/16*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)))/(-d)^(5/2)/e^(1/2)+3/16*b*po lylog(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) ))/(-d)^(5/2)/e^(1/2)+1/16*(a+b*arccsch(c*x))*e^(1/2)/(-d)^(3/2)/(-d/x+(-d )^(1/2)*e^(1/2))^2-5/16*(a+b*arccsch(c*x))/d^2/(-d/x+(-d)^(1/2)*e^(1/2)...
Result contains complex when optimal does not.
Time = 6.06 (sec) , antiderivative size = 2038, normalized size of antiderivative = 1.86 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=\text {Result too large to show} \]
(a*x)/(4*d*(d + e*x^2)^2) + (3*a*x)/(8*d^2*(d + e*x^2)) + (3*a*ArcTan[(Sqr t[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]) + b*(((I/16)*((I*c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d - e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCsch [c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - ArcSinh[1/(c*x)]/(d*Sqrt[e] ) + (I*(2*c^2*d - e)*Log[(4*d*Sqrt[c^2*d - e]*Sqrt[e]*(Sqrt[e] + I*c*(c*Sq rt[d] - Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])*x))/((2*c^2*d - e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d*(c^2*d - e)^(3/2))))/d^(3/2) - ((I/16)*(((-I)*c*Sqrt [e]*Sqrt[1 + 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d - e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCsch[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - ArcSinh[1/(c*x)]/(d* Sqrt[e]) + (I*(2*c^2*d - e)*Log[((4*I)*d*Sqrt[c^2*d - e]*Sqrt[e]*(I*Sqrt[e ] + c*(c*Sqrt[d] + Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])*x))/((2*c^2*d - e)*(Sqrt[d] - I*Sqrt[e]*x))])/(d*(c^2*d - e)^(3/2))))/d^(3/2) - (3*(-(ArcC sch[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSinh[1/(c*x)]/Sqrt[e] - Log[( 2*Sqrt[d]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(I*Sqrt[d] + Sqrt[e]*x))]/Sqrt[-(c ^2*d) + e]))/Sqrt[d]))/(16*d^2) - (3*(-(ArcCsch[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSinh[1/(c*x)]/Sqrt[e] - Log[(-2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-( c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) + e]))/Sqrt[d]))/(16*d ^2) + (((3*I)/128)*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 + 3...
Time = 3.90 (sec) , antiderivative size = 1160, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6848, 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.
\(\displaystyle \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 6848 |
\(\displaystyle -\int \frac {a+b \text {arcsinh}\left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right )^3 x^4}d\frac {1}{x}\) |
\(\Big \downarrow \) 6238 |
\(\displaystyle -\int \left (\frac {\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) e^2}{d^2 \left (\frac {d}{x^2}+e\right )^3}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) e}{d^2 \left (\frac {d}{x^2}+e\right )^2}+\frac {a+b \text {arcsinh}\left (\frac {1}{c x}\right )}{d^2 \left (\frac {d}{x^2}+e\right )}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} c}{16 (-d)^{3/2} \left (c^2 d-e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} c}{16 (-d)^{3/2} \left (c^2 d-e\right ) \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}-\frac {5 \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {5 \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{16 d^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {\sqrt {e} \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {\sqrt {e} \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{16 (-d)^{3/2} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}+\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 )}{16 d^{5/2} \left (c^2 d-e\right )^{3/2}}+\frac {5 b \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 )}{16 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 )}{16 d^{5/2} \left (c^2 d-e\right )^{3/2}}+\frac {5 b \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 )}{16 d^{5/2} \sqrt {c^2 d-e}}+\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 b \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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 b \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 b \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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 b \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 )}{16 (-d)^{5/2} \sqrt {e}}\) |
-1/16*(b*c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)])/((-d)^(3/2)*(c^2*d - e)*(Sqrt[-d ]*Sqrt[e] - d/x)) - (b*c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)])/(16*(-d)^(3/2)*(c^ 2*d - e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (Sqrt[e]*(a + b*ArcSinh[1/(c*x)]))/(1 6*(-d)^(3/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) - (5*(a + b*ArcSinh[1/(c*x)]))/(1 6*d^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (Sqrt[e]*(a + b*ArcSinh[1/(c*x)]))/(16*( -d)^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) + (5*(a + b*ArcSinh[1/(c*x)]))/(16*d ^2*(Sqrt[-d]*Sqrt[e] + d/x)) + (5*b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x) /(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(16*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)])])/(16*d^(5/2)*(c^2*d - e)^(3/2)) + (5*b*ArcT anh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/( c^2*x^2)])])/(16*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)])])/(16*d^(5 /2)*(c^2*d - e)^(3/2)) + (3*(a + b*ArcSinh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E ^ArcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(16*(-d)^(5/2)*Sqrt[e] ) - (3*(a + b*ArcSinh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(S qrt[e] - Sqrt[-(c^2*d) + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*(a + b*ArcSinh [1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d ) + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcSinh[1/(c*x)])*Log[1 + (c *Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(16*(-d)...
3.2.17.3.1 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_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcSinh[x/c])^n/x^(2*(p + 1) )), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p ]
\[\int \frac {a +b \,\operatorname {arccsch}\left (c x \right )}{\left (e \,x^{2}+d \right )^{3}}d x\]
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]