Integrand size = 21, antiderivative size = 553 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{b e^2}+\frac {b \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{2 \sqrt {c^2 d-e} e^{3/2}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )}{e^2}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,e^{-2 \text {csch}^{-1}(c x)}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e^2} \]
1/2*(-a-b*arccsch(c*x))/e/(e+d/x^2)-(a+b*arccsch(c*x))^2/b/e^2-(a+b*arccsc h(c*x))*ln(1-1/(1/c/x+(1+1/c^2/x^2)^(1/2))^2)/e^2+1/2*(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 ^2+1/2*(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^2+1/2*(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^2+1/2*(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^2+1/2*b*polylog(2,1/(1/c/x+(1+1/c^2/x^2)^(1/2))^2)/e^2+1/2*b*polyl og(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^2+1/2*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^2+1/2*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^2+1/2*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^2+1/2*b*arctan((c^2*d-e)^ (1/2)/c/x/e^(1/2)/(1+1/c^2/x^2)^(1/2))/e^(3/2)/(c^2*d-e)^(1/2)
Result contains complex when optimal does not.
Time = 2.94 (sec) , antiderivative size = 1410, normalized size of antiderivative = 2.55 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx =\text {Too large to display} \]
(b*Pi^2 + (4*a*d)/(d + e*x^2) - (4*I)*b*Pi*ArcCsch[c*x] + (2*b*Sqrt[d]*Arc Csch[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (2*b*Sqrt[d]*ArcCsch[c*x])/(Sqrt[d] + I*Sqrt[e]*x) - 8*b*ArcCsch[c*x]^2 - 4*b*ArcSinh[1/(c*x)] + 16*b*ArcSin[Sq rt[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]] - 16*b*ArcSin[Sqrt[1 - Sqrt [e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*Ar cCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*b*ArcCsch[c*x]*Log[1 - E^(-2*ArcCs ch[c*x])] + (2*I)*b*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsc h[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2* d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (8*I)*b*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])] + (2*I)*b*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])* E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqr t[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (8*I)*b*ArcSin[Sqrt[1 - Sq rt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^A rcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (8*I)*b*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])] + (2*I)*b*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[...
Time = 1.65 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.12, 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.
| \(\displaystyle \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6858 |
| \(\displaystyle -\int \frac {x \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{\left (\frac {d}{x^2}+e\right )^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 6238 |
| \(\displaystyle -\int \left (\frac {x \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{e^2}-\frac {d \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{e^2 \left (\frac {d}{x^2}+e\right ) x}-\frac {d \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{e \left (\frac {d}{x^2}+e\right )^2 x}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 e^2}+\frac {\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {e-c^2 d}}+1\right )}{2 e^2}+\frac {\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e-c^2 d}+\sqrt {e}}\right )}{2 e^2}+\frac {\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e-c^2 d}+\sqrt {e}}+1\right )}{2 e^2}-\frac {a+b \text {arcsinh}\left (\frac {1}{c x}\right )}{2 e \left (\frac {d}{x^2}+e\right )}-\frac {\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )^2}{b e^2}-\frac {\log \left (1-e^{-2 \text {arcsinh}\left (\frac {1}{c x}\right )}\right ) \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{e^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\frac {1}{c x}\right )}\right )}{2 e^2}+\frac {b \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}+1}}\right )}{2 e^{3/2} \sqrt {c^2 d-e}}\) |
-1/2*(a + b*ArcSinh[1/(c*x)])/(e*(e + d/x^2)) - (a + b*ArcSinh[1/(c*x)])^2 /(b*e^2) + (b*ArcTan[Sqrt[c^2*d - e]/(c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)]) /(2*Sqrt[c^2*d - e]*e^(3/2)) - ((a + b*ArcSinh[1/(c*x)])*Log[1 - E^(-2*Arc Sinh[1/(c*x)])])/e^2 + ((a + b*ArcSinh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^Arc Sinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(2*e^2) + ((a + b*ArcSinh[ 1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(2*e^2) + ((a + b*ArcSinh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcSinh [1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(2*e^2) + ((a + b*ArcSinh[1/(c *x)])*Log[1 + (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e ])])/(2*e^2) + (b*PolyLog[2, E^(-2*ArcSinh[1/(c*x)])])/(2*e^2) + (b*PolyLo g[2, -((c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e]))])/( 2*e^2) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c ^2*d) + e])])/(2*e^2) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(S qrt[e] + Sqrt[-(c^2*d) + e]))])/(2*e^2) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcS inh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(2*e^2)
3.2.4.3.1 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]
\[\int \frac {x^{3} \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{2}}d x\]
\[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
\[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^{3} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
\[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
1/2*a*(d/(e^3*x^2 + d*e^2) + log(e*x^2 + d)/e^2) + b*integrate(x^3*log(sqr t(1/(c^2*x^2) + 1) + 1/(c*x))/(e^2*x^4 + 2*d*e*x^2 + d^2), x)
\[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]