\(\int \frac {x^5 (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^3} \, dx\) [111]

Optimal result

Integrand size = 21, antiderivative size = 694 \[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {b c d \sqrt {1+\frac {1}{c^2 x^2}}}{8 \left (c^2 d-e\right ) e^2 \left (e+\frac {d}{x^2}\right ) x}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {csch}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{b e^3}+\frac {b \left (c^2 d-2 e\right ) \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{8 \left (c^2 d-e\right )^{3/2} e^{5/2}}+\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^{5/2}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )}{e^3}+\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^3}+\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^3}+\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^3}+\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^3}+\frac {b \operatorname {PolyLog}\left (2,e^{-2 \text {csch}^{-1}(c x)}\right )}{2 e^3}+\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^3}+\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^3}+\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^3}+\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^3} \]

[Out]

1/4*(-a-b*arccsch(c*x))/e/(e+d/x^2)^2+1/2*(-a-b*arccsch(c*x))/e^2/(e+d/x^2)-(a+b*arccsch(c*x))^2/b/e^3+1/8*b*(
c^2*d-2*e)*arctan((c^2*d-e)^(1/2)/c/x/e^(1/2)/(1+1/c^2/x^2)^(1/2))/(c^2*d-e)^(3/2)/e^(5/2)-(a+b*arccsch(c*x))*
ln(1-1/(1/c/x+(1+1/c^2/x^2)^(1/2))^2)/e^3+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^3+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^3+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^3+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^3+1/2*b*polylog(2,1/(1/c/x+(1+1/c^2/x^2)^(1/2))^2)/e^3+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^3+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^3+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^3+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^3+1/2*b*arctan((c^2*d-e)^
(1/2)/c/x/e^(1/2)/(1+1/c^2/x^2)^(1/2))/e^(5/2)/(c^2*d-e)^(1/2)+1/8*b*c*d*(1+1/c^2/x^2)^(1/2)/(c^2*d-e)/e^2/(e+
d/x^2)/x

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {6439, 5823, 5775, 3797, 2221, 2317, 2438, 5821, 390, 385, 211, 5827, 5680} \[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\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 {e-c^2 d}}\right )}{2 e^3}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}+1\right )}{2 e^3}+\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-c^2 d}+\sqrt {e}}\right )}{2 e^3}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}+1\right )}{2 e^3}-\frac {a+b \text {csch}^{-1}(c x)}{2 e^2 \left (\frac {d}{x^2}+e\right )}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (\frac {d}{x^2}+e\right )^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{b e^3}-\frac {\log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\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^{5/2} \sqrt {c^2 d-e}}+\frac {b \left (c^2 d-2 e\right ) \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}+1}}\right )}{8 e^{5/2} \left (c^2 d-e\right )^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{2 e^3}+\frac {b c d \sqrt {\frac {1}{c^2 x^2}+1}}{8 e^2 x \left (c^2 d-e\right ) \left (\frac {d}{x^2}+e\right )}+\frac {b \operatorname {PolyLog}\left (2,e^{-2 \text {csch}^{-1}(c x)}\right )}{2 e^3} \]

[In]

Int[(x^5*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]

[Out]

(b*c*d*Sqrt[1 + 1/(c^2*x^2)])/(8*(c^2*d - e)*e^2*(e + d/x^2)*x) - (a + b*ArcCsch[c*x])/(4*e*(e + d/x^2)^2) - (
a + b*ArcCsch[c*x])/(2*e^2*(e + d/x^2)) - (a + b*ArcCsch[c*x])^2/(b*e^3) + (b*(c^2*d - 2*e)*ArcTan[Sqrt[c^2*d
- e]/(c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)])/(8*(c^2*d - e)^(3/2)*e^(5/2)) + (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^(5/2)) - ((a + b*ArcCsch[c*x])*Log[1 - E^(-2*ArcCsch[c*x])])
/e^3 + ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(2*e^3) + ((
a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(2*e^3) + ((a + b*Arc
Csch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(2*e^3) + ((a + b*ArcCsch[c*x]
)*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(2*e^3) + (b*PolyLog[2, E^(-2*ArcCsch[c
*x])])/(2*e^3) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e]))])/(2*e^3) + (b*Po
lyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(2*e^3) + (b*PolyLog[2, -((c*Sqrt[-d]*E^
ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e]))])/(2*e^3) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] +
Sqrt[-(c^2*d) + e])])/(2*e^3)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5680

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d
*x))), x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 5821

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1
)*((a + b*ArcSinh[c*x])/(2*e*(p + 1))), x] - Dist[b*(c/(2*e*(p + 1))), Int[(d + e*x^2)^(p + 1)/Sqrt[1 + c^2*x^
2], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[e, c^2*d] && NeQ[p, -1]

Rule 5823

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]

Rule 5827

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Subst[Int[(a + b*x)^n*(Cosh[x
]/(c*d + e*Sinh[x])), x], x, ArcSinh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 6439

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]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{x \left (e+d x^2\right )^3} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (\frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{e^3 x}-\frac {d x \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{e \left (e+d x^2\right )^3}-\frac {d x \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{e^2 \left (e+d x^2\right )^2}-\frac {d x \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{e^3 \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^3}+\frac {d \text {Subst}\left (\int \frac {x \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{e^3}+\frac {d \text {Subst}\left (\int \frac {x \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{e^2}+\frac {d \text {Subst}\left (\int \frac {x \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac {1}{x}\right )}{e} \\ & = -\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {csch}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {\text {Subst}\left (\int x \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {csch}^{-1}(c x)\right )}{b e^3}+\frac {d \text {Subst}\left (\int \left (-\frac {\sqrt {-d} \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{e^3}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}} \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )}{2 c e^2}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}} \left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{4 c e} \\ & = \frac {b c d \sqrt {1+\frac {1}{c^2 x^2}}}{8 \left (c^2 d-e\right ) e^2 \left (e+\frac {d}{x^2}\right ) x}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {csch}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b e^3}-\frac {2 \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {csch}^{-1}(c x)\right )}{b e^3}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 e^3}+\frac {\sqrt {-d} \text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 e^3}+\frac {b \text {Subst}\left (\int \frac {1}{e-\left (-d+\frac {e}{c^2}\right ) x^2} \, dx,x,\frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{2 c e^2}+\frac {\left (b \left (c^2 d-2 e\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}} \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )}{8 c \left (c^2 d-e\right ) e^2} \\ & = \frac {b c d \sqrt {1+\frac {1}{c^2 x^2}}}{8 \left (c^2 d-e\right ) e^2 \left (e+\frac {d}{x^2}\right ) x}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {csch}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b e^3}+\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^{5/2}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )}{e^3}+\frac {\text {Subst}\left (\int \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \cosh (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \sinh (x)} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^3}+\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \cosh (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \sinh (x)} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^3}+\frac {\left (b \left (c^2 d-2 e\right )\right ) \text {Subst}\left (\int \frac {1}{e-\left (-d+\frac {e}{c^2}\right ) x^2} \, dx,x,\frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{8 c \left (c^2 d-e\right ) e^2} \\ & = \frac {b c d \sqrt {1+\frac {1}{c^2 x^2}}}{8 \left (c^2 d-e\right ) e^2 \left (e+\frac {d}{x^2}\right ) x}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {csch}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{b e^3}+\frac {b \left (c^2 d-2 e\right ) \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{8 \left (c^2 d-e\right )^{3/2} e^{5/2}}+\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^{5/2}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )}{e^3}-\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {csch}^{-1}(c x)}{b}\right )}\right )}{2 e^3}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^3}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^3}+\frac {\sqrt {-d} \text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^3}+\frac {\sqrt {-d} \text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^3} \\ & = \frac {b c d \sqrt {1+\frac {1}{c^2 x^2}}}{8 \left (c^2 d-e\right ) e^2 \left (e+\frac {d}{x^2}\right ) x}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {csch}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{b e^3}+\frac {b \left (c^2 d-2 e\right ) \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{8 \left (c^2 d-e\right )^{3/2} e^{5/2}}+\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^{5/2}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )}{e^3}+\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^3}+\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^3}+\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^3}+\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^3}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {a+b \text {csch}^{-1}(c x)}{b}\right )}\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^3} \\ & = \frac {b c d \sqrt {1+\frac {1}{c^2 x^2}}}{8 \left (c^2 d-e\right ) e^2 \left (e+\frac {d}{x^2}\right ) x}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {csch}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{b e^3}+\frac {b \left (c^2 d-2 e\right ) \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{8 \left (c^2 d-e\right )^{3/2} e^{5/2}}+\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^{5/2}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )}{e^3}+\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^3}+\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^3}+\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^3}+\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^3}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {a+b \text {csch}^{-1}(c x)}{b}\right )}\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 e^3} \\ & = \frac {b c d \sqrt {1+\frac {1}{c^2 x^2}}}{8 \left (c^2 d-e\right ) e^2 \left (e+\frac {d}{x^2}\right ) x}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {csch}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{b e^3}+\frac {b \left (c^2 d-2 e\right ) \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{8 \left (c^2 d-e\right )^{3/2} e^{5/2}}+\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^{5/2}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )}{e^3}+\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^3}+\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^3}+\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^3}+\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^3}+\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^3}+\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^3}+\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^3}+\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^3}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {a+b \text {csch}^{-1}(c x)}{b}\right )}\right )}{2 e^3} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.89 (sec) , antiderivative size = 2023, normalized size of antiderivative = 2.91 \[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Result too large to show} \]

[In]

Integrate[(x^5*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]

[Out]

-1/4*(a*d^2)/(e^3*(d + e*x^2)^2) + (a*d)/(e^3*(d + e*x^2)) + (a*Log[d + e*x^2])/(2*e^3) + b*(-1/16*(d*((I*c*Sq
rt[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]*(Sq
rt[e] + I*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))))/e^(5/2) - (d*(((-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))))/(16*e^(5/2)) - (((7*I)/16)*Sqrt[d]*(-(Ar
cCsch[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]))/e^(5/2) + (((7*I)/16)*Sqrt[d]*(-(ArcCsch[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(A
rcSinh[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]))/e^(5/2) + (Pi^2 - (
4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 + 32*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]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])]
+ (4*I)*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 - (
I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (16*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/S
qrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (4*I)*Pi*Log[1 + (I*(Sqrt[e]
+ Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])
*E^ArcCsch[c*x])/(c*Sqrt[d])] - (16*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqr
t[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (4*I)*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*Arc
Csch[c*x])] + 8*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*PolyLog[2, ((-I
)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/(16*e^3) + (Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*Ar
cCsch[c*x]^2 - 32*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]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 + (I*(-Sq
rt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d)
+ e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (16*I)*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])] + (4*I)*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^Ar
cCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d]
)] - (16*I)*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])] - (4*I)*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2,
((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d)
+ e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/(16*e^3))

Maple [F]

\[\int \frac {x^{5} \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{3}}d x\]

[In]

int(x^5*(a+b*arccsch(c*x))/(e*x^2+d)^3,x)

[Out]

int(x^5*(a+b*arccsch(c*x))/(e*x^2+d)^3,x)

Fricas [F]

\[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

[In]

integrate(x^5*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

integral((b*x^5*arccsch(c*x) + a*x^5)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(x**5*(a+b*acsch(c*x))/(e*x**2+d)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

[In]

integrate(x^5*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

1/4*a*((4*d*e*x^2 + 3*d^2)/(e^5*x^4 + 2*d*e^4*x^2 + d^2*e^3) + 2*log(e*x^2 + d)/e^3) + b*integrate(x^5*log(sqr
t(1/(c^2*x^2) + 1) + 1/(c*x))/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

Giac [F]

\[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

[In]

integrate(x^5*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)*x^5/(e*x^2 + d)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]

[In]

int((x^5*(a + b*asinh(1/(c*x))))/(d + e*x^2)^3,x)

[Out]

int((x^5*(a + b*asinh(1/(c*x))))/(d + e*x^2)^3, x)