3.2.15 \(\int \frac {x^4 (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^3} \, dx\) [115]

Optimal. Leaf size=1106 \[ -\frac {b c \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}}}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}}}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {3 b \tanh ^{-1}\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 \sqrt {d} \sqrt {c^2 d-e} e^2}+\frac {b \tanh ^{-1}\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 \sqrt {d} \left (c^2 d-e\right )^{3/2} e}-\frac {3 b \tanh ^{-1}\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 \sqrt {d} \sqrt {c^2 d-e} e^2}+\frac {b \tanh ^{-1}\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 \sqrt {d} \left (c^2 d-e\right )^{3/2} 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 \sqrt {-d} e^{5/2}}-\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 \sqrt {-d} e^{5/2}}+\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 \sqrt {-d} e^{5/2}}-\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 \sqrt {-d} e^{5/2}}-\frac {3 b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}} \]

[Out]

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)))/e^(5/2)/(-d)
^(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)))/e^(5/
2)/(-d)^(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))
)/e^(5/2)/(-d)^(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)))/e^(5/2)/(-d)^(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)))/e^(5/2)/(-d)^(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)))/
e^(5/2)/(-d)^(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)))/e^(5
/2)/(-d)^(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)))/e^(5/2)/(
-d)^(1/2)+1/16*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))/(c^2*d-e)
^(3/2)/e/d^(1/2)+1/16*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))/(c
^2*d-e)^(3/2)/e/d^(1/2)-3/16*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))/e^2/d^(1/2)/(c^2*d-e)^(1/2)-3/16*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))/e^2/d^(1/2)/(c^2*d-e)^(1/2)+1/16*(a+b*arccsch(c*x))*(-d)^(1/2)/e^(3/2)/(-d/x+(-d)^(1/2)*e^(1/2))
^2+3/16*(a+b*arccsch(c*x))/e^2/(-d/x+(-d)^(1/2)*e^(1/2))-1/16*(a+b*arccsch(c*x))*(-d)^(1/2)/e^(3/2)/(d/x+(-d)^
(1/2)*e^(1/2))^2-3/16*(a+b*arccsch(c*x))/e^2/(d/x+(-d)^(1/2)*e^(1/2))-1/16*b*c*(-d)^(1/2)*(1+1/c^2/x^2)^(1/2)/
(c^2*d-e)/e^(3/2)/(-d/x+(-d)^(1/2)*e^(1/2))-1/16*b*c*(-d)^(1/2)*(1+1/c^2/x^2)^(1/2)/(c^2*d-e)/e^(3/2)/(d/x+(-d
)^(1/2)*e^(1/2))

________________________________________________________________________________________

Rubi [A]
time = 1.06, antiderivative size = 1106, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {6439, 5793, 5828, 745, 739, 212, 5827, 5680, 2221, 2317, 2438} \begin {gather*} -\frac {b \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}} c}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}} c}{16 \left (c^2 d-e\right ) e^{3/2} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}+\frac {b \tanh ^{-1}\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 \sqrt {d} \left (c^2 d-e\right )^{3/2} e}-\frac {3 b \tanh ^{-1}\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 \sqrt {d} \sqrt {c^2 d-e} e^2}+\frac {b \tanh ^{-1}\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 \sqrt {d} \left (c^2 d-e\right )^{3/2} e}-\frac {3 b \tanh ^{-1}\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 \sqrt {d} \sqrt {c^2 d-e} e^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 {e-c^2 d}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {\sqrt {-d} e^{\text {csch}^{-1}(c x)} c}{\sqrt {e}-\sqrt {e-c^2 d}}+1\right )}{16 \sqrt {-d} e^{5/2}}+\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 {e-c^2 d}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {\sqrt {-d} e^{\text {csch}^{-1}(c x)} c}{\sqrt {e}+\sqrt {e-c^2 d}}+1\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{16 \sqrt {-d} e^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]

[Out]

-1/16*(b*c*Sqrt[-d]*Sqrt[1 + 1/(c^2*x^2)])/((c^2*d - e)*e^(3/2)*(Sqrt[-d]*Sqrt[e] - d/x)) - (b*c*Sqrt[-d]*Sqrt
[1 + 1/(c^2*x^2)])/(16*(c^2*d - e)*e^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)) + (Sqrt[-d]*(a + b*ArcCsch[c*x]))/(16*e^(
3/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (3*(a + b*ArcCsch[c*x]))/(16*e^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (Sqrt[-d]*(a +
b*ArcCsch[c*x]))/(16*e^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (3*(a + b*ArcCsch[c*x]))/(16*e^2*(Sqrt[-d]*Sqrt[e]
+ d/x)) - (3*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
*Sqrt[d]*Sqrt[c^2*d - e]*e^2) + (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*Sqrt[d]*(c^2*d - e)^(3/2)*e) - (3*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*Sqrt[d]*Sqrt[c^2*d - e]*e^2) + (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*Sqrt[d]*(c^2*d - e)^(3/2)*e) + (3*(a + b*ArcCsch[c
*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*(a + b*Ar
cCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(16*Sqrt[-d]*e^(5/2)) + (3*(a
+ b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(16*Sqrt[-d]*e^(5/2))
- (3*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(16*Sqrt[-d]*e^
(5/2)) - (3*b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e]))])/(16*Sqrt[-d]*e^(5/2))
+ (3*b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*b*P
olyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e]))])/(16*Sqrt[-d]*e^(5/2)) + (3*b*PolyLog
[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(16*Sqrt[-d]*e^(5/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 745

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*((a + c*x^2)^(p
+ 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c*(d/(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

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 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 5793

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] &&
(p > 0 || IGtQ[n, 0])

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 5828

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*
((a + b*ArcSinh[c*x])^n/(e*(m + 1))), x] - Dist[b*c*(n/(e*(m + 1))), Int[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x
])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

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*} \int \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=-\text {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac {1}{x}\right )\\ &=-\text {Subst}\left (\int \left (-\frac {d^3 \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{8 (-d)^{3/2} e^{3/2} \left (\sqrt {-d} \sqrt {e}-d x\right )^3}-\frac {3 d \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-d x\right )^2}-\frac {d^3 \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{8 (-d)^{3/2} e^{3/2} \left (\sqrt {-d} \sqrt {e}+d x\right )^3}-\frac {3 d \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}+d x\right )^2}-\frac {3 d \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{8 e^2 \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {(3 d) \text {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^2} \, dx,x,\frac {1}{x}\right )}{16 e^2}+\frac {(3 d) \text {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^2} \, dx,x,\frac {1}{x}\right )}{16 e^2}+\frac {(3 d) \text {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac {1}{x}\right )}{8 e^2}-\frac {(-d)^{3/2} \text {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^3} \, dx,x,\frac {1}{x}\right )}{8 e^{3/2}}-\frac {(-d)^{3/2} \text {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^3} \, dx,x,\frac {1}{x}\right )}{8 e^{3/2}}\\ &=\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}-d x\right ) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c e^2}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}+d x\right ) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c e^2}+\frac {(3 d) \text {Subst}\left (\int \left (-\frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}-\frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{8 e^2}-\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}-d x\right )^2 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c e^{3/2}}+\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}+d x\right )^2 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c e^{3/2}}\\ &=-\frac {b c \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}}}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}}}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {3 \text {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{16 e^{5/2}}-\frac {3 \text {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{16 e^{5/2}}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{d^2-\frac {d e}{c^2}-x^2} \, dx,x,\frac {-d-\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1+\frac {1}{c^2 x^2}}}\right )}{16 c e^2}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{d^2-\frac {d e}{c^2}-x^2} \, dx,x,\frac {d-\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1+\frac {1}{c^2 x^2}}}\right )}{16 c e^2}+\frac {b \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}-d x\right ) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c \left (c^2 d-e\right ) e}-\frac {b \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}+d x\right ) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c \left (c^2 d-e\right ) e}\\ &=-\frac {b c \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}}}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}}}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {3 b \tanh ^{-1}\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 \sqrt {d} \sqrt {c^2 d-e} e^2}-\frac {3 b \tanh ^{-1}\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 \sqrt {d} \sqrt {c^2 d-e} e^2}-\frac {3 \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 )}{16 e^{5/2}}-\frac {3 \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 )}{16 e^{5/2}}-\frac {b \text {Subst}\left (\int \frac {1}{d^2-\frac {d e}{c^2}-x^2} \, dx,x,\frac {-d-\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1+\frac {1}{c^2 x^2}}}\right )}{16 c \left (c^2 d-e\right ) e}+\frac {b \text {Subst}\left (\int \frac {1}{d^2-\frac {d e}{c^2}-x^2} \, dx,x,\frac {d-\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1+\frac {1}{c^2 x^2}}}\right )}{16 c \left (c^2 d-e\right ) e}\\ &=-\frac {b c \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}}}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}}}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {3 b \tanh ^{-1}\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 \sqrt {d} \sqrt {c^2 d-e} e^2}+\frac {b \tanh ^{-1}\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 \sqrt {d} \left (c^2 d-e\right )^{3/2} e}-\frac {3 b \tanh ^{-1}\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 \sqrt {d} \sqrt {c^2 d-e} e^2}+\frac {b \tanh ^{-1}\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 \sqrt {d} \left (c^2 d-e\right )^{3/2} e}-\frac {3 \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 )}{16 e^{5/2}}-\frac {3 \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 )}{16 e^{5/2}}-\frac {3 \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 )}{16 e^{5/2}}-\frac {3 \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 )}{16 e^{5/2}}\\ &=-\frac {b c \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}}}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}}}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {3 b \tanh ^{-1}\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 \sqrt {d} \sqrt {c^2 d-e} e^2}+\frac {b \tanh ^{-1}\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 \sqrt {d} \left (c^2 d-e\right )^{3/2} e}-\frac {3 b \tanh ^{-1}\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 \sqrt {d} \sqrt {c^2 d-e} e^2}+\frac {b \tanh ^{-1}\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 \sqrt {d} \left (c^2 d-e\right )^{3/2} 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 \sqrt {-d} e^{5/2}}-\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 \sqrt {-d} e^{5/2}}+\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 \sqrt {-d} e^{5/2}}-\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 \sqrt {-d} e^{5/2}}-\frac {(3 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 )}{16 \sqrt {-d} e^{5/2}}+\frac {(3 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 )}{16 \sqrt {-d} e^{5/2}}-\frac {(3 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 )}{16 \sqrt {-d} e^{5/2}}+\frac {(3 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 )}{16 \sqrt {-d} e^{5/2}}\\ &=-\frac {b c \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}}}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}}}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {3 b \tanh ^{-1}\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 \sqrt {d} \sqrt {c^2 d-e} e^2}+\frac {b \tanh ^{-1}\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 \sqrt {d} \left (c^2 d-e\right )^{3/2} e}-\frac {3 b \tanh ^{-1}\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 \sqrt {d} \sqrt {c^2 d-e} e^2}+\frac {b \tanh ^{-1}\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 \sqrt {d} \left (c^2 d-e\right )^{3/2} 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 \sqrt {-d} e^{5/2}}-\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 \sqrt {-d} e^{5/2}}+\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 \sqrt {-d} e^{5/2}}-\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 \sqrt {-d} e^{5/2}}-\frac {(3 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 )}{16 \sqrt {-d} e^{5/2}}+\frac {(3 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 )}{16 \sqrt {-d} e^{5/2}}-\frac {(3 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 )}{16 \sqrt {-d} e^{5/2}}+\frac {(3 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 )}{16 \sqrt {-d} e^{5/2}}\\ &=-\frac {b c \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}}}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {-d} \sqrt {1+\frac {1}{c^2 x^2}}}{16 \left (c^2 d-e\right ) e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {3 \left (a+b \text {csch}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {3 b \tanh ^{-1}\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 \sqrt {d} \sqrt {c^2 d-e} e^2}+\frac {b \tanh ^{-1}\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 \sqrt {d} \left (c^2 d-e\right )^{3/2} e}-\frac {3 b \tanh ^{-1}\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 \sqrt {d} \sqrt {c^2 d-e} e^2}+\frac {b \tanh ^{-1}\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 \sqrt {d} \left (c^2 d-e\right )^{3/2} 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 \sqrt {-d} e^{5/2}}-\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 \sqrt {-d} e^{5/2}}+\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 \sqrt {-d} e^{5/2}}-\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 \sqrt {-d} e^{5/2}}-\frac {3 b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 6.05, size = 2045, normalized size = 1.85 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]

[Out]

(a*d*x)/(4*e^2*(d + e*x^2)^2) - (5*a*x)/(8*e^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[d]*e^(
5/2)) + b*(((I/16)*Sqrt[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)*L
og[(4*d*Sqrt[c^2*d - e]*Sqrt[e]*(Sqrt[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^2 - ((I/16)*Sqrt[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)
)))/e^2 + (5*(-(ArcCsch[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*e^2) + (5*(-(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*e^2) + (((3*I
)/128)*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 + 32*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Arc
Tan[((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*ArcCsc
h[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^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqr
t[-(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*Poly
Log[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])]))/(Sqrt[d]*e^(5/2)) - (((3*I)/12
8)*(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*Ar
cCsch[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])]/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^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*(S
qrt[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])]))/(Sqrt[d]*e^(5/2)))

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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

1/8*(3*arctan(x*e^(1/2)/sqrt(d))*e^(-5/2)/sqrt(d) - (5*x^3*e + 3*d*x)/(x^4*e^4 + 2*d*x^2*e^3 + d^2*e^2))*a + b
*integrate(x^4*log(sqrt(1/(c^2*x^2) + 1) + 1/(c*x))/(x^6*e^3 + 3*d*x^4*e^2 + 3*d^2*x^2*e + d^3), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*acsch(c*x))/(e*x**2+d)**3,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a + b*asinh(1/(c*x))))/(d + e*x^2)^3,x)

[Out]

int((x^4*(a + b*asinh(1/(c*x))))/(d + e*x^2)^3, x)

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