3.2.3 \(\int \frac {x^5 (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^2} \, dx\) [103]

Optimal. Leaf size=591 \[ \frac {b \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c e^2}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 e^2}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )^2}{b e^3}-\frac {b d \text {ArcTan}\left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{2 \sqrt {c^2 d-e} e^{5/2}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )}{e^3}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{e^3}-\frac {b d \text {PolyLog}\left (2,e^{-2 \text {csch}^{-1}(c x)}\right )}{e^3}-\frac {b d \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{e^3}-\frac {b d \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{e^3}-\frac {b d \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{e^3}-\frac {b d \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{e^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.91, antiderivative size = 591, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 14, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6439, 5823, 5776, 270, 5775, 3797, 2221, 2317, 2438, 5821, 385, 211, 5827, 5680} \begin {gather*} -\frac {d \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 )}{e^3}-\frac {d \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 )}{e^3}-\frac {d \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 )}{e^3}-\frac {d \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 )}{e^3}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )^2}{b e^3}+\frac {2 d \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 e^2 \left (\frac {d}{x^2}+e\right )}+\frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 e^2}-\frac {b d \text {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 d \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{e^3}-\frac {b d \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{e^3}-\frac {b d \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{e^3}-\frac {b d \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{e^3}+\frac {b x \sqrt {\frac {1}{c^2 x^2}+1}}{2 c e^2}-\frac {b d \text {Li}_2\left (e^{-2 \text {csch}^{-1}(c x)}\right )}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*Sqrt[1 + 1/(c^2*x^2)]*x)/(2*c*e^2) + (d*(a + b*ArcCsch[c*x]))/(2*e^2*(e + d/x^2)) + (x^2*(a + b*ArcCsch[c*x
]))/(2*e^2) + (2*d*(a + b*ArcCsch[c*x])^2)/(b*e^3) - (b*d*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)) + (2*d*(a + b*ArcCsch[c*x])*Log[1 - E^(-2*ArcCsch[c*x])])/e^3 - (d*(a + b
*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/e^3 - (d*(a + b*ArcCsch[c*
x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/e^3 - (d*(a + b*ArcCsch[c*x])*Log[1 -
(c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/e^3 - (d*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d
]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/e^3 - (b*d*PolyLog[2, E^(-2*ArcCsch[c*x])])/e^3 - (b*d*Poly
Log[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e]))])/e^3 - (b*d*PolyLog[2, (c*Sqrt[-d]*E^Arc
Csch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/e^3 - (b*d*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqr
t[-(c^2*d) + e]))])/e^3 - (b*d*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/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 270

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

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 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 5776

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

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

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Mathematica [C] Result contains complex when optimal does not.
time = 3.91, size = 1447, normalized size = 2.45 \begin {gather*} -\frac {-2 a e x^2+\frac {2 a d^2}{d+e x^2}+4 a d \log \left (d+e x^2\right )+b \left (d \pi ^2-\frac {2 e \sqrt {1+\frac {1}{c^2 x^2}} x}{c}-4 i d \pi \text {csch}^{-1}(c x)-2 e x^2 \text {csch}^{-1}(c x)+\frac {d^{3/2} \text {csch}^{-1}(c x)}{\sqrt {d}-i \sqrt {e} x}+\frac {d^{3/2} \text {csch}^{-1}(c x)}{\sqrt {d}+i \sqrt {e} x}-8 d \text {csch}^{-1}(c x)^2-2 d \sinh ^{-1}\left (\frac {1}{c x}\right )+16 d \text {ArcSin}\left (\frac {\sqrt {1+\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {\left (c \sqrt {d}-\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c x)\right )\right )}{\sqrt {-c^2 d+e}}\right )-16 d \text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {\left (c \sqrt {d}+\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c x)\right )\right )}{\sqrt {-c^2 d+e}}\right )-8 d \text {csch}^{-1}(c x) \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )+2 i d \pi \log \left (1-\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 d \text {csch}^{-1}(c x) \log \left (1-\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+8 i d \text {ArcSin}\left (\frac {\sqrt {1+\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+2 i d \pi \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 d \text {csch}^{-1}(c x) \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+8 i d \text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+2 i d \pi \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 d \text {csch}^{-1}(c x) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )-8 i d \text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+2 i d \pi \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 d \text {csch}^{-1}(c x) \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )-8 i d \text {ArcSin}\left (\frac {\sqrt {1+\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )-2 i d \pi \log \left (\sqrt {e}-\frac {i \sqrt {d}}{x}\right )-2 i d \pi \log \left (\sqrt {e}+\frac {i \sqrt {d}}{x}\right )+\frac {d \sqrt {e} \log \left (\frac {2 \sqrt {d} \sqrt {e} \left (i \sqrt {e}+c \left (c \sqrt {d}+i \sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{\sqrt {-c^2 d+e} \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt {-c^2 d+e}}+\frac {d \sqrt {e} \log \left (-\frac {2 \sqrt {d} \sqrt {e} \left (\sqrt {e}+c \left (i c \sqrt {d}+\sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{\sqrt {-c^2 d+e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d+e}}+4 d \text {PolyLog}\left (2,e^{-2 \text {csch}^{-1}(c x)}\right )+4 d \text {PolyLog}\left (2,-\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 d \text {PolyLog}\left (2,\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 d \text {PolyLog}\left (2,-\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 d \text {PolyLog}\left (2,\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )\right )}{4 e^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*(-2*a*e*x^2 + (2*a*d^2)/(d + e*x^2) + 4*a*d*Log[d + e*x^2] + b*(d*Pi^2 - (2*e*Sqrt[1 + 1/(c^2*x^2)]*x)/c
- (4*I)*d*Pi*ArcCsch[c*x] - 2*e*x^2*ArcCsch[c*x] + (d^(3/2)*ArcCsch[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (d^(3/2)*A
rcCsch[c*x])/(Sqrt[d] + I*Sqrt[e]*x) - 8*d*ArcCsch[c*x]^2 - 2*d*ArcSinh[1/(c*x)] + 16*d*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]] -
16*d*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*d*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (2*I)*d*Pi*Log[1 - (I*(-Sqrt[e] + S
qrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*d*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*
E^ArcCsch[c*x])/(c*Sqrt[d])] + (8*I)*d*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sq
rt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*d*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCs
ch[c*x])/(c*Sqrt[d])] + 4*d*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d]
)] + (8*I)*d*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsc
h[c*x])/(c*Sqrt[d])] + (2*I)*d*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*d
*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (8*I)*d*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)*
d*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*d*ArcCsch[c*x]*Log[1 + (I*(Sqr
t[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (8*I)*d*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)*d*Pi*Log[Sqrt[e] - (I*Sqrt[d]
)/x] - (2*I)*d*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + (d*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] + (d*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] + 4*d*PolyLog[2, E^(-2*ArcCsch[c*x])]
+ 4*d*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*d*PolyLog[2, (I*(-Sqrt
[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*d*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E
^ArcCsch[c*x])/(c*Sqrt[d])] + 4*d*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])]))/
e^3

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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/2*(x^2*e^(-2) - 2*d*e^(-3)*log(x^2*e + d) - d^2/(x^2*e^4 + d*e^3))*a + b*integrate(x^5*log(sqrt(1/(c^2*x^2)
+ 1) + 1/(c*x))/(x^4*e^2 + 2*d*x^2*e + d^2), 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^5*(a+b*arccsch(c*x))/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

integral((b*x^5*arccsch(c*x) + a*x^5)/(x^4*e^2 + 2*d*x^2*e + d^2), 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**5*(a+b*acsch(c*x))/(e*x**2+d)**2,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^5*(a+b*arccsch(c*x))/(e*x^2+d)^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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