3.2.8 \(\int \frac {x^2 (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^2} \, dx\) [108]

Optimal. Leaf size=719 \[ \frac {a+b \text {csch}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} \sqrt {c^2 d-e} e}-\frac {b \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} \sqrt {c^2 d-e} e}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}} \]

[Out]

1/4*(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^(3/2)/(-d)^
(1/2)-1/4*(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^(3/2)
/(-d)^(1/2)+1/4*(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
^(3/2)/(-d)^(1/2)-1/4*(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^(3/2)/(-d)^(1/2)-1/4*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^(3/2)/(-d)^(1/2)+1/4*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^(3/2
)/(-d)^(1/2)-1/4*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^(3/2)/(-d
)^(1/2)+1/4*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^(3/2)/(-d)^(1/2
)-1/4*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/d^(1/2)/(c^2*d-e
)^(1/2)-1/4*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/d^(1/2)/(c
^2*d-e)^(1/2)+1/4*(a+b*arccsch(c*x))/e/(-d/x+(-d)^(1/2)*e^(1/2))+1/4*(-a-b*arccsch(c*x))/e/(d/x+(-d)^(1/2)*e^(
1/2))

________________________________________________________________________________________

Rubi [A]
time = 0.84, antiderivative size = 719, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {6439, 5793, 5828, 739, 212, 5827, 5680, 2221, 2317, 2438} \begin {gather*} \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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d-e}}\right )}{4 \sqrt {d} e \sqrt {c^2 d-e}}-\frac {b \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d-e}}\right )}{4 \sqrt {d} e \sqrt {c^2 d-e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*ArcCsch[c*x])/(4*e*(Sqrt[-d]*Sqrt[e] - d/x)) - (a + b*ArcCsch[c*x])/(4*e*(Sqrt[-d]*Sqrt[e] + d/x)) - (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)])])/(4*Sqrt[d]*Sqrt[c^
2*d - e]*e) - (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)])])/(4
*Sqrt[d]*Sqrt[c^2*d - e]*e) + ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*
d) + e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(
c^2*d) + e])])/(4*Sqrt[-d]*e^(3/2)) + ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqr
t[-(c^2*d) + e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] +
Sqrt[-(c^2*d) + e])])/(4*Sqrt[-d]*e^(3/2)) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^
2*d) + e]))])/(4*Sqrt[-d]*e^(3/2)) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])]
)/(4*Sqrt[-d]*e^(3/2)) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e]))])/(4*Sqrt
[-d]*e^(3/2)) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(4*Sqrt[-d]*e^(3/2)
)

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

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Mathematica [C] Result contains complex when optimal does not.
time = 1.65, size = 1442, normalized size = 2.01 \begin {gather*} \frac {-\frac {4 a \sqrt {e} x}{d+e x^2}+\frac {4 a \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+b \left (\frac {2 \text {csch}^{-1}(c x)}{i \sqrt {d}-\sqrt {e} x}-\frac {2 \text {csch}^{-1}(c x)}{i \sqrt {d}+\sqrt {e} x}+\frac {8 i \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 )}{\sqrt {d}}+\frac {8 i \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 )}{\sqrt {d}}-\frac {\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 )}{\sqrt {d}}+\frac {2 i \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 )}{\sqrt {d}}-\frac {4 \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 )}{\sqrt {d}}+\frac {\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 )}{\sqrt {d}}-\frac {2 i \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 )}{\sqrt {d}}+\frac {4 \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 )}{\sqrt {d}}+\frac {\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 )}{\sqrt {d}}-\frac {2 i \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 )}{\sqrt {d}}-\frac {4 \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 )}{\sqrt {d}}-\frac {\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 )}{\sqrt {d}}+\frac {2 i \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 )}{\sqrt {d}}+\frac {4 \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 )}{\sqrt {d}}-\frac {\pi \log \left (\sqrt {e}-\frac {i \sqrt {d}}{x}\right )}{\sqrt {d}}+\frac {\pi \log \left (\sqrt {e}+\frac {i \sqrt {d}}{x}\right )}{\sqrt {d}}+\frac {2 i \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 {d} \sqrt {-c^2 d+e}}-\frac {2 i \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 {d} \sqrt {-c^2 d+e}}-\frac {2 i \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 )}{\sqrt {d}}+\frac {2 i \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 )}{\sqrt {d}}+\frac {2 i \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 )}{\sqrt {d}}-\frac {2 i \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 )}{\sqrt {d}}\right )}{8 e^{3/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-4*a*Sqrt[e]*x)/(d + e*x^2) + (4*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + b*((2*ArcCsch[c*x])/(I*Sqrt[d] - S
qrt[e]*x) - (2*ArcCsch[c*x])/(I*Sqrt[d] + Sqrt[e]*x) + ((8*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Ar
cTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]])/Sqrt[d] + ((8*I)*ArcSin[Sqr
t[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]])/Sqrt[d] - (Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] + ((
2*I)*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] - (4*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])
])/Sqrt[d] + (Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] - ((2*I)*Arc
Csch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] + (4*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])])/Sqrt[
d] + (Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] - ((2*I)*ArcCsch[c*x]
*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] - (4*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])])/Sqrt[d] - (Pi*Lo
g[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] + ((2*I)*ArcCsch[c*x]*Log[1 + (I
*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] + (4*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])])/Sqrt[d] - (Pi*Log[Sqrt[e] -
(I*Sqrt[d])/x])/Sqrt[d] + (Pi*Log[Sqrt[e] + (I*Sqrt[d])/x])/Sqrt[d] + ((2*I)*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] + Sq
rt[e]*x))])/(Sqrt[d]*Sqrt[-(c^2*d) + e]) - ((2*I)*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[d]*Sqrt[-(c
^2*d) + e]) - ((2*I)*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] +
((2*I)*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] + ((2*I)*PolyLog[2,
((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] - ((2*I)*PolyLog[2, (I*(Sqrt[e] +
Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d]))/(8*e^(3/2))

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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(a + b*acsch(c*x))/(d + e*x**2)**2, x)

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

[Out]

integrate((b*arccsch(c*x) + a)*x^2/(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^2\,\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^2*(a + b*asinh(1/(c*x))))/(d + e*x^2)^2,x)

[Out]

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

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