3.1.98 \(\int \frac {x^2 (a+b \text {csch}^{-1}(c x))}{d+e x^2} \, dx\) [98]

Optimal. Leaf size=512 \[ \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c e}+\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e^{3/2}}-\frac {b \sqrt {-d} \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e^{3/2}}-\frac {b \sqrt {-d} \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.84, antiderivative size = 512, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 12, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6439, 5823, 5776, 272, 65, 214, 5793, 5827, 5680, 2221, 2317, 2438} \begin {gather*} \frac {\sqrt {-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 )}{2 e^{3/2}}-\frac {\sqrt {-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 )}{2 e^{3/2}}+\frac {\sqrt {-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 )}{2 e^{3/2}}-\frac {\sqrt {-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 )}{2 e^{3/2}}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 e^{3/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{2 e^{3/2}}+\frac {b \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{c e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/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 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 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 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 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^2 \left (a+b \text {csch}^{-1}(c x)\right )}{d+e x^2} \, dx &=-\text {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{x^2 \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )\\ &=-\text {Subst}\left (\int \left (\frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{e x^2}-\frac {d \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{e \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\text {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{x^2} \, dx,x,\frac {1}{x}\right )}{e}+\frac {d \text {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{e}\\ &=\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e}-\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c e}+\frac {d \text {Subst}\left (\int \left (\frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{e}\\ &=\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {d \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 )}{2 e^{3/2}}+\frac {d \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 )}{2 e^{3/2}}-\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 c e}\\ &=\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {d \text {Subst}\left (\int \frac {(a+b x) \cosh (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \sinh (x)} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^{3/2}}+\frac {d \text {Subst}\left (\int \frac {(a+b x) \cosh (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \sinh (x)} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^{3/2}}-\frac {(b c) \text {Subst}\left (\int \frac {1}{-c^2+c^2 x^2} \, dx,x,\sqrt {1+\frac {1}{c^2 x^2}}\right )}{e}\\ &=\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c e}+\frac {d \text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^{3/2}}+\frac {d \text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^{3/2}}+\frac {d \text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^{3/2}}+\frac {d \text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^{3/2}}\\ &=\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c e}+\frac {\sqrt {-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 )}{2 e^{3/2}}-\frac {\sqrt {-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 )}{2 e^{3/2}}+\frac {\sqrt {-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 )}{2 e^{3/2}}-\frac {\sqrt {-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 )}{2 e^{3/2}}-\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^{3/2}}+\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^{3/2}}-\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^{3/2}}+\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 e^{3/2}}\\ &=\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c e}+\frac {\sqrt {-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 )}{2 e^{3/2}}-\frac {\sqrt {-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 )}{2 e^{3/2}}+\frac {\sqrt {-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 )}{2 e^{3/2}}-\frac {\sqrt {-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 )}{2 e^{3/2}}-\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 e^{3/2}}+\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 e^{3/2}}-\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 e^{3/2}}+\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 e^{3/2}}\\ &=\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c e}+\frac {\sqrt {-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 )}{2 e^{3/2}}-\frac {\sqrt {-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 )}{2 e^{3/2}}+\frac {\sqrt {-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 )}{2 e^{3/2}}-\frac {\sqrt {-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 )}{2 e^{3/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e^{3/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e^{3/2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.13, size = 1221, normalized size = 2.38 \begin {gather*} \frac {4 a c \sqrt {e} x+4 b c \sqrt {e} x \text {csch}^{-1}(c x)-4 a c \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-8 i b c \sqrt {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 i b c \sqrt {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 )+b c \sqrt {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 )-2 i b c \sqrt {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 )+4 b c \sqrt {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 )-b c \sqrt {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 )+2 i b c \sqrt {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 )-4 b c \sqrt {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 )-b c \sqrt {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 )+2 i b c \sqrt {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 )+4 b c \sqrt {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 )+b c \sqrt {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 )-2 i b c \sqrt {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 )-4 b c \sqrt {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 )+b c \sqrt {d} \pi \log \left (\sqrt {e}-\frac {i \sqrt {d}}{x}\right )-b c \sqrt {d} \pi \log \left (\sqrt {e}+\frac {i \sqrt {d}}{x}\right )-4 b \sqrt {e} \log \left (\tanh \left (\frac {1}{2} \text {csch}^{-1}(c x)\right )\right )+2 i b c \sqrt {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 )-2 i b c \sqrt {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 )-2 i b c \sqrt {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 )+2 i b c \sqrt {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 c e^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*c*Sqrt[e]*x + 4*b*c*Sqrt[e]*x*ArcCsch[c*x] - 4*a*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - (8*I)*b*c*Sqrt[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*I)*b*c*Sqrt[d]*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + S
qrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] + b*c*Sqrt[d]*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c
^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*c*Sqrt[d]*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d)
+ e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*c*Sqrt[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])] - b*c*Sqrt[d]*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2
*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*c*Sqrt[d]*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) +
e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - 4*b*c*Sqrt[d]*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])] - b*c*Sqrt[d]*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d)
+ e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*c*Sqrt[d]*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])
*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*c*Sqrt[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])] + b*c*Sqrt[d]*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e]
)*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*c*Sqrt[d]*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^Ar
cCsch[c*x])/(c*Sqrt[d])] - 4*b*c*Sqrt[d]*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + S
qrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + b*c*Sqrt[d]*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] - b*c*Sqrt[d]*Pi
*Log[Sqrt[e] + (I*Sqrt[d])/x] - 4*b*Sqrt[e]*Log[Tanh[ArcCsch[c*x]/2]] + (2*I)*b*c*Sqrt[d]*PolyLog[2, ((-I)*(-S
qrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*c*Sqrt[d]*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-
(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*c*Sqrt[d]*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])
*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*c*Sqrt[d]*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])
/(c*Sqrt[d])])/(4*c*e^(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

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

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