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

Optimal. Leaf size=178 \[ -b d e \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )-\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{2 x^2}-2 d e \log \left (\frac{1}{x}\right ) \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{2} e^2 x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b c d^2 \sqrt{\frac{1}{c^2 x^2}+1}}{4 x}-\frac{1}{4} b c^2 d^2 \text{csch}^{-1}(c x)+\frac{b e^2 x \sqrt{\frac{1}{c^2 x^2}+1}}{2 c}+b d e \text{csch}^{-1}(c x)^2-2 b d e \text{csch}^{-1}(c x) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+2 b d e \log \left (\frac{1}{x}\right ) \text{csch}^{-1}(c x) \]

[Out]

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

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Rubi [A]  time = 0.424422, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 15, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {6304, 266, 43, 5789, 12, 6742, 264, 321, 215, 2325, 5659, 3716, 2190, 2279, 2391} \[ -b d e \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )-\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{2 x^2}-2 d e \log \left (\frac{1}{x}\right ) \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{2} e^2 x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b c d^2 \sqrt{\frac{1}{c^2 x^2}+1}}{4 x}-\frac{1}{4} b c^2 d^2 \text{csch}^{-1}(c x)+\frac{b e^2 x \sqrt{\frac{1}{c^2 x^2}+1}}{2 c}+b d e \text{csch}^{-1}(c x)^2-2 b d e \text{csch}^{-1}(c x) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+2 b d e \log \left (\frac{1}{x}\right ) \text{csch}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6304

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]

Rule 266

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 5789

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1
+ c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (GtQ[p, 0] || (
IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 264

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 321

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 2325

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(ArcSinh[(Rt[e, 2]*x)/S
qrt[d]]*(a + b*Log[c*x^n]))/Rt[e, 2], x] - Dist[(b*n)/Rt[e, 2], Int[ArcSinh[(Rt[e, 2]*x)/Sqrt[d]]/x, x], x] /;
FreeQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && PosQ[e]

Rule 5659

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

Rule 3716

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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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]

Rubi steps

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

Mathematica [A]  time = 0.902077, size = 187, normalized size = 1.05 \[ \frac{1}{4} \left (4 b d e \text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(c x)}\right )-\frac{2 a d^2}{x^2}+8 a d e \log (x)+2 a e^2 x^2-\frac{b d^2 \left (-c^2 x^2+c^2 x^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )-1\right )}{c x^3 \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{2 b e^2 x \left (\sqrt{\frac{1}{c^2 x^2}+1}+c x \text{csch}^{-1}(c x)\right )}{c}-\frac{2 b d^2 \text{csch}^{-1}(c x)}{x^2}-4 b d e \text{csch}^{-1}(c x) \left (\text{csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text{csch}^{-1}(c x)}\right )\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*a*d^2)/x^2 + 2*a*e^2*x^2 - (2*b*d^2*ArcCsch[c*x])/x^2 + (2*b*e^2*x*(Sqrt[1 + 1/(c^2*x^2)] + c*x*ArcCsch[c
*x]))/c - (b*d^2*(-1 - c^2*x^2 + c^2*x^2*Sqrt[1 + c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]]))/(c*Sqrt[1 + 1/(c^2*x^2
)]*x^3) - 4*b*d*e*ArcCsch[c*x]*(ArcCsch[c*x] + 2*Log[1 - E^(-2*ArcCsch[c*x])]) + 8*a*d*e*Log[x] + 4*b*d*e*Poly
Log[2, E^(-2*ArcCsch[c*x])])/4

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Maple [F]  time = 0.368, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( e{x}^{2}+d \right ) ^{2} \left ( a+b{\rm arccsch} \left (cx\right ) \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 4 \, b c^{2} d e \int \frac{x \log \left (x\right )}{2 \,{\left (\sqrt{c^{2} x^{2} + 1} c^{2} x^{2} + c^{2} x^{2} + \sqrt{c^{2} x^{2} + 1} + 1\right )}}\,{d x} - \frac{1}{2} \, b e^{2} x^{2} \log \left (c\right ) - \frac{1}{2} \, b e^{2} x^{2} \log \left (x\right ) + \frac{1}{2} \, a e^{2} x^{2} - 2 \, b d e \log \left (c\right ) \log \left (x\right ) - b d e \log \left (x\right )^{2} - \frac{1}{2} \,{\left (2 \, \log \left (c^{2} x^{2} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-c^{2} x^{2}\right )\right )} b d e + \frac{1}{8} \, b d^{2}{\left (\frac{\frac{2 \, c^{4} x \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c^{2} x^{2}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} - 1} - c^{3} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right ) + c^{3} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c} - \frac{4 \, \operatorname{arcsch}\left (c x\right )}{x^{2}}\right )} + 2 \, a d e \log \left (x\right ) + \frac{b e^{2}{\left (2 \, \sqrt{c^{2} x^{2} + 1} - \log \left (c^{2} x^{2} + 1\right )\right )}}{4 \, c^{2}} + \frac{b e^{2} \log \left (c^{2} x^{2} + 1\right )}{4 \, c^{2}} + \frac{1}{2} \,{\left (b e^{2} x^{2} + 4 \, b d e \log \left (x\right )\right )} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right ) - \frac{a d^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} +{\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \operatorname{arcsch}\left (c x\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{2}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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