3.241 \(\int \frac{(a+b \sinh ^{-1}(c x))^2}{x^3 (d+c^2 d x^2)^2} \, dx\)

Optimal. Leaf size=253 \[ \frac{2 b c^2 \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac{2 b c^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac{b^2 c^2 \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 c^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (c^2 x^2+1\right )}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{c^2 x^2+1}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac{4 c^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2}-\frac{b^2 c^2 \log \left (c^2 x^2+1\right )}{2 d^2}+\frac{b^2 c^2 \log (x)}{d^2} \]

[Out]

-((b*c*(a + b*ArcSinh[c*x]))/(d^2*x*Sqrt[1 + c^2*x^2])) - (c^2*(a + b*ArcSinh[c*x])^2)/(d^2*(1 + c^2*x^2)) - (
a + b*ArcSinh[c*x])^2/(2*d^2*x^2*(1 + c^2*x^2)) + (4*c^2*(a + b*ArcSinh[c*x])^2*ArcTanh[E^(2*ArcSinh[c*x])])/d
^2 + (b^2*c^2*Log[x])/d^2 - (b^2*c^2*Log[1 + c^2*x^2])/(2*d^2) + (2*b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, -E^(
2*ArcSinh[c*x])])/d^2 - (2*b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])])/d^2 - (b^2*c^2*PolyLog[3
, -E^(2*ArcSinh[c*x])])/d^2 + (b^2*c^2*PolyLog[3, E^(2*ArcSinh[c*x])])/d^2

________________________________________________________________________________________

Rubi [A]  time = 0.580781, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 15, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.577, Rules used = {5747, 5755, 5720, 5461, 4182, 2531, 2282, 6589, 5687, 260, 271, 191, 5732, 446, 72} \[ \frac{2 b c^2 \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac{2 b c^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac{b^2 c^2 \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 c^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (c^2 x^2+1\right )}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{c^2 x^2+1}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac{4 c^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2}-\frac{b^2 c^2 \log \left (c^2 x^2+1\right )}{2 d^2}+\frac{b^2 c^2 \log (x)}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c*(a + b*ArcSinh[c*x]))/(d^2*x*Sqrt[1 + c^2*x^2])) - (c^2*(a + b*ArcSinh[c*x])^2)/(d^2*(1 + c^2*x^2)) - (
a + b*ArcSinh[c*x])^2/(2*d^2*x^2*(1 + c^2*x^2)) + (4*c^2*(a + b*ArcSinh[c*x])^2*ArcTanh[E^(2*ArcSinh[c*x])])/d
^2 + (b^2*c^2*Log[x])/d^2 - (b^2*c^2*Log[1 + c^2*x^2])/(2*d^2) + (2*b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, -E^(
2*ArcSinh[c*x])])/d^2 - (2*b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])])/d^2 - (b^2*c^2*PolyLog[3
, -E^(2*ArcSinh[c*x])])/d^2 + (b^2*c^2*PolyLog[3, E^(2*ArcSinh[c*x])])/d^2

Rule 5747

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n)/(d*f*(m + 1)), x] + (-Dist[(c^2*(m + 2*p + 3))/(f^2
*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^
2)^FracPart[p])/(f*(m + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSin
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[m, -1] && Int
egerQ[m]

Rule 5755

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> -Simp
[((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n)/(2*d*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d*(p
+ 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^F
racPart[p])/(2*f*(p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[
c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ
[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])

Rule 5720

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Dist[1/d, Subst[Int[(
a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n
, 0]

Rule 5461

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 5687

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(x*(a + b*ArcSinh
[c*x])^n)/(d*Sqrt[d + e*x^2]), x] - Dist[(b*c*n*Sqrt[1 + c^2*x^2])/(d*Sqrt[d + e*x^2]), Int[(x*(a + b*ArcSinh[
c*x])^(n - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

Rule 260

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

Rule 271

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

Rule 191

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

Rule 5732

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(1 + c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcSinh[c*x]), u, x] - Dist[b*c*d^p, Int[SimplifyIntegrand[u/Sqrt[1 +
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2,
0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d, 0]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^3 \left (d+c^2 d x^2\right )^2} \, dx &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\left (2 c^2\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )^2} \, dx+\frac{(b c) \int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2}}-\frac{2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{1+c^2 x^2}}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\frac{\left (b^2 c^2\right ) \int \frac{-1-2 c^2 x^2}{x \left (1+c^2 x^2\right )} \, dx}{d^2}+\frac{\left (2 b c^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2}-\frac{\left (2 c^2\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )} \, dx}{d}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2}}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}-\frac{\left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{-1-2 c^2 x}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d^2}-\frac{\left (2 b^2 c^4\right ) \int \frac{x}{1+c^2 x^2} \, dx}{d^2}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2}}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\frac{b^2 c^2 \log \left (1+c^2 x^2\right )}{d^2}-\frac{\left (4 c^2\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}-\frac{\left (b^2 c^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{x}-\frac{c^2}{1+c^2 x}\right ) \, dx,x,x^2\right )}{2 d^2}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2}}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 c^2 \log (x)}{d^2}-\frac{b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac{\left (4 b c^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}-\frac{\left (4 b c^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2}}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 c^2 \log (x)}{d^2}-\frac{b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac{2 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac{2 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac{\left (2 b^2 c^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}+\frac{\left (2 b^2 c^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2}}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 c^2 \log (x)}{d^2}-\frac{b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac{2 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac{2 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac{\left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{\left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{d^2}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2}}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 c^2 \log (x)}{d^2}-\frac{b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac{2 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac{2 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac{b^2 c^2 \text{Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 c^2 \text{Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}\\ \end{align*}

Mathematica [B]  time = 0.925183, size = 594, normalized size = 2.35 \[ \frac{-4 a b c^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )+8 b c^2 \text{PolyLog}\left (2,\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right ) \left (a+b \sinh ^{-1}(c x)\right )+8 b c^2 \text{PolyLog}\left (2,\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right ) \left (a+b \sinh ^{-1}(c x)\right )-4 b^2 c^2 \sinh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )-8 b^2 c^2 \text{PolyLog}\left (3,\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right )-8 b^2 c^2 \text{PolyLog}\left (3,\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )+2 b^2 c^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )+\frac{a^2}{c^2 x^4+x^2}+2 a^2 c^2 \log \left (c^2 x^2+1\right )+4 a^2 c^2 \sinh ^{-1}(c x)-4 a^2 c^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\frac{2 a^2}{x^2}-\frac{2 a b c}{x \sqrt{c^2 x^2+1}}+\frac{2 a b \sinh ^{-1}(c x)}{c^2 x^4+x^2}+8 a b c^2 \sinh ^{-1}(c x) \log \left (\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right )+8 a b c^2 \sinh ^{-1}(c x) \log \left (\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right )-8 a b c^2 \sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\frac{4 a b \sinh ^{-1}(c x)}{x^2}-b^2 c^2 \log \left (c^2 x^2+1\right )+\frac{b^2 \sinh ^{-1}(c x)^2}{c^2 x^4+x^2}-\frac{2 b^2 c \sinh ^{-1}(c x)}{x \sqrt{c^2 x^2+1}}+2 b^2 c^2 \log (x)+4 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left (\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right )+4 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left (\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right )-4 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\frac{2 b^2 \sinh ^{-1}(c x)^2}{x^2}}{2 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*a^2)/x^2 - (2*a*b*c)/(x*Sqrt[1 + c^2*x^2]) + a^2/(x^2 + c^2*x^4) + 4*a^2*c^2*ArcSinh[c*x] - (4*a*b*ArcSin
h[c*x])/x^2 - (2*b^2*c*ArcSinh[c*x])/(x*Sqrt[1 + c^2*x^2]) + (2*a*b*ArcSinh[c*x])/(x^2 + c^2*x^4) - (2*b^2*Arc
Sinh[c*x]^2)/x^2 + (b^2*ArcSinh[c*x]^2)/(x^2 + c^2*x^4) + 8*a*b*c^2*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sq
rt[-c^2]] + 4*b^2*c^2*ArcSinh[c*x]^2*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 8*a*b*c^2*ArcSinh[c*x]*Log[1 + (
Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 4*b^2*c^2*ArcSinh[c*x]^2*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 4*a^2*c^2*Lo
g[1 - E^(2*ArcSinh[c*x])] - 8*a*b*c^2*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] - 4*b^2*c^2*ArcSinh[c*x]^2*Log[
1 - E^(2*ArcSinh[c*x])] + 2*b^2*c^2*Log[x] + 2*a^2*c^2*Log[1 + c^2*x^2] - b^2*c^2*Log[1 + c^2*x^2] + 8*b*c^2*(
a + b*ArcSinh[c*x])*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 8*b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[
-c^2]*E^ArcSinh[c*x])/c] - 4*a*b*c^2*PolyLog[2, E^(2*ArcSinh[c*x])] - 4*b^2*c^2*ArcSinh[c*x]*PolyLog[2, E^(2*A
rcSinh[c*x])] - 8*b^2*c^2*PolyLog[3, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 8*b^2*c^2*PolyLog[3, (Sqrt[-c^2]*E^ArcSi
nh[c*x])/c] + 2*b^2*c^2*PolyLog[3, E^(2*ArcSinh[c*x])])/(2*d^2)

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Maple [B]  time = 0.157, size = 799, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^2/d^2/x^2-1/2*c^2*a^2/d^2/(c^2*x^2+1)+c^2*b^2/d^2*ln(1+c*x+(c^2*x^2+1)^(1/2))+c^2*a^2/d^2*ln(c^2*x^2+1)
+4*c^2*b^2/d^2*polylog(3,-c*x-(c^2*x^2+1)^(1/2))+4*c^2*b^2/d^2*polylog(3,c*x+(c^2*x^2+1)^(1/2))-2*c^2*a^2/d^2*
ln(c*x)-c^2*b^2/d^2*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)+c^2*b^2/d^2*ln(c*x+(c^2*x^2+1)^(1/2)-1)-c^2*b^2/d^2*arcsin
h(c*x)^2/(c^2*x^2+1)+2*c^2*b^2/d^2*arcsinh(c*x)^2*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)+2*c^2*b^2/d^2*arcsinh(c*x)*p
olylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)-2*c^2*b^2/d^2*arcsinh(c*x)^2*ln(1+c*x+(c^2*x^2+1)^(1/2))-4*c^2*b^2/d^2*ar
csinh(c*x)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))-2*c^2*b^2/d^2*arcsinh(c*x)^2*ln(1-c*x-(c^2*x^2+1)^(1/2))-4*c^2*b^
2/d^2*arcsinh(c*x)*polylog(2,c*x+(c^2*x^2+1)^(1/2))+2*c^2*a*b/d^2*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)-4*c^2*
a*b/d^2*polylog(2,-c*x-(c^2*x^2+1)^(1/2))-4*c^2*a*b/d^2*polylog(2,c*x+(c^2*x^2+1)^(1/2))-b^2*c^2*polylog(3,-(c
*x+(c^2*x^2+1)^(1/2))^2)/d^2-c*b^2/d^2*arcsinh(c*x)/x/(c^2*x^2+1)^(1/2)-c*a*b/d^2/x/(c^2*x^2+1)^(1/2)-2*c^2*a*
b/d^2*arcsinh(c*x)/(c^2*x^2+1)+4*c^2*a*b/d^2*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)-4*c^2*a*b/d^2*arcsin
h(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))-4*c^2*a*b/d^2*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))-a*b/d^2*arcsinh(c*x)
/x^2/(c^2*x^2+1)-1/2*b^2/d^2*arcsinh(c*x)^2/x^2/(c^2*x^2+1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{2}{\left (\frac{2 \, c^{2} \log \left (c^{2} x^{2} + 1\right )}{d^{2}} - \frac{4 \, c^{2} \log \left (x\right )}{d^{2}} - \frac{2 \, c^{2} x^{2} + 1}{c^{2} d^{2} x^{4} + d^{2} x^{2}}\right )} + \int \frac{b^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{c^{4} d^{2} x^{7} + 2 \, c^{2} d^{2} x^{5} + d^{2} x^{3}} + \frac{2 \, a b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{4} d^{2} x^{7} + 2 \, c^{2} d^{2} x^{5} + d^{2} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*(2*c^2*log(c^2*x^2 + 1)/d^2 - 4*c^2*log(x)/d^2 - (2*c^2*x^2 + 1)/(c^2*d^2*x^4 + d^2*x^2)) + integrate(
b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^4*d^2*x^7 + 2*c^2*d^2*x^5 + d^2*x^3) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1)
)/(c^4*d^2*x^7 + 2*c^2*d^2*x^5 + d^2*x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}}{c^{4} d^{2} x^{7} + 2 \, c^{2} d^{2} x^{5} + d^{2} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(c^4*d^2*x^7 + 2*c^2*d^2*x^5 + d^2*x^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a^{2}}{c^{4} x^{7} + 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{c^{4} x^{7} + 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{c^{4} x^{7} + 2 c^{2} x^{5} + x^{3}}\, dx}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a**2/(c**4*x**7 + 2*c**2*x**5 + x**3), x) + Integral(b**2*asinh(c*x)**2/(c**4*x**7 + 2*c**2*x**5 + x
**3), x) + Integral(2*a*b*asinh(c*x)/(c**4*x**7 + 2*c**2*x**5 + x**3), x))/d**2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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