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

Optimal. Leaf size=573 \[ \frac{3 b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}-\frac{3 b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}-\frac{2 i b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}+\frac{2 i b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}-\frac{3 b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}+\frac{3 b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{d x \sqrt{c^2 d x^2+d}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{c^2 d x^2+d}}+\frac{4 b c^2 \sqrt{c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}+\frac{3 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{d \sqrt{c^2 d x^2+d}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.925848, antiderivative size = 573, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 15, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.536, Rules used = {5747, 5755, 5764, 5760, 4182, 2531, 2282, 6589, 5693, 4180, 2279, 2391, 266, 63, 208} \[ \frac{3 b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}-\frac{3 b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}-\frac{2 i b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}+\frac{2 i b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}-\frac{3 b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}+\frac{3 b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{d x \sqrt{c^2 d x^2+d}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{c^2 d x^2+d}}+\frac{4 b c^2 \sqrt{c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}+\frac{3 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{d \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(d*x*Sqrt[d + c^2*d*x^2])) - (3*c^2*(a + b*ArcSinh[c*x])^2)/(2*
d*Sqrt[d + c^2*d*x^2]) - (a + b*ArcSinh[c*x])^2/(2*d*x^2*Sqrt[d + c^2*d*x^2]) + (4*b*c^2*Sqrt[1 + c^2*x^2]*(a
+ b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) + (3*c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*
x])^2*ArcTanh[E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) - (b^2*c^2*Sqrt[1 + c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]]
)/(d*Sqrt[d + c^2*d*x^2]) + (3*b*c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/(d*Sq
rt[d + c^2*d*x^2]) - ((2*I)*b^2*c^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2])
+ ((2*I)*b^2*c^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) - (3*b*c^2*Sqrt[1 +
c^2*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) - (3*b^2*c^2*Sqrt[1 + c^2*x^
2]*PolyLog[3, -E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) + (3*b^2*c^2*Sqrt[1 + c^2*x^2]*PolyLog[3, E^ArcSinh[c*
x]])/(d*Sqrt[d + c^2*d*x^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 5764

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

Rule 5760

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

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 5693

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

Rule 4180

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

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 63

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 208

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

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

Mathematica [A]  time = 7.49549, size = 884, normalized size = 1.54 \[ -\frac{3 a^2 \log (x) c^2}{2 d^{3/2}}+\frac{3 a^2 \log \left (d+\sqrt{d \left (c^2 x^2+1\right )} \sqrt{d}\right ) c^2}{2 d^{3/2}}+\frac{a b \left (-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{sech}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-8 \sinh ^{-1}(c x)+16 \sqrt{c^2 x^2+1} \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )-2 \sqrt{c^2 x^2+1} \coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-12 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )+12 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1+e^{-\sinh ^{-1}(c x)}\right )-12 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )+12 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+2 \sqrt{c^2 x^2+1} \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right ) c^2}{4 d \sqrt{d \left (c^2 x^2+1\right )}}+\frac{b^2 \left (-\sqrt{c^2 x^2+1} \text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)^2-\sqrt{c^2 x^2+1} \text{sech}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)^2-12 \sqrt{c^2 x^2+1} \log \left (1-e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)^2+12 \sqrt{c^2 x^2+1} \log \left (1+e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)^2-8 \sinh ^{-1}(c x)^2-4 \sqrt{c^2 x^2+1} \coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)-16 i \sqrt{c^2 x^2+1} \log \left (1-i e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)+16 i \sqrt{c^2 x^2+1} \log \left (1+i e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)-24 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)+24 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)+4 \sqrt{c^2 x^2+1} \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)+8 \sqrt{c^2 x^2+1} \log \left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )-16 i \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )+16 i \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )-24 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,-e^{-\sinh ^{-1}(c x)}\right )+24 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,e^{-\sinh ^{-1}(c x)}\right )\right ) c^2}{8 d \sqrt{d \left (c^2 x^2+1\right )}}+\sqrt{d \left (c^2 x^2+1\right )} \left (-\frac{c^2 a^2}{d^2 \left (c^2 x^2+1\right )}-\frac{a^2}{2 d^2 x^2}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[d*(1 + c^2*x^2)]*(-a^2/(2*d^2*x^2) - (a^2*c^2)/(d^2*(1 + c^2*x^2))) - (3*a^2*c^2*Log[x])/(2*d^(3/2)) + (3
*a^2*c^2*Log[d + Sqrt[d]*Sqrt[d*(1 + c^2*x^2)]])/(2*d^(3/2)) + (a*b*c^2*(-8*ArcSinh[c*x] + 16*Sqrt[1 + c^2*x^2
]*ArcTan[Tanh[ArcSinh[c*x]/2]] - 2*Sqrt[1 + c^2*x^2]*Coth[ArcSinh[c*x]/2] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Csc
h[ArcSinh[c*x]/2]^2 - 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 12*Sqrt[1 + c^2*x^2]*ArcS
inh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 12*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-ArcSinh[c*x])] + 12*Sqrt[1 + c^2*x^
2]*PolyLog[2, E^(-ArcSinh[c*x])] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Sqrt[1 + c^2*x^2]
*Tanh[ArcSinh[c*x]/2]))/(4*d*Sqrt[d*(1 + c^2*x^2)]) + (b^2*c^2*(-8*ArcSinh[c*x]^2 - 4*Sqrt[1 + c^2*x^2]*ArcSin
h[c*x]*Coth[ArcSinh[c*x]/2] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^2 - 12*Sqrt[1 + c^2*x^2]*A
rcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c*x])] - (16*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + (
16*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] + 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Log[1 + E
^(-ArcSinh[c*x])] + 8*Sqrt[1 + c^2*x^2]*Log[Tanh[ArcSinh[c*x]/2]] - 24*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*PolyLog[
2, -E^(-ArcSinh[c*x])] - (16*I)*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (16*I)*Sqrt[1 + c^2*x^2]*P
olyLog[2, I/E^ArcSinh[c*x]] + 24*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] - 24*Sqrt[1 + c^
2*x^2]*PolyLog[3, -E^(-ArcSinh[c*x])] + 24*Sqrt[1 + c^2*x^2]*PolyLog[3, E^(-ArcSinh[c*x])] - Sqrt[1 + c^2*x^2]
*ArcSinh[c*x]^2*Sech[ArcSinh[c*x]/2]^2 + 4*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Tanh[ArcSinh[c*x]/2]))/(8*d*Sqrt[d*(
1 + c^2*x^2)])

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Maple [F]  time = 0.338, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{{x}^{3}} \left ({c}^{2}d{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \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)^(3/2),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}\right )}}{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)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*d*x^2 + d)*(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*} \int \frac{\left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}{x^{3} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \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)**(3/2),x)

[Out]

Integral((a + b*asinh(c*x))**2/(x**3*(d*(c**2*x**2 + 1))**(3/2)), x)

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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 )}^{\frac{3}{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)^(3/2),x, algorithm="giac")

[Out]

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