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

Optimal. Leaf size=498 \[ -\frac{2 b d \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{2 b d \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{2 b^2 d \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{2 b^2 d \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{2 a b c d x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}-\frac{2 b c^3 d x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{c^2 x^2+1}}-\frac{2 b c d x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{c^2 x^2+1}}+\frac{1}{3} \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 d \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 x^2+1}}+\frac{22}{9} b^2 d \sqrt{c^2 d x^2+d}+\frac{2}{27} b^2 d \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}-\frac{2 b^2 c d x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}} \]

[Out]

(22*b^2*d*Sqrt[d + c^2*d*x^2])/9 - (2*a*b*c*d*x*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] + (2*b^2*d*(1 + c^2*x^2
)*Sqrt[d + c^2*d*x^2])/27 - (2*b^2*c*d*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (2*b*c*d*x*Sqrt
[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*Sqrt[1 + c^2*x^2]) - (2*b*c^3*d*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSin
h[c*x]))/(9*Sqrt[1 + c^2*x^2]) + d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2 + ((d + c^2*d*x^2)^(3/2)*(a + b*
ArcSinh[c*x])^2)/3 - (2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2
] - (2*b*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (2*b*d*Sq
rt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (2*b^2*d*Sqrt[d + c^2*d
*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (2*b^2*d*Sqrt[d + c^2*d*x^2]*PolyLog[3, E^ArcSinh[c*x]]
)/Sqrt[1 + c^2*x^2]

________________________________________________________________________________________

Rubi [A]  time = 0.594588, antiderivative size = 498, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5744, 5742, 5760, 4182, 2531, 2282, 6589, 5653, 261, 5679, 444, 43} \[ -\frac{2 b d \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{2 b d \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{2 b^2 d \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{2 b^2 d \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{2 a b c d x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}-\frac{2 b c^3 d x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{c^2 x^2+1}}-\frac{2 b c d x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{c^2 x^2+1}}+\frac{1}{3} \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 d \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 x^2+1}}+\frac{22}{9} b^2 d \sqrt{c^2 d x^2+d}+\frac{2}{27} b^2 d \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}-\frac{2 b^2 c d x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(22*b^2*d*Sqrt[d + c^2*d*x^2])/9 - (2*a*b*c*d*x*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] + (2*b^2*d*(1 + c^2*x^2
)*Sqrt[d + c^2*d*x^2])/27 - (2*b^2*c*d*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (2*b*c*d*x*Sqrt
[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*Sqrt[1 + c^2*x^2]) - (2*b*c^3*d*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSin
h[c*x]))/(9*Sqrt[1 + c^2*x^2]) + d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2 + ((d + c^2*d*x^2)^(3/2)*(a + b*
ArcSinh[c*x])^2)/3 - (2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2
] - (2*b*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (2*b*d*Sq
rt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (2*b^2*d*Sqrt[d + c^2*d
*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (2*b^2*d*Sqrt[d + c^2*d*x^2]*PolyLog[3, E^ArcSinh[c*x]]
)/Sqrt[1 + c^2*x^2]

Rule 5744

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*(a + b*ArcSinh[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*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)^FracPart[p]
)/(f*(m + 2*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] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[n, 1])

Rule 5742

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(
(f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(f*(m + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1
+ c^2*x^2]), Int[((f*x)^m*(a + b*ArcSinh[c*x])^n)/Sqrt[1 + c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*
(m + 2)*Sqrt[1 + c^2*x^2]), Int[(f*x)^(m + 1)*(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[m, -1] && (RationalQ[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 5653

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

Rule 261

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

Rule 5679

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(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}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

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])

Rubi steps

\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx &=\frac{1}{3} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+d \int \frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx-\frac{\left (2 b c d \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{2 b c d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{3} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \sqrt{1+c^2 x^2}} \, dx}{\sqrt{1+c^2 x^2}}-\frac{\left (2 b c d \sqrt{d+c^2 d x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}+\frac{\left (2 b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x \left (1+\frac{c^2 x^2}{3}\right )}{\sqrt{1+c^2 x^2}} \, dx}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{2 a b c d x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}-\frac{2 b c d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{3} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (2 b^2 c d \sqrt{d+c^2 d x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{\sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{c^2 x}{3}}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{2 a b c d x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}-\frac{2 b^2 c d x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}}-\frac{2 b c d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{3} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (2 b d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (2 b d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1+c^2 x}}+\frac{1}{3} \sqrt{1+c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt{1+c^2 x^2}}+\frac{\left (2 b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{22}{9} b^2 d \sqrt{d+c^2 d x^2}-\frac{2 a b c d x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\frac{2}{27} b^2 d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}-\frac{2 b^2 c d x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}}-\frac{2 b c d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{3} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{2 b d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{2 b d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (2 b^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (2 b^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\\ &=\frac{22}{9} b^2 d \sqrt{d+c^2 d x^2}-\frac{2 a b c d x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\frac{2}{27} b^2 d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}-\frac{2 b^2 c d x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}}-\frac{2 b c d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{3} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{2 b d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{2 b d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (2 b^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (2 b^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ &=\frac{22}{9} b^2 d \sqrt{d+c^2 d x^2}-\frac{2 a b c d x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\frac{2}{27} b^2 d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}-\frac{2 b^2 c d x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}}-\frac{2 b c d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{3} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{2 b d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{2 b d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{2 b^2 d \sqrt{d+c^2 d x^2} \text{Li}_3\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{2 b^2 d \sqrt{d+c^2 d x^2} \text{Li}_3\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 2.43784, size = 520, normalized size = 1.04 \[ \frac{2 a b d \sqrt{c^2 d x^2+d} \left (\text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-c x+\sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )}{\sqrt{c^2 x^2+1}}+\frac{b^2 d \sqrt{c^2 d x^2+d} \left (2 \sinh ^{-1}(c x) \left (\text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )\right )+2 \left (\text{PolyLog}\left (3,-e^{-\sinh ^{-1}(c x)}\right )-\text{PolyLog}\left (3,e^{-\sinh ^{-1}(c x)}\right )\right )+2 \sqrt{c^2 x^2+1}+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2-2 c x \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2 \left (\log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )\right )}{\sqrt{c^2 x^2+1}}-a^2 d^{3/2} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+d\right )+\frac{1}{3} a^2 d \left (c^2 x^2+4\right ) \sqrt{c^2 d x^2+d}+a^2 d^{3/2} \log (c x)-\frac{2 a b d \sqrt{c^2 d x^2+d} \left (c^3 x^3-3 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)+3 c x\right )}{9 \sqrt{c^2 x^2+1}}+\frac{b^2 d \sqrt{c^2 d x^2+d} \left (27 \sqrt{c^2 x^2+1} \left (\sinh ^{-1}(c x)^2+2\right )-6 \sinh ^{-1}(c x) \left (9 c x+\sinh \left (3 \sinh ^{-1}(c x)\right )\right )+\left (9 \sinh ^{-1}(c x)^2+2\right ) \cosh \left (3 \sinh ^{-1}(c x)\right )\right )}{108 \sqrt{c^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*d*(4 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/3 - (2*a*b*d*Sqrt[d + c^2*d*x^2]*(3*c*x + c^3*x^3 - 3*(1 + c^2*x^2)^
(3/2)*ArcSinh[c*x]))/(9*Sqrt[1 + c^2*x^2]) + a^2*d^(3/2)*Log[c*x] - a^2*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d + c^2*d
*x^2]] + (2*a*b*d*Sqrt[d + c^2*d*x^2]*(-(c*x) + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^(-ArcS
inh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh
[c*x])]))/Sqrt[1 + c^2*x^2] + (b^2*d*Sqrt[d + c^2*d*x^2]*(2*Sqrt[1 + c^2*x^2] - 2*c*x*ArcSinh[c*x] + Sqrt[1 +
c^2*x^2]*ArcSinh[c*x]^2 + ArcSinh[c*x]^2*(Log[1 - E^(-ArcSinh[c*x])] - Log[1 + E^(-ArcSinh[c*x])]) + 2*ArcSinh
[c*x]*(PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]) + 2*(PolyLog[3, -E^(-ArcSinh[c*x])] - P
olyLog[3, E^(-ArcSinh[c*x])])))/Sqrt[1 + c^2*x^2] + (b^2*d*Sqrt[d + c^2*d*x^2]*(27*Sqrt[1 + c^2*x^2]*(2 + ArcS
inh[c*x]^2) + (2 + 9*ArcSinh[c*x]^2)*Cosh[3*ArcSinh[c*x]] - 6*ArcSinh[c*x]*(9*c*x + Sinh[3*ArcSinh[c*x]])))/(1
08*Sqrt[1 + c^2*x^2])

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Maple [B]  time = 0.291, size = 1053, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10/3*a*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*arcsinh(c*x)*x^2*c^2+1/3*(c^2*d*x^2+d)^(3/2)*a^2+2/3*a*b*(d*(c^2*
x^2+1))^(1/2)*d/(c^2*x^2+1)*arcsinh(c*x)*x^4*c^4-2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(3,c*x+(
c^2*x^2+1)^(1/2))*d+2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(3,-c*x-(c^2*x^2+1)^(1/2))*d+4/3*b^2*
(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*arcsinh(c*x)^2-2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*
polylog(2,-c*x-(c^2*x^2+1)^(1/2))*d+8/3*a*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*arcsinh(c*x)-2*a*b*(d*(c^2*x^2
+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*d+2*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)
*polylog(2,c*x+(c^2*x^2+1)^(1/2))*d+2/27*b^2*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*c^4*x^4-b^2*(d*(c^2*x^2+1))^(
1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*ln(1+c*x+(c^2*x^2+1)^(1/2))*d+70/27*b^2*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2
+1)*c^2*x^2+2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*polylog(2,c*x+(c^2*x^2+1)^(1/2))*d+b^2*
(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*ln(1-c*x-(c^2*x^2+1)^(1/2))*d-a^2*d^(3/2)*ln((2*d+2*d^(
1/2)*(c^2*d*x^2+d)^(1/2))/x)+a^2*(c^2*d*x^2+d)^(1/2)*d+68/27*b^2*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)-2/9*b^2*(
d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*x^3*c^3-8/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)^(1/2
)*arcsinh(c*x)*x*c-2/9*a*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)^(1/2)*c^3*x^3-8/3*a*b*(d*(c^2*x^2+1))^(1/2)*d/(
c^2*x^2+1)^(1/2)*c*x+1/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*arcsinh(c*x)^2*x^4*c^4+5/3*b^2*(d*(c^2*x^2+1)
)^(1/2)*d/(c^2*x^2+1)*arcsinh(c*x)^2*x^2*c^2+2*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*ln(1-c
*x-(c^2*x^2+1)^(1/2))*d-2*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))
*d

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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((c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2/x,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{{\left (a^{2} c^{2} d x^{2} + a^{2} d +{\left (b^{2} c^{2} d x^{2} + b^{2} d\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} + a b d\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^2*c^2*d*x^2 + a^2*d + (b^2*c^2*d*x^2 + b^2*d)*arcsinh(c*x)^2 + 2*(a*b*c^2*d*x^2 + a*b*d)*arcsinh(c
*x))*sqrt(c^2*d*x^2 + d)/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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