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

Optimal. Leaf size=272 \[ -2 b c^2 d^2 \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-b^2 c^2 d^2 \text{PolyLog}\left (3,e^{-2 \sinh ^{-1}(c x)}\right )+\frac{1}{2} b c^3 d^2 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+c^2 d^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b c d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac{d^2 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac{2 c^2 d^2 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+\frac{1}{4} c^2 d^2 \left (a+b \sinh ^{-1}(c x)\right )^2+2 c^2 d^2 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} b^2 c^4 d^2 x^2+b^2 c^2 d^2 \log (x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.501433, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {5739, 5744, 5659, 3716, 2190, 2531, 2282, 6589, 5682, 5675, 30, 14} \[ 2 b c^2 d^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-b^2 c^2 d^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )+\frac{1}{2} b c^3 d^2 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+c^2 d^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b c d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac{d^2 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{2 c^2 d^2 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+\frac{1}{4} c^2 d^2 \left (a+b \sinh ^{-1}(c x)\right )^2+2 c^2 d^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} b^2 c^4 d^2 x^2+b^2 c^2 d^2 \log (x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 5739

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 + 1)), x] + (-Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x
)^(m + 2)*(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 + 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}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

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 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 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 5682

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

Rule 5675

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.879375, size = 305, normalized size = 1.12 \[ \frac{1}{2} d^2 \left (4 a b c^2 \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+2 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )-\text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )\right )-\frac{2}{3} b^2 c^2 \left (-6 \sinh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )+3 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )+2 \sinh ^{-1}(c x)^2 \left (\sinh ^{-1}(c x)-3 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )\right )\right )+a^2 c^4 x^2+4 a^2 c^2 \log (x)-\frac{a^2}{x^2}+a b c^2 \left (\left (2 c^2 x^2+1\right ) \sinh ^{-1}(c x)-c x \sqrt{c^2 x^2+1}\right )-\frac{2 a b \left (c x \sqrt{c^2 x^2+1}+\sinh ^{-1}(c x)\right )}{x^2}-\frac{b^2 \left (-2 c^2 x^2 \log (c x)+2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2\right )}{x^2}+\frac{1}{4} b^2 c^2 \left (\left (2 \sinh ^{-1}(c x)^2+1\right ) \cosh \left (2 \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^2*(-(a^2/x^2) + a^2*c^4*x^2 - (2*a*b*(c*x*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/x^2 + a*b*c^2*(-(c*x*Sqrt[1 +
c^2*x^2]) + (1 + 2*c^2*x^2)*ArcSinh[c*x]) + 4*a^2*c^2*Log[x] - (b^2*(2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + Ar
cSinh[c*x]^2 - 2*c^2*x^2*Log[c*x]))/x^2 + 4*a*b*c^2*(ArcSinh[c*x]*(ArcSinh[c*x] + 2*Log[1 - E^(-2*ArcSinh[c*x]
)]) - PolyLog[2, E^(-2*ArcSinh[c*x])]) - (2*b^2*c^2*(2*ArcSinh[c*x]^2*(ArcSinh[c*x] - 3*Log[1 - E^(2*ArcSinh[c
*x])]) - 6*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] + 3*PolyLog[3, E^(2*ArcSinh[c*x])]))/3 + (b^2*c^2*((1 +
2*ArcSinh[c*x]^2)*Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]]))/4))/2

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Maple [B]  time = 0.342, size = 719, normalized size = 2.6 \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)^2*(a+b*arcsinh(c*x))^2/x^3,x)

[Out]

d^2*a*b*c^2+4*c^2*d^2*b^2*arcsinh(c*x)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+2*c^2*d^2*b^2*arcsinh(c*x)^2*ln(1-c*x
-(c^2*x^2+1)^(1/2))-d^2*a*b*arcsinh(c*x)/x^2-2*c^2*d^2*a*b*arcsinh(c*x)^2+1/2*c^2*d^2*a*b*arcsinh(c*x)+4*c^2*d
^2*b^2*arcsinh(c*x)*polylog(2,c*x+(c^2*x^2+1)^(1/2))+1/4*b^2*c^4*d^2*x^2-1/2*d^2*a^2/x^2+1/8*d^2*b^2*c^2-c*d^2
*b^2*arcsinh(c*x)/x*(c^2*x^2+1)^(1/2)-1/2*c^3*d^2*a*b*x*(c^2*x^2+1)^(1/2)+c^4*d^2*a*b*arcsinh(c*x)*x^2-1/2*c^3
*d^2*b^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x+4*c^2*d^2*a*b*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))+4*c^2*d^2*a*b
*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))-c*d^2*a*b/x*(c^2*x^2+1)^(1/2)-2*c^2*d^2*b^2*ln(c*x+(c^2*x^2+1)^(1/2)
)+c^2*d^2*b^2*ln(1+c*x+(c^2*x^2+1)^(1/2))+c^2*d^2*b^2*ln(c*x+(c^2*x^2+1)^(1/2)-1)+2*c^2*d^2*a^2*ln(c*x)+1/4*c^
2*d^2*b^2*arcsinh(c*x)^2-2/3*c^2*d^2*b^2*arcsinh(c*x)^3-4*c^2*d^2*b^2*polylog(3,c*x+(c^2*x^2+1)^(1/2))-1/2*d^2
*b^2*arcsinh(c*x)^2/x^2+1/2*c^4*d^2*a^2*x^2-4*c^2*d^2*b^2*polylog(3,-c*x-(c^2*x^2+1)^(1/2))+c^2*d^2*b^2*arcsin
h(c*x)+4*c^2*d^2*a*b*polylog(2,c*x+(c^2*x^2+1)^(1/2))+4*c^2*d^2*a*b*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+1/2*c^4*
d^2*b^2*arcsinh(c*x)^2*x^2+2*c^2*d^2*b^2*arcsinh(c*x)^2*ln(1+c*x+(c^2*x^2+1)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**2*(Integral(a**2/x**3, x) + Integral(2*a**2*c**2/x, x) + Integral(a**2*c**4*x, x) + Integral(b**2*asinh(c*x
)**2/x**3, x) + Integral(2*a*b*asinh(c*x)/x**3, x) + Integral(2*b**2*c**2*asinh(c*x)**2/x, x) + Integral(b**2*
c**4*x*asinh(c*x)**2, x) + Integral(4*a*b*c**2*asinh(c*x)/x, x) + Integral(2*a*b*c**4*x*asinh(c*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 )}^{2}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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