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

Optimal. Leaf size=354 \[ -3 b c^2 d^3 \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{3}{2} b^2 c^2 d^3 \text{PolyLog}\left (3,e^{-2 \sinh ^{-1}(c x)}\right )+\frac{7}{8} b c^3 d^3 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{3}{16} b c^3 d^3 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{4} c^2 d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac{d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac{c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{b}-\frac{3}{32} c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+3 c^2 d^3 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{32} b^2 c^6 d^3 x^4+\frac{21}{32} b^2 c^4 d^3 x^2+b^2 c^2 d^3 \log (x) \]

[Out]

(21*b^2*c^4*d^3*x^2)/32 + (b^2*c^6*d^3*x^4)/32 - (3*b*c^3*d^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/16 + (
7*b*c^3*d^3*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/8 - (b*c*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))
/x - (3*c^2*d^3*(a + b*ArcSinh[c*x])^2)/32 + (3*c^2*d^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/2 + (3*c^2*d^3*(
1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/4 - (d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/(2*x^2) + (c^2*d^3*(a
+ b*ArcSinh[c*x])^3)/b + 3*c^2*d^3*(a + b*ArcSinh[c*x])^2*Log[1 - E^(-2*ArcSinh[c*x])] + b^2*c^2*d^3*Log[x] -
3*b*c^2*d^3*(a + b*ArcSinh[c*x])*PolyLog[2, E^(-2*ArcSinh[c*x])] - (3*b^2*c^2*d^3*PolyLog[3, E^(-2*ArcSinh[c*x
])])/2

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Rubi [A]  time = 0.749885, antiderivative size = 355, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 15, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.577, Rules used = {5739, 5744, 5659, 3716, 2190, 2531, 2282, 6589, 5682, 5675, 30, 5684, 14, 266, 43} \[ 3 b c^2 d^3 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{3}{2} b^2 c^2 d^3 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )+\frac{7}{8} b c^3 d^3 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{3}{16} b c^3 d^3 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{4} c^2 d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac{d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{b}-\frac{3}{32} c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+3 c^2 d^3 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{32} b^2 c^6 d^3 x^4+\frac{21}{32} b^2 c^4 d^3 x^2+b^2 c^2 d^3 \log (x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(21*b^2*c^4*d^3*x^2)/32 + (b^2*c^6*d^3*x^4)/32 - (3*b*c^3*d^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/16 + (
7*b*c^3*d^3*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/8 - (b*c*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))
/x - (3*c^2*d^3*(a + b*ArcSinh[c*x])^2)/32 + (3*c^2*d^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/2 + (3*c^2*d^3*(
1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/4 - (d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/(2*x^2) - (c^2*d^3*(a
+ b*ArcSinh[c*x])^3)/b + 3*c^2*d^3*(a + b*ArcSinh[c*x])^2*Log[1 - E^(2*ArcSinh[c*x])] + b^2*c^2*d^3*Log[x] + 3
*b*c^2*d^3*(a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])] - (3*b^2*c^2*d^3*PolyLog[3, E^(2*ArcSinh[c*x])]
)/2

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 5684

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

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]]

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])

Rubi steps

\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x^3} \, dx &=-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\left (3 c^2 d\right ) \int \frac{\left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx+\left (b c d^3\right ) \int \frac{\left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx\\ &=-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac{3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\left (3 c^2 d^2\right ) \int \frac{\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx+\left (b^2 c^2 d^3\right ) \int \frac{\left (1+c^2 x^2\right )^2}{x} \, dx-\frac{1}{2} \left (3 b c^3 d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\left (5 b c^3 d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=\frac{7}{8} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac{3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\left (3 c^2 d^3\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx+\frac{1}{2} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\left (1+c^2 x\right )^2}{x} \, dx,x,x^2\right )-\frac{1}{8} \left (9 b c^3 d^3\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\left (3 b c^3 d^3\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac{1}{4} \left (15 b c^3 d^3\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac{1}{8} \left (3 b^2 c^4 d^3\right ) \int x \left (1+c^2 x^2\right ) \, dx-\frac{1}{4} \left (5 b^2 c^4 d^3\right ) \int x \left (1+c^2 x^2\right ) \, dx\\ &=-\frac{3}{16} b c^3 d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{7}{8} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac{3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\left (3 c^2 d^3\right ) \operatorname{Subst}\left (\int (a+b x)^2 \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )+\frac{1}{2} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \left (2 c^2+\frac{1}{x}+c^4 x\right ) \, dx,x,x^2\right )-\frac{1}{16} \left (9 b c^3 d^3\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^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx+\frac{1}{8} \left (15 b c^3 d^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx+\frac{1}{8} \left (3 b^2 c^4 d^3\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac{1}{16} \left (9 b^2 c^4 d^3\right ) \int x \, dx-\frac{1}{4} \left (5 b^2 c^4 d^3\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac{1}{2} \left (3 b^2 c^4 d^3\right ) \int x \, dx-\frac{1}{8} \left (15 b^2 c^4 d^3\right ) \int x \, dx\\ &=\frac{21}{32} b^2 c^4 d^3 x^2+\frac{1}{32} b^2 c^6 d^3 x^4-\frac{3}{16} b c^3 d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{7}{8} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac{3}{32} c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{b}+b^2 c^2 d^3 \log (x)-\left (6 c^2 d^3\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{21}{32} b^2 c^4 d^3 x^2+\frac{1}{32} b^2 c^6 d^3 x^4-\frac{3}{16} b c^3 d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{7}{8} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac{3}{32} c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{b}+3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)-\left (6 b c^2 d^3\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac{21}{32} b^2 c^4 d^3 x^2+\frac{1}{32} b^2 c^6 d^3 x^4-\frac{3}{16} b c^3 d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{7}{8} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac{3}{32} c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{b}+3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)+3 b c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\left (3 b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac{21}{32} b^2 c^4 d^3 x^2+\frac{1}{32} b^2 c^6 d^3 x^4-\frac{3}{16} b c^3 d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{7}{8} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac{3}{32} c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{b}+3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)+3 b c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\frac{1}{2} \left (3 b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )\\ &=\frac{21}{32} b^2 c^4 d^3 x^2+\frac{1}{32} b^2 c^6 d^3 x^4-\frac{3}{16} b c^3 d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{7}{8} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac{3}{32} c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{b}+3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)+3 b c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\frac{3}{2} b^2 c^2 d^3 \text{Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 1.17451, size = 459, normalized size = 1.3 \[ \frac{1}{256} d^3 \left (-768 a b c^2 \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )+768 b^2 c^2 \sinh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )-384 b^2 c^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )+64 a^2 c^6 x^4+384 a^2 c^4 x^2+768 a^2 c^2 \log (x)-\frac{128 a^2}{x^2}-32 a b c^5 x^3 \sqrt{c^2 x^2+1}-336 a b c^3 x \sqrt{c^2 x^2+1}-\frac{256 a b c \sqrt{c^2 x^2+1}}{x}+128 a b c^6 x^4 \sinh ^{-1}(c x)+768 a b c^4 x^2 \sinh ^{-1}(c x)+768 a b c^2 \sinh ^{-1}(c x)^2+336 a b c^2 \sinh ^{-1}(c x)+1536 a b c^2 \sinh ^{-1}(c x) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-\frac{256 a b \sinh ^{-1}(c x)}{x^2}-\frac{256 b^2 c \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{x}+256 b^2 c^2 \log (c x)-256 b^2 c^2 \sinh ^{-1}(c x)^3-160 b^2 c^2 \sinh ^{-1}(c x) \sinh \left (2 \sinh ^{-1}(c x)\right )-4 b^2 c^2 \sinh ^{-1}(c x) \sinh \left (4 \sinh ^{-1}(c x)\right )+768 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+80 b^2 c^2 \cosh \left (2 \sinh ^{-1}(c x)\right )+160 b^2 c^2 \sinh ^{-1}(c x)^2 \cosh \left (2 \sinh ^{-1}(c x)\right )+b^2 c^2 \cosh \left (4 \sinh ^{-1}(c x)\right )+8 b^2 c^2 \sinh ^{-1}(c x)^2 \cosh \left (4 \sinh ^{-1}(c x)\right )-\frac{128 b^2 \sinh ^{-1}(c x)^2}{x^2}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^3*((-128*a^2)/x^2 + 384*a^2*c^4*x^2 + 64*a^2*c^6*x^4 - (256*a*b*c*Sqrt[1 + c^2*x^2])/x - 336*a*b*c^3*x*Sqrt
[1 + c^2*x^2] - 32*a*b*c^5*x^3*Sqrt[1 + c^2*x^2] + 336*a*b*c^2*ArcSinh[c*x] - (256*a*b*ArcSinh[c*x])/x^2 + 768
*a*b*c^4*x^2*ArcSinh[c*x] + 128*a*b*c^6*x^4*ArcSinh[c*x] - (256*b^2*c*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/x + 768*
a*b*c^2*ArcSinh[c*x]^2 - (128*b^2*ArcSinh[c*x]^2)/x^2 - 256*b^2*c^2*ArcSinh[c*x]^3 + 80*b^2*c^2*Cosh[2*ArcSinh
[c*x]] + 160*b^2*c^2*ArcSinh[c*x]^2*Cosh[2*ArcSinh[c*x]] + b^2*c^2*Cosh[4*ArcSinh[c*x]] + 8*b^2*c^2*ArcSinh[c*
x]^2*Cosh[4*ArcSinh[c*x]] + 1536*a*b*c^2*ArcSinh[c*x]*Log[1 - E^(-2*ArcSinh[c*x])] + 768*b^2*c^2*ArcSinh[c*x]^
2*Log[1 - E^(2*ArcSinh[c*x])] + 768*a^2*c^2*Log[x] + 256*b^2*c^2*Log[c*x] - 768*a*b*c^2*PolyLog[2, E^(-2*ArcSi
nh[c*x])] + 768*b^2*c^2*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] - 384*b^2*c^2*PolyLog[3, E^(2*ArcSinh[c*x]
)] - 160*b^2*c^2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]] - 4*b^2*c^2*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/256

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Maple [B]  time = 0.411, size = 838, normalized size = 2.4 \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*(a+b*arcsinh(c*x))^2/x^3,x)

[Out]

d^3*a*b*c^2-1/2*d^3*a^2/x^2+81/256*d^3*b^2*c^2+21/32*b^2*c^4*d^3*x^2+1/32*b^2*c^6*d^3*x^4+1/4*c^6*d^3*a^2*x^4+
3/2*c^4*d^3*a^2*x^2+c^2*d^3*b^2*ln(1+c*x+(c^2*x^2+1)^(1/2))+c^2*d^3*b^2*ln(c*x+(c^2*x^2+1)^(1/2)-1)-2*c^2*d^3*
b^2*ln(c*x+(c^2*x^2+1)^(1/2))+3*c^2*d^3*a^2*ln(c*x)+21/32*c^2*d^3*b^2*arcsinh(c*x)^2-6*c^2*d^3*b^2*polylog(3,c
*x+(c^2*x^2+1)^(1/2))-6*c^2*d^3*b^2*polylog(3,-c*x-(c^2*x^2+1)^(1/2))-c^2*d^3*b^2*arcsinh(c*x)^3+c^2*d^3*b^2*a
rcsinh(c*x)-1/2*d^3*b^2*arcsinh(c*x)^2/x^2+1/2*c^6*d^3*a*b*arcsinh(c*x)*x^4+3*c^4*d^3*a*b*arcsinh(c*x)*x^2-1/8
*c^5*d^3*a*b*x^3*(c^2*x^2+1)^(1/2)-21/16*c^3*d^3*a*b*x*(c^2*x^2+1)^(1/2)-c*d^3*a*b/x*(c^2*x^2+1)^(1/2)-c*d^3*b
^2*arcsinh(c*x)/x*(c^2*x^2+1)^(1/2)+6*c^2*d^3*a*b*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))+6*c^2*d^3*a*b*arcsi
nh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))-1/8*c^5*d^3*b^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^3-21/16*c^3*d^3*b^2*arcsi
nh(c*x)*(c^2*x^2+1)^(1/2)*x+6*c^2*d^3*a*b*polylog(2,c*x+(c^2*x^2+1)^(1/2))-d^3*a*b*arcsinh(c*x)/x^2+6*c^2*d^3*
a*b*polylog(2,-c*x-(c^2*x^2+1)^(1/2))-3*c^2*d^3*a*b*arcsinh(c*x)^2+21/16*c^2*d^3*a*b*arcsinh(c*x)+6*c^2*d^3*b^
2*arcsinh(c*x)*polylog(2,c*x+(c^2*x^2+1)^(1/2))+3*c^2*d^3*b^2*arcsinh(c*x)^2*ln(1-c*x-(c^2*x^2+1)^(1/2))+3*c^2
*d^3*b^2*arcsinh(c*x)^2*ln(1+c*x+(c^2*x^2+1)^(1/2))+6*c^2*d^3*b^2*arcsinh(c*x)*polylog(2,-c*x-(c^2*x^2+1)^(1/2
))+1/4*c^6*d^3*b^2*arcsinh(c*x)^2*x^4+3/2*c^4*d^3*b^2*arcsinh(c*x)^2*x^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, a^{2} c^{6} d^{3} x^{4} + \frac{3}{2} \, a^{2} c^{4} d^{3} x^{2} + 3 \, a^{2} c^{2} d^{3} \log \left (x\right ) - a b d^{3}{\left (\frac{\sqrt{c^{2} x^{2} + 1} c}{x} + \frac{\operatorname{arsinh}\left (c x\right )}{x^{2}}\right )} - \frac{a^{2} d^{3}}{2 \, x^{2}} + \int b^{2} c^{6} d^{3} x^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \, a b c^{6} d^{3} x^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 3 \, b^{2} c^{4} d^{3} x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 6 \, a b c^{4} d^{3} x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + \frac{3 \, b^{2} c^{2} d^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{x} + \frac{6 \, a b c^{2} d^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{x} + \frac{b^{2} d^{3} \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)^3*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="maxima")

[Out]

1/4*a^2*c^6*d^3*x^4 + 3/2*a^2*c^4*d^3*x^2 + 3*a^2*c^2*d^3*log(x) - a*b*d^3*(sqrt(c^2*x^2 + 1)*c/x + arcsinh(c*
x)/x^2) - 1/2*a^2*d^3/x^2 + integrate(b^2*c^6*d^3*x^3*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*a*b*c^6*d^3*x^3*log(c
*x + sqrt(c^2*x^2 + 1)) + 3*b^2*c^4*d^3*x*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6*a*b*c^4*d^3*x*log(c*x + sqrt(c^2*
x^2 + 1)) + 3*b^2*c^2*d^3*log(c*x + sqrt(c^2*x^2 + 1))^2/x + 6*a*b*c^2*d^3*log(c*x + sqrt(c^2*x^2 + 1))/x + b^
2*d^3*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^{6} d^{3} x^{6} + 3 \, a^{2} c^{4} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{2} + a^{2} d^{3} +{\left (b^{2} c^{6} d^{3} x^{6} + 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} + b^{2} d^{3}\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{6} d^{3} x^{6} + 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2} + a b d^{3}\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)^3*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="fricas")

[Out]

integral((a^2*c^6*d^3*x^6 + 3*a^2*c^4*d^3*x^4 + 3*a^2*c^2*d^3*x^2 + a^2*d^3 + (b^2*c^6*d^3*x^6 + 3*b^2*c^4*d^3
*x^4 + 3*b^2*c^2*d^3*x^2 + b^2*d^3)*arcsinh(c*x)^2 + 2*(a*b*c^6*d^3*x^6 + 3*a*b*c^4*d^3*x^4 + 3*a*b*c^2*d^3*x^
2 + a*b*d^3)*arcsinh(c*x))/x^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d^{3} \left (\int \frac{a^{2}}{x^{3}}\, dx + \int \frac{3 a^{2} c^{2}}{x}\, dx + \int 3 a^{2} c^{4} x\, dx + \int a^{2} c^{6} x^{3}\, 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{3 b^{2} c^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{x}\, dx + \int 3 b^{2} c^{4} x \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{6} x^{3} \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int \frac{6 a b c^{2} \operatorname{asinh}{\left (c x \right )}}{x}\, dx + \int 6 a b c^{4} x \operatorname{asinh}{\left (c x \right )}\, dx + \int 2 a b c^{6} x^{3} \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)**3*(a+b*asinh(c*x))**2/x**3,x)

[Out]

d**3*(Integral(a**2/x**3, x) + Integral(3*a**2*c**2/x, x) + Integral(3*a**2*c**4*x, x) + Integral(a**2*c**6*x*
*3, x) + Integral(b**2*asinh(c*x)**2/x**3, x) + Integral(2*a*b*asinh(c*x)/x**3, x) + Integral(3*b**2*c**2*asin
h(c*x)**2/x, x) + Integral(3*b**2*c**4*x*asinh(c*x)**2, x) + Integral(b**2*c**6*x**3*asinh(c*x)**2, x) + Integ
ral(6*a*b*c**2*asinh(c*x)/x, x) + Integral(6*a*b*c**4*x*asinh(c*x), x) + Integral(2*a*b*c**6*x**3*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 )}^{3}{\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)^3*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="giac")

[Out]

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