3.3.21 \(\int \frac {(d+c^2 d x^2)^3 (a+b \sinh ^{-1}(c x))^2}{x} \, dx\) [221]

Optimal. Leaf size=337 \[ \frac {71}{144} b^2 c^2 d^3 x^2+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {19}{48} d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-b d^3 \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )-\frac {1}{2} b^2 d^3 \text {PolyLog}\left (3,e^{-2 \sinh ^{-1}(c x)}\right ) \]

[Out]

71/144*b^2*c^2*d^3*x^2+7/144*b^2*c^4*d^3*x^4+1/108*b^2*d^3*(c^2*x^2+1)^3-7/36*b*c*d^3*x*(c^2*x^2+1)^(3/2)*(a+b
*arcsinh(c*x))-1/18*b*c*d^3*x*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))-19/48*d^3*(a+b*arcsinh(c*x))^2+1/2*d^3*(c^2
*x^2+1)*(a+b*arcsinh(c*x))^2+1/4*d^3*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2+1/6*d^3*(c^2*x^2+1)^3*(a+b*arcsinh(c*x
))^2+1/3*d^3*(a+b*arcsinh(c*x))^3/b+d^3*(a+b*arcsinh(c*x))^2*ln(1-1/(c*x+(c^2*x^2+1)^(1/2))^2)-b*d^3*(a+b*arcs
inh(c*x))*polylog(2,1/(c*x+(c^2*x^2+1)^(1/2))^2)-1/2*b^2*d^3*polylog(3,1/(c*x+(c^2*x^2+1)^(1/2))^2)-19/24*b*c*
d^3*x*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.49, antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 13, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5808, 5775, 3797, 2221, 2611, 2320, 6724, 5785, 5783, 30, 5786, 14, 267} \begin {gather*} -\frac {1}{18} b c d^3 x \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {19}{24} b c d^3 x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-b d^3 \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}-\frac {19}{48} d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+d^3 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {71}{144} b^2 c^2 d^3 x^2+\frac {1}{108} b^2 d^3 \left (c^2 x^2+1\right )^3-\frac {1}{2} b^2 d^3 \text {Li}_3\left (e^{-2 \sinh ^{-1}(c x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(71*b^2*c^2*d^3*x^2)/144 + (7*b^2*c^4*d^3*x^4)/144 + (b^2*d^3*(1 + c^2*x^2)^3)/108 - (19*b*c*d^3*x*Sqrt[1 + c^
2*x^2]*(a + b*ArcSinh[c*x]))/24 - (7*b*c*d^3*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/36 - (b*c*d^3*x*(1 +
c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/18 - (19*d^3*(a + b*ArcSinh[c*x])^2)/48 + (d^3*(1 + c^2*x^2)*(a + b*ArcSi
nh[c*x])^2)/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)/
6 + (d^3*(a + b*ArcSinh[c*x])^3)/(3*b) + d^3*(a + b*ArcSinh[c*x])^2*Log[1 - E^(-2*ArcSinh[c*x])] - b*d^3*(a +
b*ArcSinh[c*x])*PolyLog[2, E^(-2*ArcSinh[c*x])] - (b^2*d^3*PolyLog[3, E^(-2*ArcSinh[c*x])])/2

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 30

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

Rule 267

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2320

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 2611

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 3797

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))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 5783

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

Rule 5785

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[(1/2)*Simp[Sqrt[d + e*x^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/2)*Simp[Sqrt[d + e*x^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 5786

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/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*A
rcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]

Rule 5808

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/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^
p/(1 + c^2*x^2)^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]

Rule 6724

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]

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} \, dx &=\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+d \int \frac {\left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx-\frac {1}{3} \left (b c d^3\right ) \int \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 \int \frac {\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx-\frac {1}{18} \left (5 b c d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac {1}{2} \left (b c d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {1}{18} \left (b^2 c^2 d^3\right ) \int x \left (1+c^2 x^2\right )^2 \, dx\\ &=\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+d^3 \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx-\frac {1}{24} \left (5 b c 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 c d^3\right ) \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\left (b c d^3\right ) \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {1}{72} \left (5 b^2 c^2 d^3\right ) \int x \left (1+c^2 x^2\right ) \, dx+\frac {1}{8} \left (b^2 c^2 d^3\right ) \int x \left (1+c^2 x^2\right ) \, dx\\ &=\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+d^3 \text {Subst}\left (\int (a+b x)^2 \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )-\frac {1}{48} \left (5 b c d^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{16} \left (3 b c d^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{2} \left (b c d^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{72} \left (5 b^2 c^2 d^3\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac {1}{48} \left (5 b^2 c^2 d^3\right ) \int x \, dx+\frac {1}{8} \left (b^2 c^2 d^3\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac {1}{16} \left (3 b^2 c^2 d^3\right ) \int x \, dx+\frac {1}{2} \left (b^2 c^2 d^3\right ) \int x \, dx\\ &=\frac {71}{144} b^2 c^2 d^3 x^2+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {19}{48} d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}-\left (2 d^3\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)^2}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac {71}{144} b^2 c^2 d^3 x^2+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {19}{48} d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\left (2 b d^3\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac {71}{144} b^2 c^2 d^3 x^2+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {19}{48} d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b d^3 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\left (b^2 d^3\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac {71}{144} b^2 c^2 d^3 x^2+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {19}{48} d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b 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 (b^2 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )\\ &=\frac {71}{144} b^2 c^2 d^3 x^2+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {19}{48} d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b d^3 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\frac {1}{2} b^2 d^3 \text {Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.46, size = 429, normalized size = 1.27 \begin {gather*} \frac {d^3 \left (5184 a^2 c^2 x^2+2592 a^2 c^4 x^4+576 a^2 c^6 x^6-3600 a b c x \sqrt {1+c^2 x^2}-1056 a b c^3 x^3 \sqrt {1+c^2 x^2}-192 a b c^5 x^5 \sqrt {1+c^2 x^2}+10368 a b c^2 x^2 \sinh ^{-1}(c x)+5184 a b c^4 x^4 \sinh ^{-1}(c x)+1152 a b c^6 x^6 \sinh ^{-1}(c x)+3456 a b \sinh ^{-1}(c x)^2-1152 b^2 \sinh ^{-1}(c x)^3+3600 a b \tanh ^{-1}\left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )+783 b^2 \cosh \left (2 \sinh ^{-1}(c x)\right )+1566 b^2 \sinh ^{-1}(c x)^2 \cosh \left (2 \sinh ^{-1}(c x)\right )+27 b^2 \cosh \left (4 \sinh ^{-1}(c x)\right )+216 b^2 \sinh ^{-1}(c x)^2 \cosh \left (4 \sinh ^{-1}(c x)\right )+b^2 \cosh \left (6 \sinh ^{-1}(c x)\right )+18 b^2 \sinh ^{-1}(c x)^2 \cosh \left (6 \sinh ^{-1}(c x)\right )+6912 a b \sinh ^{-1}(c x) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )+3456 b^2 \sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+3456 a^2 \log (c x)-3456 a b \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )+3456 b^2 \sinh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )-1728 b^2 \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )-1566 b^2 \sinh ^{-1}(c x) \sinh \left (2 \sinh ^{-1}(c x)\right )-108 b^2 \sinh ^{-1}(c x) \sinh \left (4 \sinh ^{-1}(c x)\right )-6 b^2 \sinh ^{-1}(c x) \sinh \left (6 \sinh ^{-1}(c x)\right )\right )}{3456} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*(5184*a^2*c^2*x^2 + 2592*a^2*c^4*x^4 + 576*a^2*c^6*x^6 - 3600*a*b*c*x*Sqrt[1 + c^2*x^2] - 1056*a*b*c^3*x^
3*Sqrt[1 + c^2*x^2] - 192*a*b*c^5*x^5*Sqrt[1 + c^2*x^2] + 10368*a*b*c^2*x^2*ArcSinh[c*x] + 5184*a*b*c^4*x^4*Ar
cSinh[c*x] + 1152*a*b*c^6*x^6*ArcSinh[c*x] + 3456*a*b*ArcSinh[c*x]^2 - 1152*b^2*ArcSinh[c*x]^3 + 3600*a*b*ArcT
anh[(c*x)/Sqrt[1 + c^2*x^2]] + 783*b^2*Cosh[2*ArcSinh[c*x]] + 1566*b^2*ArcSinh[c*x]^2*Cosh[2*ArcSinh[c*x]] + 2
7*b^2*Cosh[4*ArcSinh[c*x]] + 216*b^2*ArcSinh[c*x]^2*Cosh[4*ArcSinh[c*x]] + b^2*Cosh[6*ArcSinh[c*x]] + 18*b^2*A
rcSinh[c*x]^2*Cosh[6*ArcSinh[c*x]] + 6912*a*b*ArcSinh[c*x]*Log[1 - E^(-2*ArcSinh[c*x])] + 3456*b^2*ArcSinh[c*x
]^2*Log[1 - E^(2*ArcSinh[c*x])] + 3456*a^2*Log[c*x] - 3456*a*b*PolyLog[2, E^(-2*ArcSinh[c*x])] + 3456*b^2*ArcS
inh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] - 1728*b^2*PolyLog[3, E^(2*ArcSinh[c*x])] - 1566*b^2*ArcSinh[c*x]*Sinh
[2*ArcSinh[c*x]] - 108*b^2*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]] - 6*b^2*ArcSinh[c*x]*Sinh[6*ArcSinh[c*x]]))/3456

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(705\) vs. \(2(334)=668\).
time = 4.64, size = 706, normalized size = 2.09

method	result	size
derivativedivides	\(\frac {25 b^{2} c^{2} d^{3} x^{2}}{48}+\frac {11 b^{2} c^{4} d^{3} x^{4}}{144}+2 d^{3} a b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+d^{3} a^{2} \ln \left (c x \right )-\frac {11 d^{3} a b \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{36}-\frac {25 d^{3} a b c x \sqrt {c^{2} x^{2}+1}}{24}+\frac {3 d^{3} a b \arcsinh \left (c x \right ) c^{4} x^{4}}{2}+3 d^{3} a b \arcsinh \left (c x \right ) c^{2} x^{2}+2 d^{3} a b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} a b \sqrt {c^{2} x^{2}+1}\, c^{5} x^{5}}{18}+\frac {3 d^{3} a^{2} c^{4} x^{4}}{4}+\frac {3 d^{3} a^{2} c^{2} x^{2}}{2}+\frac {d^{3} a^{2} c^{6} x^{6}}{6}+\frac {b^{2} d^{3} c^{6} x^{6}}{108}+\frac {b^{2} d^{3} \arcsinh \left (c x \right )^{2} c^{6} x^{6}}{6}+\frac {3 b^{2} d^{3} \arcsinh \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {3 b^{2} d^{3} \arcsinh \left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {d^{3} a b \arcsinh \left (c x \right ) c^{6} x^{6}}{3}-\frac {b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{5} x^{5}}{18}-\frac {11 b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{36}-\frac {25 b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{24}+\frac {811 b^{2} d^{3}}{3456}+2 b^{2} d^{3} \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d^{3} \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {25 b^{2} d^{3} \arcsinh \left (c x \right )^{2}}{48}-2 b^{2} d^{3} \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {b^{2} d^{3} \arcsinh \left (c x \right )^{3}}{3}+b^{2} d^{3} \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {25 d^{3} a b \arcsinh \left (c x \right )}{24}+2 d^{3} a b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-d^{3} a b \arcsinh \left (c x \right )^{2}+2 d^{3} a b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d^{3} \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+b^{2} d^{3} \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )\)	\(706\)
default	\(\frac {25 b^{2} c^{2} d^{3} x^{2}}{48}+\frac {11 b^{2} c^{4} d^{3} x^{4}}{144}+2 d^{3} a b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+d^{3} a^{2} \ln \left (c x \right )-\frac {11 d^{3} a b \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{36}-\frac {25 d^{3} a b c x \sqrt {c^{2} x^{2}+1}}{24}+\frac {3 d^{3} a b \arcsinh \left (c x \right ) c^{4} x^{4}}{2}+3 d^{3} a b \arcsinh \left (c x \right ) c^{2} x^{2}+2 d^{3} a b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} a b \sqrt {c^{2} x^{2}+1}\, c^{5} x^{5}}{18}+\frac {3 d^{3} a^{2} c^{4} x^{4}}{4}+\frac {3 d^{3} a^{2} c^{2} x^{2}}{2}+\frac {d^{3} a^{2} c^{6} x^{6}}{6}+\frac {b^{2} d^{3} c^{6} x^{6}}{108}+\frac {b^{2} d^{3} \arcsinh \left (c x \right )^{2} c^{6} x^{6}}{6}+\frac {3 b^{2} d^{3} \arcsinh \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {3 b^{2} d^{3} \arcsinh \left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {d^{3} a b \arcsinh \left (c x \right ) c^{6} x^{6}}{3}-\frac {b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{5} x^{5}}{18}-\frac {11 b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{36}-\frac {25 b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{24}+\frac {811 b^{2} d^{3}}{3456}+2 b^{2} d^{3} \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d^{3} \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {25 b^{2} d^{3} \arcsinh \left (c x \right )^{2}}{48}-2 b^{2} d^{3} \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {b^{2} d^{3} \arcsinh \left (c x \right )^{3}}{3}+b^{2} d^{3} \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {25 d^{3} a b \arcsinh \left (c x \right )}{24}+2 d^{3} a b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-d^{3} a b \arcsinh \left (c x \right )^{2}+2 d^{3} a b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d^{3} \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+b^{2} d^{3} \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )\)	\(706\)

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,x,method=_RETURNVERBOSE)

[Out]

25/48*b^2*c^2*d^3*x^2+11/144*b^2*c^4*d^3*x^4+1/3*d^3*a*b*arcsinh(c*x)*c^6*x^6+d^3*a^2*ln(c*x)-11/36*d^3*a*b*(c
^2*x^2+1)^(1/2)*c^3*x^3-25/24*d^3*a*b*c*x*(c^2*x^2+1)^(1/2)-1/18*b^2*d^3*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^5*x^
5-11/36*b^2*d^3*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^3*x^3-25/24*b^2*d^3*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c*x-1/18*d
^3*a*b*(c^2*x^2+1)^(1/2)*c^5*x^5+3/4*d^3*a^2*c^4*x^4+3/2*d^3*a^2*c^2*x^2+1/6*d^3*a^2*c^6*x^6+25/24*d^3*a*b*arc
sinh(c*x)+2*d^3*a*b*polylog(2,-c*x-(c^2*x^2+1)^(1/2))-d^3*a*b*arcsinh(c*x)^2+2*d^3*a*b*polylog(2,c*x+(c^2*x^2+
1)^(1/2))+2*b^2*d^3*arcsinh(c*x)*polylog(2,c*x+(c^2*x^2+1)^(1/2))+b^2*d^3*arcsinh(c*x)^2*ln(1+c*x+(c^2*x^2+1)^
(1/2))+2*b^2*d^3*arcsinh(c*x)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+b^2*d^3*arcsinh(c*x)^2*ln(1-c*x-(c^2*x^2+1)^(1
/2))+1/108*b^2*d^3*c^6*x^6+3/2*d^3*a*b*arcsinh(c*x)*c^4*x^4+3*d^3*a*b*arcsinh(c*x)*c^2*x^2+2*d^3*a*b*arcsinh(c
*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))+2*d^3*a*b*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))+1/6*b^2*d^3*arcsinh(c*x)^2*
c^6*x^6+3/4*b^2*d^3*arcsinh(c*x)^2*c^4*x^4+3/2*b^2*d^3*arcsinh(c*x)^2*c^2*x^2+811/3456*b^2*d^3-2*b^2*d^3*polyl
og(3,-c*x-(c^2*x^2+1)^(1/2))+25/48*b^2*d^3*arcsinh(c*x)^2-2*b^2*d^3*polylog(3,c*x+(c^2*x^2+1)^(1/2))-1/3*b^2*d
^3*arcsinh(c*x)^3

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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,x, algorithm="maxima")

[Out]

1/6*a^2*c^6*d^3*x^6 + 3/4*a^2*c^4*d^3*x^4 + 3/2*a^2*c^2*d^3*x^2 + a^2*d^3*log(x) + integrate(b^2*c^6*d^3*x^5*l
og(c*x + sqrt(c^2*x^2 + 1))^2 + 2*a*b*c^6*d^3*x^5*log(c*x + sqrt(c^2*x^2 + 1)) + 3*b^2*c^4*d^3*x^3*log(c*x + s
qrt(c^2*x^2 + 1))^2 + 6*a*b*c^4*d^3*x^3*log(c*x + sqrt(c^2*x^2 + 1)) + 3*b^2*c^2*d^3*x*log(c*x + sqrt(c^2*x^2
+ 1))^2 + 6*a*b*c^2*d^3*x*log(c*x + sqrt(c^2*x^2 + 1)) + b^2*d^3*log(c*x + sqrt(c^2*x^2 + 1))^2/x + 2*a*b*d^3*
log(c*x + sqrt(c^2*x^2 + 1))/x, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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,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, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} d^{3} \left (\int \frac {a^{2}}{x}\, dx + \int 3 a^{2} c^{2} x\, dx + \int 3 a^{2} c^{4} x^{3}\, dx + \int a^{2} c^{6} x^{5}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 3 b^{2} c^{2} x \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 3 b^{2} c^{4} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{6} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{2} x \operatorname {asinh}{\left (c x \right )}\, dx + \int 6 a b c^{4} x^{3} \operatorname {asinh}{\left (c x \right )}\, dx + \int 2 a b c^{6} x^{5} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \end {gather*}

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,x)

[Out]

d**3*(Integral(a**2/x, x) + Integral(3*a**2*c**2*x, x) + Integral(3*a**2*c**4*x**3, x) + Integral(a**2*c**6*x*
*5, x) + Integral(b**2*asinh(c*x)**2/x, x) + Integral(2*a*b*asinh(c*x)/x, x) + Integral(3*b**2*c**2*x*asinh(c*
x)**2, x) + Integral(3*b**2*c**4*x**3*asinh(c*x)**2, x) + Integral(b**2*c**6*x**5*asinh(c*x)**2, x) + Integral
(6*a*b*c**2*x*asinh(c*x), x) + Integral(6*a*b*c**4*x**3*asinh(c*x), x) + Integral(2*a*b*c**6*x**5*asinh(c*x),
x))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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