\(\int \frac {(d+c^2 d x^2)^2 (a+b \text {arcsinh}(c x))^2}{x} \, dx\) [212]

Optimal result

Integrand size = 26, antiderivative size = 257 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {13}{32} b^2 c^2 d^2 x^2+\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {11}{32} d^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {d^2 (a+b \text {arcsinh}(c x))^3}{3 b}+d^2 (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b d^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c x)}\right ) \]

[Out]

13/32*b^2*c^2*d^2*x^2+1/32*b^2*c^4*d^2*x^4-1/8*b*c*d^2*x*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))-11/32*d^2*(a+b*a
rcsinh(c*x))^2+1/2*d^2*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2+1/4*d^2*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2+1/3*d^2*(a+
b*arcsinh(c*x))^3/b+d^2*(a+b*arcsinh(c*x))^2*ln(1-1/(c*x+(c^2*x^2+1)^(1/2))^2)-b*d^2*(a+b*arcsinh(c*x))*polylo
g(2,1/(c*x+(c^2*x^2+1)^(1/2))^2)-1/2*b^2*d^2*polylog(3,1/(c*x+(c^2*x^2+1)^(1/2))^2)-11/16*b*c*d^2*x*(a+b*arcsi
nh(c*x))*(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5808, 5775, 3797, 2221, 2611, 2320, 6724, 5785, 5783, 30, 5786, 14} \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x} \, dx=-\frac {1}{8} b c d^2 x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {11}{16} b c d^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-b d^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+\frac {d^2 (a+b \text {arcsinh}(c x))^3}{3 b}-\frac {11}{32} d^2 (a+b \text {arcsinh}(c x))^2+d^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c x)}\right )+\frac {1}{32} b^2 c^4 d^2 x^4+\frac {13}{32} b^2 c^2 d^2 x^2 \]

[In]

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

[Out]

(13*b^2*c^2*d^2*x^2)/32 + (b^2*c^4*d^2*x^4)/32 - (11*b*c*d^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/16 - (b
*c*d^2*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/8 - (11*d^2*(a + b*ArcSinh[c*x])^2)/32 + (d^2*(1 + c^2*x^2)
*(a + b*ArcSinh[c*x])^2)/2 + (d^2*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/4 + (d^2*(a + b*ArcSinh[c*x])^3)/(3*
b) + d^2*(a + b*ArcSinh[c*x])^2*Log[1 - E^(-2*ArcSinh[c*x])] - b*d^2*(a + b*ArcSinh[c*x])*PolyLog[2, E^(-2*Arc
Sinh[c*x])] - (b^2*d^2*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 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*} \text {integral}& = \frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+d \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x} \, dx-\frac {1}{2} \left (b c d^2\right ) \int \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx \\ & = -\frac {1}{8} b c d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+d^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x} \, dx-\frac {1}{8} \left (3 b c d^2\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx-\left (b c d^2\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx+\frac {1}{8} \left (b^2 c^2 d^2\right ) \int x \left (1+c^2 x^2\right ) \, dx \\ & = -\frac {11}{16} b c d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {d^2 \text {Subst}\left (\int x^2 \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}-\frac {1}{16} \left (3 b c d^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{2} \left (b c d^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{8} \left (b^2 c^2 d^2\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac {1}{16} \left (3 b^2 c^2 d^2\right ) \int x \, dx+\frac {1}{2} \left (b^2 c^2 d^2\right ) \int x \, dx \\ & = \frac {13}{32} b^2 c^2 d^2 x^2+\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {11}{32} d^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {d^2 (a+b \text {arcsinh}(c x))^3}{3 b}+\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x^2}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b} \\ & = \frac {13}{32} b^2 c^2 d^2 x^2+\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {11}{32} d^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {d^2 (a+b \text {arcsinh}(c x))^3}{3 b}+d^2 (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-\left (2 d^2\right ) \text {Subst}\left (\int x \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right ) \\ & = \frac {13}{32} b^2 c^2 d^2 x^2+\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {11}{32} d^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {d^2 (a+b \text {arcsinh}(c x))^3}{3 b}+d^2 (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b d^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )+\left (b d^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right ) \\ & = \frac {13}{32} b^2 c^2 d^2 x^2+\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {11}{32} d^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {d^2 (a+b \text {arcsinh}(c x))^3}{3 b}+d^2 (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b d^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \left (b^2 d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ & = \frac {13}{32} b^2 c^2 d^2 x^2+\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {11}{32} d^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {d^2 (a+b \text {arcsinh}(c x))^3}{3 b}+d^2 (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b d^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.27 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {1}{768} d^2 \left (768 a^2 c^2 x^2+192 a^2 c^4 x^4-624 a b c x \sqrt {1+c^2 x^2}-96 a b c^3 x^3 \sqrt {1+c^2 x^2}+1536 a b c^2 x^2 \text {arcsinh}(c x)+384 a b c^4 x^4 \text {arcsinh}(c x)-768 a b \text {arcsinh}(c x)^2-256 b^2 \text {arcsinh}(c x)^3+144 b^2 \cosh (2 \text {arcsinh}(c x))+288 b^2 \text {arcsinh}(c x)^2 \cosh (2 \text {arcsinh}(c x))+3 b^2 \cosh (4 \text {arcsinh}(c x))+24 b^2 \text {arcsinh}(c x)^2 \cosh (4 \text {arcsinh}(c x))+1536 a b \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+768 b^2 \text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+768 a^2 \log (c x)-624 a b \log \left (-c x+\sqrt {1+c^2 x^2}\right )+768 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-384 b^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )-288 b^2 \text {arcsinh}(c x) \sinh (2 \text {arcsinh}(c x))-12 b^2 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))\right ) \]

[In]

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

[Out]

(d^2*(768*a^2*c^2*x^2 + 192*a^2*c^4*x^4 - 624*a*b*c*x*Sqrt[1 + c^2*x^2] - 96*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] + 1
536*a*b*c^2*x^2*ArcSinh[c*x] + 384*a*b*c^4*x^4*ArcSinh[c*x] - 768*a*b*ArcSinh[c*x]^2 - 256*b^2*ArcSinh[c*x]^3
+ 144*b^2*Cosh[2*ArcSinh[c*x]] + 288*b^2*ArcSinh[c*x]^2*Cosh[2*ArcSinh[c*x]] + 3*b^2*Cosh[4*ArcSinh[c*x]] + 24
*b^2*ArcSinh[c*x]^2*Cosh[4*ArcSinh[c*x]] + 1536*a*b*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + 768*b^2*ArcSinh
[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + 768*a^2*Log[c*x] - 624*a*b*Log[-(c*x) + Sqrt[1 + c^2*x^2]] + 768*b*(a +
b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])] - 384*b^2*PolyLog[3, E^(2*ArcSinh[c*x])] - 288*b^2*ArcSinh[c*x]
*Sinh[2*ArcSinh[c*x]] - 12*b^2*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/768

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(572\) vs. \(2(262)=524\).

Time = 0.31 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.23

method	result	size
parts	\(d^{2} a^{2} \left (\frac {c^{4} x^{4}}{4}+c^{2} x^{2}+\ln \left (x \right )\right )-\frac {d^{2} a b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{8}+\frac {13 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{32}-\frac {d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{3}}{3}-2 d^{2} b^{2} \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 d^{2} b^{2} \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {49 d^{2} b^{2}}{256}+\frac {13 b^{2} c^{2} d^{2} x^{2}}{32}+\frac {b^{2} c^{4} d^{2} x^{4}}{32}+d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-d^{2} a b \operatorname {arcsinh}\left (c x \right )^{2}+\frac {13 d^{2} a b \,\operatorname {arcsinh}\left (c x \right )}{16}+2 d^{2} a b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 d^{2} a b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {13 d^{2} a b c x \sqrt {c^{2} x^{2}+1}}{16}+\frac {d^{2} a b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{2}+2 d^{2} a b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-\frac {d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{8}-\frac {13 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{16}+\frac {d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{4}}{4}+d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{2} x^{2}+2 d^{2} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{2} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )\)	\(573\)
derivativedivides	\(-\frac {d^{2} a b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{8}+d^{2} a^{2} \left (\frac {c^{4} x^{4}}{4}+c^{2} x^{2}+\ln \left (c x \right )\right )+\frac {13 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{32}-\frac {d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{3}}{3}-2 d^{2} b^{2} \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 d^{2} b^{2} \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {49 d^{2} b^{2}}{256}+\frac {13 b^{2} c^{2} d^{2} x^{2}}{32}+\frac {b^{2} c^{4} d^{2} x^{4}}{32}+d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-d^{2} a b \operatorname {arcsinh}\left (c x \right )^{2}+\frac {13 d^{2} a b \,\operatorname {arcsinh}\left (c x \right )}{16}+2 d^{2} a b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 d^{2} a b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {13 d^{2} a b c x \sqrt {c^{2} x^{2}+1}}{16}+\frac {d^{2} a b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{2}+2 d^{2} a b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-\frac {d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{8}-\frac {13 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{16}+\frac {d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{4}}{4}+d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{2} x^{2}+2 d^{2} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{2} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )\)	\(575\)
default	\(-\frac {d^{2} a b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{8}+d^{2} a^{2} \left (\frac {c^{4} x^{4}}{4}+c^{2} x^{2}+\ln \left (c x \right )\right )+\frac {13 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{32}-\frac {d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{3}}{3}-2 d^{2} b^{2} \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 d^{2} b^{2} \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {49 d^{2} b^{2}}{256}+\frac {13 b^{2} c^{2} d^{2} x^{2}}{32}+\frac {b^{2} c^{4} d^{2} x^{4}}{32}+d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-d^{2} a b \operatorname {arcsinh}\left (c x \right )^{2}+\frac {13 d^{2} a b \,\operatorname {arcsinh}\left (c x \right )}{16}+2 d^{2} a b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 d^{2} a b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {13 d^{2} a b c x \sqrt {c^{2} x^{2}+1}}{16}+\frac {d^{2} a b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{2}+2 d^{2} a b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-\frac {d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{8}-\frac {13 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{16}+\frac {d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{4}}{4}+d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{2} x^{2}+2 d^{2} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{2} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )\)	\(575\)

[In]

int((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2/x,x,method=_RETURNVERBOSE)

[Out]

d^2*a^2*(1/4*c^4*x^4+c^2*x^2+ln(x))-1/8*d^2*a*b*c^3*x^3*(c^2*x^2+1)^(1/2)+13/32*d^2*b^2*arcsinh(c*x)^2-1/3*d^2
*b^2*arcsinh(c*x)^3-2*d^2*b^2*polylog(3,-c*x-(c^2*x^2+1)^(1/2))-2*d^2*b^2*polylog(3,c*x+(c^2*x^2+1)^(1/2))+49/
256*d^2*b^2+13/32*b^2*c^2*d^2*x^2+1/32*b^2*c^4*d^2*x^4+d^2*b^2*arcsinh(c*x)^2*ln(1-c*x-(c^2*x^2+1)^(1/2))+2*d^
2*b^2*arcsinh(c*x)*polylog(2,c*x+(c^2*x^2+1)^(1/2))+2*d^2*b^2*arcsinh(c*x)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+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)^2+13/16*d^2*a*b*arcsinh(c*x)+2*d^2*a*b*
polylog(2,-c*x-(c^2*x^2+1)^(1/2))+2*d^2*a*b*polylog(2,c*x+(c^2*x^2+1)^(1/2))-13/16*d^2*a*b*c*x*(c^2*x^2+1)^(1/
2)+1/2*d^2*a*b*arcsinh(c*x)*c^4*x^4+2*d^2*a*b*arcsinh(c*x)*c^2*x^2-1/8*d^2*b^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*
c^3*x^3-13/16*d^2*b^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c*x+1/4*d^2*b^2*arcsinh(c*x)^2*c^4*x^4+d^2*b^2*arcsinh(c*
x)^2*c^2*x^2+2*d^2*a*b*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))+2*d^2*a*b*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1
/2))

Fricas [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]

[In]

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

Sympy [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x} \, dx=d^{2} \left (\int \frac {a^{2}}{x}\, dx + \int 2 a^{2} c^{2} x\, dx + \int a^{2} c^{4} x^{3}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 2 b^{2} c^{2} x \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{4} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 4 a b c^{2} x \operatorname {asinh}{\left (c x \right )}\, dx + \int 2 a b c^{4} x^{3} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c^2*d*x^2+d)^2*(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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2}{x} \,d x \]

[In]

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

[Out]

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