3.3.23 \(\int \frac {(d+c^2 d x^2)^3 (a+b \sinh ^{-1}(c x))^2}{x^3} \, dx\) [223]

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

[Out]

21/32*b^2*c^4*d^3*x^2+1/32*b^2*c^6*d^3*x^4+7/8*b*c^3*d^3*x*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))-b*c*d^3*(c^2*x
^2+1)^(5/2)*(a+b*arcsinh(c*x))/x-3/32*c^2*d^3*(a+b*arcsinh(c*x))^2+3/2*c^2*d^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))^
2+3/4*c^2*d^3*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2-1/2*d^3*(c^2*x^2+1)^3*(a+b*arcsinh(c*x))^2/x^2+c^2*d^3*(a+b*a
rcsinh(c*x))^3/b+3*c^2*d^3*(a+b*arcsinh(c*x))^2*ln(1-1/(c*x+(c^2*x^2+1)^(1/2))^2)+b^2*c^2*d^3*ln(x)-3*b*c^2*d^
3*(a+b*arcsinh(c*x))*polylog(2,1/(c*x+(c^2*x^2+1)^(1/2))^2)-3/2*b^2*c^2*d^3*polylog(3,1/(c*x+(c^2*x^2+1)^(1/2)
)^2)-3/16*b*c^3*d^3*x*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.58, antiderivative size = 354, normalized size of antiderivative = 1.00, 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 = {5807, 5808, 5775, 3797, 2221, 2611, 2320, 6724, 5785, 5783, 30, 5786, 14, 272, 45} \begin {gather*} -3 b c^2 d^3 \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right ) \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 {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 {1}{32} b^2 c^6 d^3 x^4+\frac {21}{32} b^2 c^4 d^3 x^2-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (e^{-2 \sinh ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[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 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 45

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])

Rule 272

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 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 5807

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

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^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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.39, size = 512, normalized size = 1.45 \begin {gather*} \frac {d^3 \left (-128 a^2+384 a^2 c^4 x^4+64 a^2 c^6 x^6-256 a b c x \sqrt {1+c^2 x^2}-336 a b c^3 x^3 \sqrt {1+c^2 x^2}-32 a b c^5 x^5 \sqrt {1+c^2 x^2}-256 a b \sinh ^{-1}(c x)+768 a b c^4 x^4 \sinh ^{-1}(c x)+128 a b c^6 x^6 \sinh ^{-1}(c x)-256 b^2 c x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)-128 b^2 \sinh ^{-1}(c x)^2+768 a b c^2 x^2 \sinh ^{-1}(c x)^2-256 b^2 c^2 x^2 \sinh ^{-1}(c x)^3+336 a b c^2 x^2 \tanh ^{-1}\left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )+80 b^2 c^2 x^2 \cosh \left (2 \sinh ^{-1}(c x)\right )+160 b^2 c^2 x^2 \sinh ^{-1}(c x)^2 \cosh \left (2 \sinh ^{-1}(c x)\right )+b^2 c^2 x^2 \cosh \left (4 \sinh ^{-1}(c x)\right )+8 b^2 c^2 x^2 \sinh ^{-1}(c x)^2 \cosh \left (4 \sinh ^{-1}(c x)\right )+1536 a b c^2 x^2 \sinh ^{-1}(c x) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )+768 b^2 c^2 x^2 \sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+768 a^2 c^2 x^2 \log (x)+256 b^2 c^2 x^2 \log (c x)-768 a b c^2 x^2 \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )+768 b^2 c^2 x^2 \sinh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )-384 b^2 c^2 x^2 \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )-160 b^2 c^2 x^2 \sinh ^{-1}(c x) \sinh \left (2 \sinh ^{-1}(c x)\right )-4 b^2 c^2 x^2 \sinh ^{-1}(c x) \sinh \left (4 \sinh ^{-1}(c x)\right )\right )}{256 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(787\) vs. \(2(357)=714\).
time = 8.82, size = 788, normalized size = 2.23

method	result	size
derivativedivides	\(c^{2} \left (\frac {21 b^{2} c^{2} d^{3} x^{2}}{32}+\frac {b^{2} c^{4} d^{3} x^{4}}{32}+6 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 x}+d^{3} a b -\frac {d^{3} a b \arcsinh \left (c x \right )}{c^{2} x^{2}}-\frac {b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{c x}+3 d^{3} a^{2} \ln \left (c x \right )+b^{2} d^{3} \arcsinh \left (c x \right )-\frac {d^{3} a b \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{8}-\frac {21 d^{3} a b c x \sqrt {c^{2} x^{2}+1}}{16}+\frac {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}+6 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^{2} c^{4} x^{4}}{4}+\frac {3 d^{3} a^{2} c^{2} x^{2}}{2}+\frac {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 {b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{8}-\frac {21 b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{16}+\frac {81 b^{2} d^{3}}{256}+b^{2} d^{3} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+b^{2} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )-2 b^{2} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )-6 b^{2} d^{3} \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {21 b^{2} d^{3} \arcsinh \left (c x \right )^{2}}{32}-6 b^{2} d^{3} \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )-b^{2} d^{3} \arcsinh \left (c x \right )^{3}-\frac {d^{3} a^{2}}{2 c^{2} x^{2}}-\frac {b^{2} d^{3} \arcsinh \left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {21 d^{3} a b \arcsinh \left (c x \right )}{16}+6 d^{3} a b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-3 d^{3} a b \arcsinh \left (c x \right )^{2}+6 d^{3} a b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+6 b^{2} d^{3} \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+3 b^{2} d^{3} \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+6 b^{2} d^{3} \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+3 b^{2} d^{3} \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )\right )\)	\(788\)
default	\(c^{2} \left (\frac {21 b^{2} c^{2} d^{3} x^{2}}{32}+\frac {b^{2} c^{4} d^{3} x^{4}}{32}+6 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 x}+d^{3} a b -\frac {d^{3} a b \arcsinh \left (c x \right )}{c^{2} x^{2}}-\frac {b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{c x}+3 d^{3} a^{2} \ln \left (c x \right )+b^{2} d^{3} \arcsinh \left (c x \right )-\frac {d^{3} a b \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{8}-\frac {21 d^{3} a b c x \sqrt {c^{2} x^{2}+1}}{16}+\frac {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}+6 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^{2} c^{4} x^{4}}{4}+\frac {3 d^{3} a^{2} c^{2} x^{2}}{2}+\frac {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 {b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{8}-\frac {21 b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{16}+\frac {81 b^{2} d^{3}}{256}+b^{2} d^{3} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+b^{2} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )-2 b^{2} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )-6 b^{2} d^{3} \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {21 b^{2} d^{3} \arcsinh \left (c x \right )^{2}}{32}-6 b^{2} d^{3} \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )-b^{2} d^{3} \arcsinh \left (c x \right )^{3}-\frac {d^{3} a^{2}}{2 c^{2} x^{2}}-\frac {b^{2} d^{3} \arcsinh \left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {21 d^{3} a b \arcsinh \left (c x \right )}{16}+6 d^{3} a b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-3 d^{3} a b \arcsinh \left (c x \right )^{2}+6 d^{3} a b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+6 b^{2} d^{3} \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+3 b^{2} d^{3} \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+6 b^{2} d^{3} \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+3 b^{2} d^{3} \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )\right )\)	\(788\)

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,method=_RETURNVERBOSE)

[Out]

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

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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^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 = 0.00, 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^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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 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 {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**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(-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^3,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^3} \,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^3,x)

[Out]

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

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