3.3.14 \(\int \frac {(d+c^2 d x^2)^2 (a+b \sinh ^{-1}(c x))^2}{x^3} \, dx\) [214]

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

[Out]

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

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Rubi [A]
time = 0.39, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5807, 5808, 5775, 3797, 2221, 2611, 2320, 6724, 5785, 5783, 30, 14} \begin {gather*} -2 b c^2 d^2 \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+c^2 d^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {b c d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {d^2 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac {2 c^2 d^2 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+\frac {1}{4} c^2 d^2 \left (a+b \sinh ^{-1}(c x)\right )^2+2 c^2 d^2 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} b c^3 d^2 x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} b^2 c^4 d^2 x^2-b^2 c^2 d^2 \text {Li}_3\left (e^{-2 \sinh ^{-1}(c x)}\right )+b^2 c^2 d^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.59, size = 319, normalized size = 1.17 \begin {gather*} \frac {1}{2} d^2 \left (-\frac {a^2}{x^2}+a^2 c^4 x^2-\frac {2 a b \left (c x \sqrt {1+c^2 x^2}+\sinh ^{-1}(c x)\right )}{x^2}+a b c^2 \left (-c x \sqrt {1+c^2 x^2}+2 c^2 x^2 \sinh ^{-1}(c x)+\tanh ^{-1}\left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )\right )+4 a^2 c^2 \log (x)-\frac {b^2 \left (2 c x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2-2 c^2 x^2 \log (c x)\right )}{x^2}+4 a b c^2 \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+2 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )-\text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )\right )-\frac {2}{3} b^2 c^2 \left (2 \sinh ^{-1}(c x)^2 \left (\sinh ^{-1}(c x)-3 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )\right )-6 \sinh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )+3 \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )\right )+\frac {1}{4} b^2 c^2 \left (\left (1+2 \sinh ^{-1}(c x)^2\right ) \cosh \left (2 \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(668\) vs. \(2(285)=570\).
time = 6.55, size = 669, normalized size = 2.46

method	result	size
derivativedivides	\(c^{2} \left (4 a \,d^{2} b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+a \,d^{2} b \arcsinh \left (c x \right ) c^{2} x^{2}-\frac {b^{2} d^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{2}+b^{2} d^{2} \arcsinh \left (c x \right )+\frac {a^{2} c^{2} d^{2} x^{2}}{2}+\frac {b^{2} d^{2}}{8}-\frac {b^{2} d^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{c x}-\frac {a \,d^{2} b \arcsinh \left (c x \right )}{c^{2} x^{2}}-\frac {a \,d^{2} b \sqrt {c^{2} x^{2}+1}}{c x}+2 a^{2} d^{2} \ln \left (c x \right )-\frac {a \,d^{2} b c x \sqrt {c^{2} x^{2}+1}}{2}+\frac {a \,d^{2} b \arcsinh \left (c x \right )}{2}+4 a \,d^{2} b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+4 a \,d^{2} b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d^{2} \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+4 b^{2} d^{2} \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+4 b^{2} d^{2} \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+a \,d^{2} b +\frac {b^{2} d^{2} \arcsinh \left (c x \right )^{2} c^{2} x^{2}}{2}+4 a \,d^{2} b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {b^{2} d^{2} \arcsinh \left (c x \right )^{2}}{2 c^{2} x^{2}}-4 b^{2} d^{2} \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {b^{2} d^{2} \arcsinh \left (c x \right )^{2}}{4}-\frac {2 b^{2} d^{2} \arcsinh \left (c x \right )^{3}}{3}-4 b^{2} d^{2} \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )+b^{2} d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )+b^{2} d^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {a^{2} d^{2}}{2 c^{2} x^{2}}-2 a \,d^{2} b \arcsinh \left (c x \right )^{2}+2 b^{2} d^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {b^{2} c^{2} d^{2} x^{2}}{4}\right )\)	\(669\)
default	\(c^{2} \left (4 a \,d^{2} b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+a \,d^{2} b \arcsinh \left (c x \right ) c^{2} x^{2}-\frac {b^{2} d^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{2}+b^{2} d^{2} \arcsinh \left (c x \right )+\frac {a^{2} c^{2} d^{2} x^{2}}{2}+\frac {b^{2} d^{2}}{8}-\frac {b^{2} d^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{c x}-\frac {a \,d^{2} b \arcsinh \left (c x \right )}{c^{2} x^{2}}-\frac {a \,d^{2} b \sqrt {c^{2} x^{2}+1}}{c x}+2 a^{2} d^{2} \ln \left (c x \right )-\frac {a \,d^{2} b c x \sqrt {c^{2} x^{2}+1}}{2}+\frac {a \,d^{2} b \arcsinh \left (c x \right )}{2}+4 a \,d^{2} b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+4 a \,d^{2} b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d^{2} \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+4 b^{2} d^{2} \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+4 b^{2} d^{2} \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+a \,d^{2} b +\frac {b^{2} d^{2} \arcsinh \left (c x \right )^{2} c^{2} x^{2}}{2}+4 a \,d^{2} b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {b^{2} d^{2} \arcsinh \left (c x \right )^{2}}{2 c^{2} x^{2}}-4 b^{2} d^{2} \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {b^{2} d^{2} \arcsinh \left (c x \right )^{2}}{4}-\frac {2 b^{2} d^{2} \arcsinh \left (c x \right )^{3}}{3}-4 b^{2} d^{2} \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )+b^{2} d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )+b^{2} d^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {a^{2} d^{2}}{2 c^{2} x^{2}}-2 a \,d^{2} b \arcsinh \left (c x \right )^{2}+2 b^{2} d^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {b^{2} c^{2} d^{2} x^{2}}{4}\right )\)	\(669\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2/x^3,x,method=_RETURNVERBOSE)

[Out]

c^2*(-b^2*d^2*arcsinh(c*x)/c/x*(c^2*x^2+1)^(1/2)+1/2*a*d^2*b*arcsinh(c*x)-2*a*d^2*b*arcsinh(c*x)^2+4*a*d^2*b*p
olylog(2,-c*x-(c^2*x^2+1)^(1/2))+4*a*d^2*b*polylog(2,c*x+(c^2*x^2+1)^(1/2))+2*b^2*d^2*arcsinh(c*x)^2*ln(1-c*x-
(c^2*x^2+1)^(1/2))+4*b^2*d^2*arcsinh(c*x)*polylog(2,c*x+(c^2*x^2+1)^(1/2))+2*b^2*d^2*arcsinh(c*x)^2*ln(1+c*x+(
c^2*x^2+1)^(1/2))+4*b^2*d^2*arcsinh(c*x)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+4*a*d^2*b*arcsinh(c*x)*ln(1+c*x+(c^
2*x^2+1)^(1/2))+4*a*d^2*b*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))+1/2*a^2*c^2*d^2*x^2+1/8*b^2*d^2+a*d^2*b*arc
sinh(c*x)*c^2*x^2-a*d^2*b/c/x*(c^2*x^2+1)^(1/2)-a*d^2*b*arcsinh(c*x)/c^2/x^2-4*b^2*d^2*polylog(3,-c*x-(c^2*x^2
+1)^(1/2))+1/4*b^2*d^2*arcsinh(c*x)^2-2/3*b^2*d^2*arcsinh(c*x)^3-4*b^2*d^2*polylog(3,c*x+(c^2*x^2+1)^(1/2))+2*
a^2*d^2*ln(c*x)-1/2*b^2*d^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c*x-1/2*a*d^2*b*c*x*(c^2*x^2+1)^(1/2)+a*d^2*b+1/2*b
^2*d^2*arcsinh(c*x)^2*c^2*x^2-1/2*b^2*d^2*arcsinh(c*x)^2/c^2/x^2+b^2*d^2*ln(c*x+(c^2*x^2+1)^(1/2)-1)+b^2*d^2*a
rcsinh(c*x)+b^2*d^2*ln(1+c*x+(c^2*x^2+1)^(1/2))-2*b^2*d^2*ln(c*x+(c^2*x^2+1)^(1/2))-1/2*a^2*d^2/c^2/x^2+1/4*b^
2*c^2*d^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)^2*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, 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)^2*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-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)^2*(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 )}^2}{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)^2)/x^3,x)

[Out]

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

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