3.3.41 \(\int \frac {(a+b \sinh ^{-1}(c x))^2}{x^3 (d+c^2 d x^2)^2} \, dx\) [241]

Optimal. Leaf size=253 \[ -\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2}}-\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {2 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {2 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {b^2 c^2 \text {PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 c^2 \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )}{d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.38, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 15, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {5809, 5811, 5799, 5569, 4267, 2611, 2320, 6724, 5787, 266, 277, 197, 5804, 457, 78} \begin {gather*} \frac {2 b c^2 \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac {2 b c^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (c^2 x^2+1\right )}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {c^2 x^2+1}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac {4 c^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2}-\frac {b^2 c^2 \text {Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 c^2 \text {Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {b^2 c^2 \log \left (c^2 x^2+1\right )}{2 d^2}+\frac {b^2 c^2 \log (x)}{d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

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 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 5787

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcSinh
[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Dist[b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Int[x*((a + b*ArcS
inh[c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

Rule 5799

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Dist[1/d, Subst[Int[(
a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n
, 0]

Rule 5804

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[
SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegerQ[p -
1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rule 5809

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 + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Dist[c^2*((m + 2*p + 3)/(f^2*
(m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]

Rule 5811

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 + 1)*((a + b*ArcSinh[c*x])^n/(2*d*f*(p + 1))), x] + (Dist[(m + 2*p + 3)/(2*d*(
p + 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Dist[b*c*(n/(2*f*(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] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] ||
IntegerQ[p] || EqQ[n, 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 (a+b \sinh ^{-1}(c x)\right )^2}{x^3 \left (d+c^2 d x^2\right )^2} \, dx &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\left (2 c^2\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )^2} \, dx+\frac {(b c) \int \frac {a+b \sinh ^{-1}(c x)}{x^2 \left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2}\\ &=-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2}}-\frac {2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {1+c^2 x^2}}-\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\frac {\left (b^2 c^2\right ) \int \frac {-1-2 c^2 x^2}{x \left (1+c^2 x^2\right )} \, dx}{d^2}+\frac {\left (2 b c^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2}-\frac {\left (2 c^2\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )} \, dx}{d}\\ &=-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2}}-\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\frac {\left (2 c^2\right ) \text {Subst}\left (\int (a+b x)^2 \text {csch}(x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}-\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {-1-2 c^2 x}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d^2}-\frac {\left (2 b^2 c^4\right ) \int \frac {x}{1+c^2 x^2} \, dx}{d^2}\\ &=-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2}}-\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\frac {b^2 c^2 \log \left (1+c^2 x^2\right )}{d^2}-\frac {\left (4 c^2\right ) \text {Subst}\left (\int (a+b x)^2 \text {csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}-\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \left (-\frac {1}{x}-\frac {c^2}{1+c^2 x}\right ) \, dx,x,x^2\right )}{2 d^2}\\ &=-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2}}-\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {\left (4 b c^2\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}-\frac {\left (4 b c^2\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2}}-\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {2 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {2 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {\left (2 b^2 c^2\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}+\frac {\left (2 b^2 c^2\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2}}-\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {2 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {2 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{d^2}\\ &=-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2}}-\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \left (1+c^2 x^2\right )}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {2 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {2 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {b^2 c^2 \text {Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 c^2 \text {Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(594\) vs. \(2(253)=506\).
time = 0.63, size = 594, normalized size = 2.35 \begin {gather*} \frac {-\frac {2 a^2}{x^2}-\frac {2 a b c}{x \sqrt {1+c^2 x^2}}+\frac {a^2}{x^2+c^2 x^4}+4 a^2 c^2 \sinh ^{-1}(c x)-\frac {4 a b \sinh ^{-1}(c x)}{x^2}-\frac {2 b^2 c \sinh ^{-1}(c x)}{x \sqrt {1+c^2 x^2}}+\frac {2 a b \sinh ^{-1}(c x)}{x^2+c^2 x^4}-\frac {2 b^2 \sinh ^{-1}(c x)^2}{x^2}+\frac {b^2 \sinh ^{-1}(c x)^2}{x^2+c^2 x^4}+8 a b c^2 \sinh ^{-1}(c x) \log \left (1+\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}\right )+4 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left (1+\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}\right )+8 a b c^2 \sinh ^{-1}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )+4 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left (1+\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )-4 a^2 c^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-8 a b c^2 \sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-4 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+2 b^2 c^2 \log (x)+2 a^2 c^2 \log \left (1+c^2 x^2\right )-b^2 c^2 \log \left (1+c^2 x^2\right )+8 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}\right )+8 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )-4 a b c^2 \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )-4 b^2 c^2 \sinh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )-8 b^2 c^2 \text {PolyLog}\left (3,\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}\right )-8 b^2 c^2 \text {PolyLog}\left (3,\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )+2 b^2 c^2 \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a^2)/x^2 - (2*a*b*c)/(x*Sqrt[1 + c^2*x^2]) + a^2/(x^2 + c^2*x^4) + 4*a^2*c^2*ArcSinh[c*x] - (4*a*b*ArcSin
h[c*x])/x^2 - (2*b^2*c*ArcSinh[c*x])/(x*Sqrt[1 + c^2*x^2]) + (2*a*b*ArcSinh[c*x])/(x^2 + c^2*x^4) - (2*b^2*Arc
Sinh[c*x]^2)/x^2 + (b^2*ArcSinh[c*x]^2)/(x^2 + c^2*x^4) + 8*a*b*c^2*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sq
rt[-c^2]] + 4*b^2*c^2*ArcSinh[c*x]^2*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 8*a*b*c^2*ArcSinh[c*x]*Log[1 + (
Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 4*b^2*c^2*ArcSinh[c*x]^2*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 4*a^2*c^2*Lo
g[1 - E^(2*ArcSinh[c*x])] - 8*a*b*c^2*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] - 4*b^2*c^2*ArcSinh[c*x]^2*Log[
1 - E^(2*ArcSinh[c*x])] + 2*b^2*c^2*Log[x] + 2*a^2*c^2*Log[1 + c^2*x^2] - b^2*c^2*Log[1 + c^2*x^2] + 8*b*c^2*(
a + b*ArcSinh[c*x])*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 8*b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[
-c^2]*E^ArcSinh[c*x])/c] - 4*a*b*c^2*PolyLog[2, E^(2*ArcSinh[c*x])] - 4*b^2*c^2*ArcSinh[c*x]*PolyLog[2, E^(2*A
rcSinh[c*x])] - 8*b^2*c^2*PolyLog[3, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 8*b^2*c^2*PolyLog[3, (Sqrt[-c^2]*E^ArcSi
nh[c*x])/c] + 2*b^2*c^2*PolyLog[3, E^(2*ArcSinh[c*x])])/(2*d^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(746\) vs. \(2(292)=584\).
time = 4.39, size = 747, normalized size = 2.95

method	result	size
derivativedivides	\(c^{2} \left (-\frac {a b}{d^{2} \sqrt {c^{2} x^{2}+1}\, c x}-\frac {2 a b \arcsinh \left (c x \right )}{d^{2} \left (c^{2} x^{2}+1\right )}-\frac {4 a b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {b^{2} \arcsinh \left (c x \right )}{d^{2} \sqrt {c^{2} x^{2}+1}\, c x}-\frac {b^{2} \arcsinh \left (c x \right )^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right ) c^{2} x^{2}}+\frac {b^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )}{d^{2}}+\frac {4 b^{2} \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 b^{2} \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {b^{2} \polylog \left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{d^{2}}-\frac {a^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}-\frac {a b \arcsinh \left (c x \right )}{d^{2} \left (c^{2} x^{2}+1\right ) c^{2} x^{2}}-\frac {a^{2}}{2 d^{2} c^{2} x^{2}}-\frac {2 b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 a b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 a b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 b^{2} \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 a b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {2 b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 a b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {2 a^{2} \ln \left (c x \right )}{d^{2}}-\frac {b^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {2 a b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {b^{2} \arcsinh \left (c x \right )^{2}}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {2 b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}\right )\)	\(747\)
default	\(c^{2} \left (-\frac {a b}{d^{2} \sqrt {c^{2} x^{2}+1}\, c x}-\frac {2 a b \arcsinh \left (c x \right )}{d^{2} \left (c^{2} x^{2}+1\right )}-\frac {4 a b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {b^{2} \arcsinh \left (c x \right )}{d^{2} \sqrt {c^{2} x^{2}+1}\, c x}-\frac {b^{2} \arcsinh \left (c x \right )^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right ) c^{2} x^{2}}+\frac {b^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )}{d^{2}}+\frac {4 b^{2} \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 b^{2} \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {b^{2} \polylog \left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{d^{2}}-\frac {a^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}-\frac {a b \arcsinh \left (c x \right )}{d^{2} \left (c^{2} x^{2}+1\right ) c^{2} x^{2}}-\frac {a^{2}}{2 d^{2} c^{2} x^{2}}-\frac {2 b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 a b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 a b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 b^{2} \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 a b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {2 b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 a b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {2 a^{2} \ln \left (c x \right )}{d^{2}}-\frac {b^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {2 a b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {b^{2} \arcsinh \left (c x \right )^{2}}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {2 b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}\right )\)	\(747\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d)^2,x, algorithm="maxima")

[Out]

1/2*a^2*(2*c^2*log(c^2*x^2 + 1)/d^2 - 4*c^2*log(x)/d^2 - (2*c^2*x^2 + 1)/(c^2*d^2*x^4 + d^2*x^2)) + integrate(
b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^4*d^2*x^7 + 2*c^2*d^2*x^5 + d^2*x^3) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1)
)/(c^4*d^2*x^7 + 2*c^2*d^2*x^5 + d^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((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d)^2,x, algorithm="fricas")

[Out]

integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(c^4*d^2*x^7 + 2*c^2*d^2*x^5 + d^2*x^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2}}{c^{4} x^{7} + 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{4} x^{7} + 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{7} + 2 c^{2} x^{5} + x^{3}}\, dx}{d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a**2/(c**4*x**7 + 2*c**2*x**5 + x**3), x) + Integral(b**2*asinh(c*x)**2/(c**4*x**7 + 2*c**2*x**5 + x
**3), x) + Integral(2*a*b*asinh(c*x)/(c**4*x**7 + 2*c**2*x**5 + x**3), x))/d**2

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Giac [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((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d)^2,x, algorithm="giac")

[Out]

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

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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}{x^3\,{\left (d\,c^2\,x^2+d\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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