\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 (d+c^2 d x^2)^{3/2}} \, dx\) [308]

Optimal result

Integrand size = 28, antiderivative size = 573 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))^2}{2 d \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \sqrt {d+c^2 d x^2}}+\frac {4 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1+c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {2 i b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {2 i b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {3 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {3 b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5809, 5811, 5816, 4267, 2611, 2320, 6724, 5789, 4265, 2317, 2438, 272, 65, 214} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {4 b c^2 \sqrt {c^2 x^2+1} \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d \sqrt {c^2 d x^2+d}}+\frac {3 c^2 \sqrt {c^2 x^2+1} \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {3 b c^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d \sqrt {c^2 d x^2+d}}-\frac {3 b c^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d \sqrt {c^2 d x^2+d}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))^2}{2 d \sqrt {c^2 d x^2+d}}-\frac {b c \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{d x \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \sqrt {c^2 d x^2+d}}-\frac {2 i b^2 c^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b^2 c^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d \sqrt {c^2 d x^2+d}}-\frac {3 b^2 c^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{d \sqrt {c^2 d x^2+d}}+\frac {3 b^2 c^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {c^2 d x^2+d}}-\frac {b^2 c^2 \sqrt {c^2 x^2+1} \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )}{d \sqrt {c^2 d x^2+d}} \]

[In]

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

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[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 5789

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/(c*d), Subst[Int[(a +
b*x)^n*Sech[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 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 5816

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

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 {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \sqrt {d+c^2 d x^2}}-\frac {1}{2} \left (3 c^2\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx+\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (1+c^2 x^2\right )} \, dx}{d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))^2}{2 d \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \sqrt {d+c^2 d x^2}}-\frac {\left (3 c^2\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {d+c^2 d x^2}} \, dx}{2 d}+\frac {\left (b^2 c^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x \sqrt {1+c^2 x^2}} \, dx}{d \sqrt {d+c^2 d x^2}}-\frac {\left (b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{1+c^2 x^2} \, dx}{d \sqrt {d+c^2 d x^2}}+\frac {\left (3 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{1+c^2 x^2} \, dx}{d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))^2}{2 d \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \sqrt {d+c^2 d x^2}}-\frac {\left (3 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \text {csch}(x) \, dx,x,\text {arcsinh}(c x)\right )}{2 d \sqrt {d+c^2 d x^2}}-\frac {\left (b c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{d \sqrt {d+c^2 d x^2}}+\frac {\left (3 b c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{d \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{2 d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))^2}{2 d \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \sqrt {d+c^2 d x^2}}+\frac {4 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (3 b c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d \sqrt {d+c^2 d x^2}}-\frac {\left (3 b c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d \sqrt {d+c^2 d x^2}}-\frac {\left (i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d \sqrt {d+c^2 d x^2}}-\frac {\left (3 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (3 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))^2}{2 d \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \sqrt {d+c^2 d x^2}}+\frac {4 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1+c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {3 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {\left (i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {\left (3 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (3 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {\left (3 b^2 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (3 b^2 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))^2}{2 d \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \sqrt {d+c^2 d x^2}}+\frac {4 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1+c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {2 i b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {2 i b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {3 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {\left (3 b^2 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (3 b^2 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))^2}{2 d \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \sqrt {d+c^2 d x^2}}+\frac {4 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1+c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {2 i b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {2 i b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {3 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {3 b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.42 (sec) , antiderivative size = 884, normalized size of antiderivative = 1.54 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\sqrt {d \left (1+c^2 x^2\right )} \left (-\frac {a^2}{2 d^2 x^2}-\frac {a^2 c^2}{d^2 \left (1+c^2 x^2\right )}\right )-\frac {3 a^2 c^2 \log (x)}{2 d^{3/2}}+\frac {3 a^2 c^2 \log \left (d+\sqrt {d} \sqrt {d \left (1+c^2 x^2\right )}\right )}{2 d^{3/2}}+\frac {a b c^2 \left (-8 \text {arcsinh}(c x)+16 \sqrt {1+c^2 x^2} \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-2 \sqrt {1+c^2 x^2} \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\sqrt {1+c^2 x^2} \text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-12 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+12 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )-12 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+12 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-\sqrt {1+c^2 x^2} \text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+2 \sqrt {1+c^2 x^2} \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{4 d \sqrt {d \left (1+c^2 x^2\right )}}+\frac {b^2 c^2 \left (-8 \text {arcsinh}(c x)^2-4 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-12 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2 \log \left (1-e^{-\text {arcsinh}(c x)}\right )-16 i \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1-i e^{-\text {arcsinh}(c x)}\right )+16 i \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+12 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2 \log \left (1+e^{-\text {arcsinh}(c x)}\right )+8 \sqrt {1+c^2 x^2} \log \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-24 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-16 i \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )+16 i \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )+24 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-24 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c x)}\right )+24 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c x)}\right )-\sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2 \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{8 d \sqrt {d \left (1+c^2 x^2\right )}} \]

[In]

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

[Out]

Sqrt[d*(1 + c^2*x^2)]*(-1/2*a^2/(d^2*x^2) - (a^2*c^2)/(d^2*(1 + c^2*x^2))) - (3*a^2*c^2*Log[x])/(2*d^(3/2)) +
(3*a^2*c^2*Log[d + Sqrt[d]*Sqrt[d*(1 + c^2*x^2)]])/(2*d^(3/2)) + (a*b*c^2*(-8*ArcSinh[c*x] + 16*Sqrt[1 + c^2*x
^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] - 2*Sqrt[1 + c^2*x^2]*Coth[ArcSinh[c*x]/2] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*C
sch[ArcSinh[c*x]/2]^2 - 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 12*Sqrt[1 + c^2*x^2]*Ar
cSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 12*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-ArcSinh[c*x])] + 12*Sqrt[1 + c^2*
x^2]*PolyLog[2, E^(-ArcSinh[c*x])] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Sqrt[1 + c^2*x^
2]*Tanh[ArcSinh[c*x]/2]))/(4*d*Sqrt[d*(1 + c^2*x^2)]) + (b^2*c^2*(-8*ArcSinh[c*x]^2 - 4*Sqrt[1 + c^2*x^2]*ArcS
inh[c*x]*Coth[ArcSinh[c*x]/2] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^2 - 12*Sqrt[1 + c^2*x^2]
*ArcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c*x])] - (16*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] +
(16*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] + 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Log[1 +
E^(-ArcSinh[c*x])] + 8*Sqrt[1 + c^2*x^2]*Log[Tanh[ArcSinh[c*x]/2]] - 24*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*PolyLo
g[2, -E^(-ArcSinh[c*x])] - (16*I)*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (16*I)*Sqrt[1 + c^2*x^2]
*PolyLog[2, I/E^ArcSinh[c*x]] + 24*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] - 24*Sqrt[1 +
c^2*x^2]*PolyLog[3, -E^(-ArcSinh[c*x])] + 24*Sqrt[1 + c^2*x^2]*PolyLog[3, E^(-ArcSinh[c*x])] - Sqrt[1 + c^2*x^
2]*ArcSinh[c*x]^2*Sech[ArcSinh[c*x]/2]^2 + 4*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Tanh[ArcSinh[c*x]/2]))/(8*d*Sqrt[d
*(1 + c^2*x^2)])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(c^2*d*x^2 + d)*(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)

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{3} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral((a + b*asinh(c*x))**2/(x**3*(d*(c**2*x**2 + 1))**(3/2)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^3\,{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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