\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 (d+c^2 d x^2)^2} \, dx\) [240]

Optimal result

Integrand size = 26, antiderivative size = 287 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {b c (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 c (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 c \arctan (c x)}{d^2}-\frac {4 b c (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {3 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {3 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {3 i b^2 c \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {5809, 5788, 5789, 4265, 2611, 2320, 6724, 5798, 209, 5811, 5816, 4267, 2317, 2438} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {3 c \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{d^2}-\frac {4 b c \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d^2}-\frac {b c (a+b \text {arcsinh}(c x))}{d^2 \sqrt {c^2 x^2+1}}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))^2}{2 d^2 \left (c^2 x^2+1\right )}-\frac {(a+b \text {arcsinh}(c x))^2}{d^2 x \left (c^2 x^2+1\right )}+\frac {3 i b c \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d^2}-\frac {3 i b c \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d^2}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {3 i b^2 c \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 c \arctan (c x)}{d^2} \]

[In]

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

[Out]

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 5788

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

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 5798

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

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

Mathematica [A] (verified)

Time = 7.06 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.91 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {a^2}{d^2 x}-\frac {a^2 c^2 x}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 a^2 c \arctan (c x)}{2 d^2}+\frac {2 a b c \left (\frac {\sqrt {1+c^2 x^2}+i \text {arcsinh}(c x)}{4 (-1-i c x)}-\frac {\text {arcsinh}(c x)}{c x}-\frac {i \sqrt {1+c^2 x^2}+\text {arcsinh}(c x)}{4 (i+c x)}-\text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+\frac {3}{4} i \left (-\frac {1}{2} \text {arcsinh}(c x)^2+2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )-\frac {3}{4} i \left (-\frac {1}{2} \text {arcsinh}(c x)^2+2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )\right )}{d^2}+\frac {b^2 c \left (-\frac {2 \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {c x \text {arcsinh}(c x)^2}{1+c^2 x^2}+4 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-\text {arcsinh}(c x)^2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+3 i \text {arcsinh}(c x)^2 \log \left (1-i e^{-\text {arcsinh}(c x)}\right )-3 i \text {arcsinh}(c x)^2 \log \left (1+i e^{-\text {arcsinh}(c x)}\right )-4 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+6 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-6 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )-4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )+6 i \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )-6 i \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )+\text {arcsinh}(c x)^2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{2 d^2} \]

[In]

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

[Out]

-(a^2/(d^2*x)) - (a^2*c^2*x)/(2*d^2*(1 + c^2*x^2)) - (3*a^2*c*ArcTan[c*x])/(2*d^2) + (2*a*b*c*((Sqrt[1 + c^2*x
^2] + I*ArcSinh[c*x])/(4*(-1 - I*c*x)) - ArcSinh[c*x]/(c*x) - (I*Sqrt[1 + c^2*x^2] + ArcSinh[c*x])/(4*(I + c*x
)) - ArcTanh[Sqrt[1 + c^2*x^2]] + ((3*I)/4)*(-1/2*ArcSinh[c*x]^2 + 2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] +
2*PolyLog[2, (-I)*E^ArcSinh[c*x]]) - ((3*I)/4)*(-1/2*ArcSinh[c*x]^2 + 2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]]
+ 2*PolyLog[2, I*E^ArcSinh[c*x]])))/d^2 + (b^2*c*((-2*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (c*x*ArcSinh[c*x]^2)/
(1 + c^2*x^2) + 4*ArcTan[Tanh[ArcSinh[c*x]/2]] - ArcSinh[c*x]^2*Coth[ArcSinh[c*x]/2] + 4*ArcSinh[c*x]*Log[1 -
E^(-ArcSinh[c*x])] + (3*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] - (3*I)*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh
[c*x]] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] + (6*I)*ArcSinh[c*x]*Pol
yLog[2, (-I)/E^ArcSinh[c*x]] - (6*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] - 4*PolyLog[2, E^(-ArcSinh[c*x]
)] + (6*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] - (6*I)*PolyLog[3, I/E^ArcSinh[c*x]] + ArcSinh[c*x]^2*Tanh[ArcSinh[
c*x]/2]))/(2*d^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

integrate((a+b*arcsinh(c*x))^2/x^2/(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^6 + 2*c^2*d^2*x^4 + d^2*x^2), x)

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx}{d^{2}} \]

[In]

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

[Out]

(Integral(a**2/(c**4*x**6 + 2*c**2*x**4 + x**2), x) + Integral(b**2*asinh(c*x)**2/(c**4*x**6 + 2*c**2*x**4 + x
**2), x) + Integral(2*a*b*asinh(c*x)/(c**4*x**6 + 2*c**2*x**4 + x**2), x))/d**2

Maxima [F]

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

[In]

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

[Out]

-1/2*a^2*((3*c^2*x^2 + 2)/(c^2*d^2*x^3 + d^2*x) + 3*c*arctan(c*x)/d^2) + integrate(b^2*log(c*x + sqrt(c^2*x^2
+ 1))^2/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1))/(c^4*d^2*x^6 + 2*c^2*d^2*
x^4 + d^2*x^2), x)

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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