\(\int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^3} \, dx\) [245]

Optimal result

Integrand size = 26, antiderivative size = 318 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b (a+b \text {arcsinh}(c x))}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {(a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \arctan (c x)}{6 c^3 d^3}-\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3} \]

[Out]

1/12*b^2*x/c^2/d^3/(c^2*x^2+1)-1/6*b*(a+b*arcsinh(c*x))/c^3/d^3/(c^2*x^2+1)^(3/2)-1/4*x*(a+b*arcsinh(c*x))^2/c
^2/d^3/(c^2*x^2+1)^2+1/8*x*(a+b*arcsinh(c*x))^2/c^2/d^3/(c^2*x^2+1)+1/4*(a+b*arcsinh(c*x))^2*arctan(c*x+(c^2*x
^2+1)^(1/2))/c^3/d^3-1/6*b^2*arctan(c*x)/c^3/d^3-1/4*I*b*(a+b*arcsinh(c*x))*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2
)))/c^3/d^3+1/4*I*b*(a+b*arcsinh(c*x))*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/c^3/d^3+1/4*I*b^2*polylog(3,-I*(c*
x+(c^2*x^2+1)^(1/2)))/c^3/d^3-1/4*I*b^2*polylog(3,I*(c*x+(c^2*x^2+1)^(1/2)))/c^3/d^3+1/4*b*(a+b*arcsinh(c*x))/
c^3/d^3/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5810, 5788, 5789, 4265, 2611, 2320, 6724, 5798, 209, 205} \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{4 c^3 d^3}-\frac {i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{4 c^3 d^3}+\frac {i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{4 c^3 d^3}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (c^2 x^2+1\right )}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}+\frac {b (a+b \text {arcsinh}(c x))}{4 c^3 d^3 \sqrt {c^2 x^2+1}}-\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (c^2 x^2+1\right )^{3/2}}+\frac {i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \arctan (c x)}{6 c^3 d^3}+\frac {b^2 x}{12 c^2 d^3 \left (c^2 x^2+1\right )} \]

[In]

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

[Out]

(b^2*x)/(12*c^2*d^3*(1 + c^2*x^2)) - (b*(a + b*ArcSinh[c*x]))/(6*c^3*d^3*(1 + c^2*x^2)^(3/2)) + (b*(a + b*ArcS
inh[c*x]))/(4*c^3*d^3*Sqrt[1 + c^2*x^2]) - (x*(a + b*ArcSinh[c*x])^2)/(4*c^2*d^3*(1 + c^2*x^2)^2) + (x*(a + b*
ArcSinh[c*x])^2)/(8*c^2*d^3*(1 + c^2*x^2)) + ((a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/(4*c^3*d^3) - (b^
2*ArcTan[c*x])/(6*c^3*d^3) - ((I/4)*b*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^3*d^3) + ((I/4)
*b*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/(c^3*d^3) + ((I/4)*b^2*PolyLog[3, (-I)*E^ArcSinh[c*x]])/
(c^3*d^3) - ((I/4)*b^2*PolyLog[3, I*E^ArcSinh[c*x]])/(c^3*d^3)

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 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 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 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 5810

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] + (-Dist[f^2*((m - 1)/(2*e*(p +
1))), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*f*(n/(2*c*(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}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[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*} \text {integral}& = -\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {b \int \frac {x (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{2 c d^3}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx}{4 c^2 d} \\ & = -\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {b^2 \int \frac {1}{\left (1+c^2 x^2\right )^2} \, dx}{6 c^2 d^3}-\frac {b \int \frac {x (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{4 c d^3}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx}{8 c^2 d^2} \\ & = \frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b (a+b \text {arcsinh}(c x))}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {\text {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\text {arcsinh}(c x)\right )}{8 c^3 d^3}+\frac {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{12 c^2 d^3}-\frac {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{4 c^2 d^3} \\ & = \frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b (a+b \text {arcsinh}(c x))}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {(a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \arctan (c x)}{6 c^3 d^3}-\frac {(i b) \text {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{4 c^3 d^3}+\frac {(i b) \text {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{4 c^3 d^3} \\ & = \frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b (a+b \text {arcsinh}(c x))}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {(a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \arctan (c x)}{6 c^3 d^3}-\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{4 c^3 d^3}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{4 c^3 d^3} \\ & = \frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b (a+b \text {arcsinh}(c x))}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {(a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \arctan (c x)}{6 c^3 d^3}-\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3} \\ & = \frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b (a+b \text {arcsinh}(c x))}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {(a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \arctan (c x)}{6 c^3 d^3}-\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.03 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.73 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {-\frac {6 a^2 c x}{\left (1+c^2 x^2\right )^2}+\frac {3 a^2 c x}{1+c^2 x^2}+\frac {a b \left ((2+i c x) \sqrt {1+c^2 x^2}+3 i \text {arcsinh}(c x)\right )}{(-i+c x)^2}+\frac {3 a b \left (-i \sqrt {1+c^2 x^2}+\text {arcsinh}(c x)\right )}{-i+c x}+\frac {3 a b \left (i \sqrt {1+c^2 x^2}+\text {arcsinh}(c x)\right )}{i+c x}-\frac {i a b \left ((2 i+c x) \sqrt {1+c^2 x^2}+3 \text {arcsinh}(c x)\right )}{(i+c x)^2}+3 a^2 \arctan (c x)+\frac {3}{2} i a b \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1+i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )-\frac {3}{2} i a b \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1-i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )+b^2 \left (\frac {2 c x}{1+c^2 x^2}-\frac {4 \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}}+\frac {6 \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {6 c x \text {arcsinh}(c x)^2}{\left (1+c^2 x^2\right )^2}+\frac {3 c x \text {arcsinh}(c x)^2}{1+c^2 x^2}-8 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-3 i \text {arcsinh}(c x)^2 \log \left (1-i e^{-\text {arcsinh}(c x)}\right )+3 i \text {arcsinh}(c x)^2 \log \left (1+i e^{-\text {arcsinh}(c x)}\right )-6 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )+6 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )-6 i \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )+6 i \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )\right )}{24 c^3 d^3} \]

[In]

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

[Out]

((-6*a^2*c*x)/(1 + c^2*x^2)^2 + (3*a^2*c*x)/(1 + c^2*x^2) + (a*b*((2 + I*c*x)*Sqrt[1 + c^2*x^2] + (3*I)*ArcSin
h[c*x]))/(-I + c*x)^2 + (3*a*b*((-I)*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(-I + c*x) + (3*a*b*(I*Sqrt[1 + c^2*x^
2] + ArcSinh[c*x]))/(I + c*x) - (I*a*b*((2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(I + c*x)^2 + 3*a^2*A
rcTan[c*x] + ((3*I)/2)*a*b*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 + I*E^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^Ar
cSinh[c*x]]) - ((3*I)/2)*a*b*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^ArcSinh[c*x]]) - 4*PolyLog[2, I*E^Arc
Sinh[c*x]]) + b^2*((2*c*x)/(1 + c^2*x^2) - (4*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) + (6*ArcSinh[c*x])/Sqrt[1 + c^
2*x^2] - (6*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2)^2 + (3*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2) - 8*ArcTan[Tanh[ArcSinh
[c*x]/2]] - (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]] -
(6*I)*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (6*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] - (6*I)*P
olyLog[3, (-I)/E^ArcSinh[c*x]] + (6*I)*PolyLog[3, I/E^ArcSinh[c*x]]))/(24*c^3*d^3)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b^2*x^2*arcsinh(c*x)^2 + 2*a*b*x^2*arcsinh(c*x) + a^2*x^2)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*
x^2 + d^3), x)

Sympy [F]

\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a^{2} x^{2}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{2} \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \]

[In]

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

[Out]

(Integral(a**2*x**2/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), x) + Integral(b**2*x**2*asinh(c*x)**2/(c**6*x
**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), x) + Integral(2*a*b*x**2*asinh(c*x)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x*
*2 + 1), x))/d**3

Maxima [F]

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

[In]

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

[Out]

1/8*a^2*((c^2*x^3 - x)/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3) + arctan(c*x)/(c^3*d^3)) + integrate(b^2*x^2*lo
g(c*x + sqrt(c^2*x^2 + 1))^2/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3) + 2*a*b*x^2*log(c*x + sqrt(c^
2*x^2 + 1))/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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