\(\int \frac {(d+c^2 d x^2)^2 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx\) [215]

Optimal result

Integrand size = 26, antiderivative size = 248 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d^2}{3 x}+2 b^2 c^4 d^2 x-\frac {5}{3} b c^3 d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {8}{3} c^4 d^2 x (a+b \text {arcsinh}(c x))^2-\frac {4 c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {22}{3} b c^3 d^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {11}{3} b^2 c^3 d^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {11}{3} b^2 c^3 d^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \]

[Out]

-1/3*b^2*c^2*d^2/x+2*b^2*c^4*d^2*x-1/3*b*c*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/x^2+8/3*c^4*d^2*x*(a+b*arc
sinh(c*x))^2-4/3*c^2*d^2*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/x-1/3*d^2*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2/x^3-22/
3*b*c^3*d^2*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2))-11/3*b^2*c^3*d^2*polylog(2,-c*x-(c^2*x^2+1)^(1/2
))+11/3*b^2*c^3*d^2*polylog(2,c*x+(c^2*x^2+1)^(1/2))-5/3*b*c^3*d^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5807, 5772, 5798, 8, 5806, 5816, 4267, 2317, 2438, 14} \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {22}{3} b c^3 d^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+\frac {8}{3} c^4 d^2 x (a+b \text {arcsinh}(c x))^2-\frac {4 c^2 d^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {b c d^2 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {5}{3} b c^3 d^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {11}{3} b^2 c^3 d^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {11}{3} b^2 c^3 d^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )+2 b^2 c^4 d^2 x-\frac {b^2 c^2 d^2}{3 x} \]

[In]

Int[((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2)/x^4,x]

[Out]

-1/3*(b^2*c^2*d^2)/x + 2*b^2*c^4*d^2*x - (5*b*c^3*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/3 - (b*c*d^2*(1
+ c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*x^2) + (8*c^4*d^2*x*(a + b*ArcSinh[c*x])^2)/3 - (4*c^2*d^2*(1 + c^2*
x^2)*(a + b*ArcSinh[c*x])^2)/(3*x) - (d^2*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(3*x^3) - (22*b*c^3*d^2*(a +
b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/3 - (11*b^2*c^3*d^2*PolyLog[2, -E^ArcSinh[c*x]])/3 + (11*b^2*c^3*d^2
*PolyLog[2, E^ArcSinh[c*x]])/3

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 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 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 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 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[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 5806

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

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 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]

Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {1}{3} \left (4 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^2} \, dx+\frac {1}{3} \left (2 b c d^2\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx \\ & = -\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {4 c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {1}{3} \left (b^2 c^2 d^2\right ) \int \frac {1+c^2 x^2}{x^2} \, dx+\left (b c^3 d^2\right ) \int \frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx+\frac {1}{3} \left (8 b c^3 d^2\right ) \int \frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx+\frac {1}{3} \left (8 c^4 d^2\right ) \int (a+b \text {arcsinh}(c x))^2 \, dx \\ & = \frac {11}{3} b c^3 d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {8}{3} c^4 d^2 x (a+b \text {arcsinh}(c x))^2-\frac {4 c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {1}{3} \left (b^2 c^2 d^2\right ) \int \left (c^2+\frac {1}{x^2}\right ) \, dx+\left (b c^3 d^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx+\frac {1}{3} \left (8 b c^3 d^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx-\left (b^2 c^4 d^2\right ) \int 1 \, dx-\frac {1}{3} \left (8 b^2 c^4 d^2\right ) \int 1 \, dx-\frac {1}{3} \left (16 b c^5 d^2\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {b^2 c^2 d^2}{3 x}-\frac {10}{3} b^2 c^4 d^2 x-\frac {5}{3} b c^3 d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {8}{3} c^4 d^2 x (a+b \text {arcsinh}(c x))^2-\frac {4 c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\left (b c^3 d^2\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))+\frac {1}{3} \left (8 b c^3 d^2\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))+\frac {1}{3} \left (16 b^2 c^4 d^2\right ) \int 1 \, dx \\ & = -\frac {b^2 c^2 d^2}{3 x}+2 b^2 c^4 d^2 x-\frac {5}{3} b c^3 d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {8}{3} c^4 d^2 x (a+b \text {arcsinh}(c x))^2-\frac {4 c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {22}{3} b c^3 d^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\left (b^2 c^3 d^2\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )+\left (b^2 c^3 d^2\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )-\frac {1}{3} \left (8 b^2 c^3 d^2\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )+\frac {1}{3} \left (8 b^2 c^3 d^2\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right ) \\ & = -\frac {b^2 c^2 d^2}{3 x}+2 b^2 c^4 d^2 x-\frac {5}{3} b c^3 d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {8}{3} c^4 d^2 x (a+b \text {arcsinh}(c x))^2-\frac {4 c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {22}{3} b c^3 d^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\left (b^2 c^3 d^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )+\left (b^2 c^3 d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )-\frac {1}{3} \left (8 b^2 c^3 d^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )+\frac {1}{3} \left (8 b^2 c^3 d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right ) \\ & = -\frac {b^2 c^2 d^2}{3 x}+2 b^2 c^4 d^2 x-\frac {5}{3} b c^3 d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {8}{3} c^4 d^2 x (a+b \text {arcsinh}(c x))^2-\frac {4 c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {22}{3} b c^3 d^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {11}{3} b^2 c^3 d^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {11}{3} b^2 c^3 d^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.44 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\frac {d^2 \left (-a^2-6 a^2 c^2 x^2-b^2 c^2 x^2+3 a^2 c^4 x^4+6 b^2 c^4 x^4-a b c x \sqrt {1+c^2 x^2}-6 a b c^3 x^3 \sqrt {1+c^2 x^2}-2 a b \text {arcsinh}(c x)-12 a b c^2 x^2 \text {arcsinh}(c x)+6 a b c^4 x^4 \text {arcsinh}(c x)-b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-6 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-b^2 \text {arcsinh}(c x)^2-6 b^2 c^2 x^2 \text {arcsinh}(c x)^2+3 b^2 c^4 x^4 \text {arcsinh}(c x)^2-11 a b c^3 x^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+11 b^2 c^3 x^3 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-11 b^2 c^3 x^3 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+11 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-11 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{3 x^3} \]

[In]

Integrate[((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2)/x^4,x]

[Out]

(d^2*(-a^2 - 6*a^2*c^2*x^2 - b^2*c^2*x^2 + 3*a^2*c^4*x^4 + 6*b^2*c^4*x^4 - a*b*c*x*Sqrt[1 + c^2*x^2] - 6*a*b*c
^3*x^3*Sqrt[1 + c^2*x^2] - 2*a*b*ArcSinh[c*x] - 12*a*b*c^2*x^2*ArcSinh[c*x] + 6*a*b*c^4*x^4*ArcSinh[c*x] - b^2
*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 6*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - b^2*ArcSinh[c*x]^2 - 6*b^
2*c^2*x^2*ArcSinh[c*x]^2 + 3*b^2*c^4*x^4*ArcSinh[c*x]^2 - 11*a*b*c^3*x^3*ArcTanh[Sqrt[1 + c^2*x^2]] + 11*b^2*c
^3*x^3*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 11*b^2*c^3*x^3*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 11*b
^2*c^3*x^3*PolyLog[2, -E^(-ArcSinh[c*x])] - 11*b^2*c^3*x^3*PolyLog[2, E^(-ArcSinh[c*x])]))/(3*x^3)

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.46

method	result	size
derivativedivides	\(c^{3} \left (d^{2} a^{2} \left (c x -\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}\right )+d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c x -2 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 d^{2} b^{2} c x -\frac {2 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{c x}-\frac {d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 c^{2} x^{2}}-\frac {d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {d^{2} b^{2}}{3 c x}-\frac {11 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {11 d^{2} b^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {11 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {11 d^{2} b^{2} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}+2 d^{2} a b \left (\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(363\)
default	\(c^{3} \left (d^{2} a^{2} \left (c x -\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}\right )+d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c x -2 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 d^{2} b^{2} c x -\frac {2 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{c x}-\frac {d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 c^{2} x^{2}}-\frac {d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {d^{2} b^{2}}{3 c x}-\frac {11 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {11 d^{2} b^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {11 d^{2} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {11 d^{2} b^{2} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}+2 d^{2} a b \left (\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(363\)
parts	\(d^{2} a^{2} \left (c^{4} x -\frac {2 c^{2}}{x}-\frac {1}{3 x^{3}}\right )+d^{2} b^{2} c^{4} \operatorname {arcsinh}\left (c x \right )^{2} x -2 d^{2} b^{2} c^{3} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )+2 b^{2} c^{4} d^{2} x -\frac {2 d^{2} b^{2} c^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{x}-\frac {d^{2} b^{2} c \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 x^{2}}-\frac {b^{2} c^{2} d^{2}}{3 x}-\frac {d^{2} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{3 x^{3}}-\frac {11 d^{2} b^{2} c^{3} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {11 b^{2} c^{3} d^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {11 d^{2} b^{2} c^{3} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {11 b^{2} c^{3} d^{2} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}+2 d^{2} a b \,c^{3} \left (\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\)	\(375\)

[In]

int((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2/x^4,x,method=_RETURNVERBOSE)

[Out]

c^3*(d^2*a^2*(c*x-1/3/c^3/x^3-2/c/x)+d^2*b^2*arcsinh(c*x)^2*c*x-2*d^2*b^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*d^2
*b^2*c*x-2*d^2*b^2*arcsinh(c*x)^2/c/x-1/3*d^2*b^2/c^2/x^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)-1/3*d^2*b^2/c^3/x^3*a
rcsinh(c*x)^2-1/3*d^2*b^2/c/x-11/3*d^2*b^2*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))-11/3*d^2*b^2*polylog(2,-c*
x-(c^2*x^2+1)^(1/2))+11/3*d^2*b^2*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))+11/3*d^2*b^2*polylog(2,c*x+(c^2*x^2
+1)^(1/2))+2*d^2*a*b*(arcsinh(c*x)*c*x-1/3*arcsinh(c*x)/c^3/x^3-2*arcsinh(c*x)/c/x-(c^2*x^2+1)^(1/2)-1/6/c^2/x
^2*(c^2*x^2+1)^(1/2)-11/6*arctanh(1/(c^2*x^2+1)^(1/2))))

Fricas [F]

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

[In]

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

Sympy [F]

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

[In]

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

[Out]

d**2*(Integral(a**2*c**4, x) + Integral(a**2/x**4, x) + Integral(2*a**2*c**2/x**2, x) + Integral(b**2*c**4*asi
nh(c*x)**2, x) + Integral(b**2*asinh(c*x)**2/x**4, x) + Integral(2*a*b*c**4*asinh(c*x), x) + Integral(2*a*b*as
inh(c*x)/x**4, x) + Integral(2*b**2*c**2*asinh(c*x)**2/x**2, x) + Integral(4*a*b*c**2*asinh(c*x)/x**2, x))

Maxima [F]

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

[In]

integrate((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="maxima")

[Out]

b^2*c^4*d^2*x*arcsinh(c*x)^2 + 2*b^2*c^4*d^2*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + a^2*c^4*d^2*x + 2*(c*x*a
rcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*c^3*d^2 - 4*(c*arcsinh(1/(c*abs(x))) + arcsinh(c*x)/x)*a*b*c^2*d^2 + 1/3*
((c^2*arcsinh(1/(c*abs(x))) - sqrt(c^2*x^2 + 1)/x^2)*c - 2*arcsinh(c*x)/x^3)*a*b*d^2 - 2*a^2*c^2*d^2/x - 1/3*a
^2*d^2/x^3 - 1/3*(6*b^2*c^2*d^2*x^2 + b^2*d^2)*log(c*x + sqrt(c^2*x^2 + 1))^2/x^3 + integrate(2/3*(6*b^2*c^5*d
^2*x^4 + 7*b^2*c^3*d^2*x^2 + b^2*c*d^2 + (6*b^2*c^4*d^2*x^3 + b^2*c^2*d^2*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt
(c^2*x^2 + 1))/(c^3*x^6 + c*x^4 + (c^2*x^5 + x^3)*sqrt(c^2*x^2 + 1)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

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

Mupad [F(-1)]

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

[In]

int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^2)/x^4,x)

[Out]

int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^2)/x^4, x)