\(\int \frac {(d+c^2 d x^2)^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx\) [224]

Optimal result

Integrand size = 26, antiderivative size = 326 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d^3}{3 x}+\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {34}{3} b c^3 d^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {17}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {17}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \]

[Out]

-1/3*b^2*c^2*d^3/x+50/9*b^2*c^4*d^3*x+2/27*b^2*c^6*d^3*x^3+1/9*b*c^3*d^3*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))-
1/3*b*c*d^3*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))/x^2+16/3*c^4*d^3*x*(a+b*arcsinh(c*x))^2+8/3*c^4*d^3*x*(c^2*x^
2+1)*(a+b*arcsinh(c*x))^2-2*c^2*d^3*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2/x-1/3*d^3*(c^2*x^2+1)^3*(a+b*arcsinh(c*
x))^2/x^3-34/3*b*c^3*d^3*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2))-17/3*b^2*c^3*d^3*polylog(2,-c*x-(c^
2*x^2+1)^(1/2))+17/3*b^2*c^3*d^3*polylog(2,c*x+(c^2*x^2+1)^(1/2))-5*b*c^3*d^3*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(
1/2)

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5807, 5786, 5772, 5798, 8, 5808, 5806, 5816, 4267, 2317, 2438, 276} \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {34}{3} b c^3 d^3 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {b c d^3 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {8}{3} c^4 d^3 x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{9} b c^3 d^3 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-5 b c^3 d^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {17}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {17}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )+\frac {2}{27} b^2 c^6 d^3 x^3+\frac {50}{9} b^2 c^4 d^3 x-\frac {b^2 c^2 d^3}{3 x} \]

[In]

Int[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^4,x]

[Out]

-1/3*(b^2*c^2*d^3)/x + (50*b^2*c^4*d^3*x)/9 + (2*b^2*c^6*d^3*x^3)/27 - 5*b*c^3*d^3*Sqrt[1 + c^2*x^2]*(a + b*Ar
cSinh[c*x]) + (b*c^3*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/9 - (b*c*d^3*(1 + c^2*x^2)^(5/2)*(a + b*Arc
Sinh[c*x]))/(3*x^2) + (16*c^4*d^3*x*(a + b*ArcSinh[c*x])^2)/3 + (8*c^4*d^3*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]
)^2)/3 - (2*c^2*d^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/x - (d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/(
3*x^3) - (34*b*c^3*d^3*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/3 - (17*b^2*c^3*d^3*PolyLog[2, -E^ArcSinh
[c*x]])/3 + (17*b^2*c^3*d^3*PolyLog[2, E^ArcSinh[c*x]])/3

Rule 8

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

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 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 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 5786

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*(
(a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Dist[2*d*(p/(2*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*A
rcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 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 5808

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 + 2*p + 1))), x] + (Dist[2*d*(p/(m + 2*p + 1)), Int
[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 2*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] && 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^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\left (2 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^2} \, dx+\frac {1}{3} \left (2 b c d^3\right ) \int \frac {\left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx \\ & = -\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\left (8 c^4 d^2\right ) \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx+\frac {1}{3} \left (b^2 c^2 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^2}{x^2} \, dx+\frac {1}{3} \left (5 b c^3 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx+\left (4 b c^3 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx \\ & = \frac {17}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {1}{3} \left (b^2 c^2 d^3\right ) \int \left (2 c^2+\frac {1}{x^2}+c^4 x^2\right ) \, dx+\frac {1}{3} \left (5 b c^3 d^3\right ) \int \frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx+\left (4 b c^3 d^3\right ) \int \frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx+\frac {1}{3} \left (16 c^4 d^3\right ) \int (a+b \text {arcsinh}(c x))^2 \, dx-\frac {1}{9} \left (5 b^2 c^4 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac {1}{3} \left (4 b^2 c^4 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac {1}{3} \left (16 b c^5 d^3\right ) \int x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx \\ & = -\frac {b^2 c^2 d^3}{3 x}-\frac {11}{9} b^2 c^4 d^3 x-\frac {14}{27} b^2 c^6 d^3 x^3+\frac {17}{3} b c^3 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {1}{3} \left (5 b c^3 d^3\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx+\left (4 b c^3 d^3\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx-\frac {1}{3} \left (5 b^2 c^4 d^3\right ) \int 1 \, dx+\frac {1}{9} \left (16 b^2 c^4 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\left (4 b^2 c^4 d^3\right ) \int 1 \, dx-\frac {1}{3} \left (32 b c^5 d^3\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {b^2 c^2 d^3}{3 x}-\frac {46}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {1}{3} \left (5 b c^3 d^3\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))+\left (4 b c^3 d^3\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))+\frac {1}{3} \left (32 b^2 c^4 d^3\right ) \int 1 \, dx \\ & = -\frac {b^2 c^2 d^3}{3 x}+\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {34}{3} b c^3 d^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {1}{3} \left (5 b^2 c^3 d^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )+\frac {1}{3} \left (5 b^2 c^3 d^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )-\left (4 b^2 c^3 d^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )+\left (4 b^2 c^3 d^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right ) \\ & = -\frac {b^2 c^2 d^3}{3 x}+\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {34}{3} b c^3 d^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {1}{3} \left (5 b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )+\frac {1}{3} \left (5 b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )-\left (4 b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )+\left (4 b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right ) \\ & = -\frac {b^2 c^2 d^3}{3 x}+\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {34}{3} b c^3 d^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {17}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {17}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.41 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\frac {d^3 \left (-9 a^2-81 a^2 c^2 x^2-9 b^2 c^2 x^2+81 a^2 c^4 x^4+150 b^2 c^4 x^4+9 a^2 c^6 x^6+2 b^2 c^6 x^6-9 a b c x \sqrt {1+c^2 x^2}-150 a b c^3 x^3 \sqrt {1+c^2 x^2}-6 a b c^5 x^5 \sqrt {1+c^2 x^2}-18 a b \text {arcsinh}(c x)-162 a b c^2 x^2 \text {arcsinh}(c x)+162 a b c^4 x^4 \text {arcsinh}(c x)+18 a b c^6 x^6 \text {arcsinh}(c x)-9 b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-150 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-6 b^2 c^5 x^5 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-9 b^2 \text {arcsinh}(c x)^2-81 b^2 c^2 x^2 \text {arcsinh}(c x)^2+81 b^2 c^4 x^4 \text {arcsinh}(c x)^2+9 b^2 c^6 x^6 \text {arcsinh}(c x)^2-153 a b c^3 x^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+153 b^2 c^3 x^3 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-153 b^2 c^3 x^3 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+153 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-153 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{27 x^3} \]

[In]

Integrate[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^4,x]

[Out]

(d^3*(-9*a^2 - 81*a^2*c^2*x^2 - 9*b^2*c^2*x^2 + 81*a^2*c^4*x^4 + 150*b^2*c^4*x^4 + 9*a^2*c^6*x^6 + 2*b^2*c^6*x
^6 - 9*a*b*c*x*Sqrt[1 + c^2*x^2] - 150*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 6*a*b*c^5*x^5*Sqrt[1 + c^2*x^2] - 18*a*
b*ArcSinh[c*x] - 162*a*b*c^2*x^2*ArcSinh[c*x] + 162*a*b*c^4*x^4*ArcSinh[c*x] + 18*a*b*c^6*x^6*ArcSinh[c*x] - 9
*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 150*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 6*b^2*c^5*x^5*Sqrt[
1 + c^2*x^2]*ArcSinh[c*x] - 9*b^2*ArcSinh[c*x]^2 - 81*b^2*c^2*x^2*ArcSinh[c*x]^2 + 81*b^2*c^4*x^4*ArcSinh[c*x]
^2 + 9*b^2*c^6*x^6*ArcSinh[c*x]^2 - 153*a*b*c^3*x^3*ArcTanh[Sqrt[1 + c^2*x^2]] + 153*b^2*c^3*x^3*ArcSinh[c*x]*
Log[1 - E^(-ArcSinh[c*x])] - 153*b^2*c^3*x^3*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 153*b^2*c^3*x^3*PolyLog
[2, -E^(-ArcSinh[c*x])] - 153*b^2*c^3*x^3*PolyLog[2, E^(-ArcSinh[c*x])]))/(27*x^3)

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.44

method	result	size
derivativedivides	\(c^{3} \left (d^{3} a^{2} \left (\frac {c^{3} x^{3}}{3}+3 c x -\frac {1}{3 c^{3} x^{3}}-\frac {3}{c x}\right )-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 c^{2} x^{2}}-\frac {2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}}{9}+\frac {2 d^{3} b^{2} c^{3} x^{3}}{27}-\frac {d^{3} b^{2}}{3 c x}+\frac {50 d^{3} b^{2} c x}{9}-\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{c x}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{3} x^{3}}{3}+3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c x -\frac {50 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{9}+\frac {17 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {17 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {17 d^{3} b^{2} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {17 d^{3} b^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+3 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(468\)
default	\(c^{3} \left (d^{3} a^{2} \left (\frac {c^{3} x^{3}}{3}+3 c x -\frac {1}{3 c^{3} x^{3}}-\frac {3}{c x}\right )-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 c^{2} x^{2}}-\frac {2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}}{9}+\frac {2 d^{3} b^{2} c^{3} x^{3}}{27}-\frac {d^{3} b^{2}}{3 c x}+\frac {50 d^{3} b^{2} c x}{9}-\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{c x}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{3} x^{3}}{3}+3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c x -\frac {50 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{9}+\frac {17 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {17 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {17 d^{3} b^{2} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {17 d^{3} b^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+3 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(468\)
parts	\(d^{3} a^{2} \left (\frac {c^{6} x^{3}}{3}+3 c^{4} x -\frac {3 c^{2}}{x}-\frac {1}{3 x^{3}}\right )-\frac {d^{3} b^{2} c \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 x^{2}}-\frac {2 d^{3} b^{2} c^{5} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2}}{9}-\frac {17 b^{2} c^{3} d^{3} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {17 b^{2} c^{3} d^{3} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {b^{2} c^{2} d^{3}}{3 x}-\frac {17 d^{3} b^{2} c^{3} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {2 b^{2} c^{6} d^{3} x^{3}}{27}+\frac {17 d^{3} b^{2} c^{3} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {50 d^{3} b^{2} c^{3} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{9}+\frac {50 b^{2} c^{4} d^{3} x}{9}-\frac {3 d^{3} b^{2} c^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{x}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{3 x^{3}}+\frac {d^{3} b^{2} c^{6} \operatorname {arcsinh}\left (c x \right )^{2} x^{3}}{3}+3 d^{3} b^{2} c^{4} \operatorname {arcsinh}\left (c x \right )^{2} x +2 d^{3} a b \,c^{3} \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+3 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\)	\(480\)

[In]

int((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2/x^4,x,method=_RETURNVERBOSE)

[Out]

c^3*(d^3*a^2*(1/3*c^3*x^3+3*c*x-1/3/c^3/x^3-3/c/x)-1/3*d^3*b^2/c^2/x^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)-2/9*d^3*
b^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^2*x^2+2/27*d^3*b^2*c^3*x^3-1/3*d^3*b^2/c/x+50/9*d^3*b^2*c*x-3*d^3*b^2*arc
sinh(c*x)^2/c/x-1/3*d^3*b^2/c^3/x^3*arcsinh(c*x)^2+1/3*d^3*b^2*arcsinh(c*x)^2*c^3*x^3+3*d^3*b^2*arcsinh(c*x)^2
*c*x-50/9*d^3*b^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+17/3*d^3*b^2*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))-17/3*d^
3*b^2*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))+17/3*d^3*b^2*polylog(2,c*x+(c^2*x^2+1)^(1/2))-17/3*d^3*b^2*poly
log(2,-c*x-(c^2*x^2+1)^(1/2))+2*d^3*a*b*(1/3*arcsinh(c*x)*c^3*x^3+3*arcsinh(c*x)*c*x-1/3*arcsinh(c*x)/c^3/x^3-
3*arcsinh(c*x)/c/x-1/9*c^2*x^2*(c^2*x^2+1)^(1/2)-25/9*(c^2*x^2+1)^(1/2)-1/6/c^2/x^2*(c^2*x^2+1)^(1/2)-17/6*arc
tanh(1/(c^2*x^2+1)^(1/2))))

Fricas [F]

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

[In]

integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate((c**2*d*x**2+d)**3*(a+b*asinh(c*x))**2/x**4,x)

[Out]

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

Maxima [F]

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

[In]

integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="maxima")

[Out]

1/3*a^2*c^6*d^3*x^3 + 2/9*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*c
^6*d^3 + 3*b^2*c^4*d^3*x*arcsinh(c*x)^2 + 6*b^2*c^4*d^3*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + 3*a^2*c^4*d^3
*x + 6*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*c^3*d^3 - 6*(c*arcsinh(1/(c*abs(x))) + arcsinh(c*x)/x)*a*b*c
^2*d^3 + 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^3 - 3*a^2*c^2*
d^3/x - 1/3*a^2*d^3/x^3 + 1/3*(b^2*c^6*d^3*x^6 - 9*b^2*c^2*d^3*x^2 - b^2*d^3)*log(c*x + sqrt(c^2*x^2 + 1))^2/x
^3 - integrate(2/3*(b^2*c^9*d^3*x^8 + b^2*c^7*d^3*x^6 - 9*b^2*c^5*d^3*x^4 - 10*b^2*c^3*d^3*x^2 - b^2*c*d^3 + (
b^2*c^8*d^3*x^7 - 9*b^2*c^4*d^3*x^3 - b^2*c^2*d^3*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. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c^2*d*x^2+d)^3*(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. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3}{x^4} \,d x \]

[In]

int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3)/x^4,x)

[Out]

int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3)/x^4, x)