\(\int \frac {\text {arcsinh}(a x)^2}{(c+a^2 c x^2)^{7/2}} \, dx\) [320]

Optimal result

Integrand size = 21, antiderivative size = 366 \[ \int \frac {\text {arcsinh}(a x)^2}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=-\frac {x}{3 c^3 \sqrt {c+a^2 c x^2}}-\frac {x}{30 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {\text {arcsinh}(a x)}{10 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {4 \text {arcsinh}(a x)}{15 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^2}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {16 \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \log \left (1+e^{2 \text {arcsinh}(a x)}\right )}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(a x)}\right )}{15 a c^3 \sqrt {c+a^2 c x^2}} \]

[Out]

1/5*x*arcsinh(a*x)^2/c/(a^2*c*x^2+c)^(5/2)+4/15*x*arcsinh(a*x)^2/c^2/(a^2*c*x^2+c)^(3/2)-1/3*x/c^3/(a^2*c*x^2+
c)^(1/2)-1/30*x/c^3/(a^2*x^2+1)/(a^2*c*x^2+c)^(1/2)+1/10*arcsinh(a*x)/a/c^3/(a^2*x^2+1)^(3/2)/(a^2*c*x^2+c)^(1
/2)+8/15*x*arcsinh(a*x)^2/c^3/(a^2*c*x^2+c)^(1/2)+4/15*arcsinh(a*x)/a/c^3/(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2
)+8/15*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)/a/c^3/(a^2*c*x^2+c)^(1/2)-16/15*arcsinh(a*x)*ln(1+(a*x+(a^2*x^2+1)^(1/
2))^2)*(a^2*x^2+1)^(1/2)/a/c^3/(a^2*c*x^2+c)^(1/2)-8/15*polylog(2,-(a*x+(a^2*x^2+1)^(1/2))^2)*(a^2*x^2+1)^(1/2
)/a/c^3/(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5788, 5787, 5797, 3799, 2221, 2317, 2438, 5798, 197, 198} \[ \int \frac {\text {arcsinh}(a x)^2}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=-\frac {8 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(a x)}\right )}{15 a c^3 \sqrt {a^2 c x^2+c}}+\frac {8 x \text {arcsinh}(a x)^2}{15 c^3 \sqrt {a^2 c x^2+c}}+\frac {8 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{15 a c^3 \sqrt {a^2 c x^2+c}}+\frac {4 \text {arcsinh}(a x)}{15 a c^3 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}+\frac {\text {arcsinh}(a x)}{10 a c^3 \left (a^2 x^2+1\right )^{3/2} \sqrt {a^2 c x^2+c}}-\frac {16 \sqrt {a^2 x^2+1} \text {arcsinh}(a x) \log \left (e^{2 \text {arcsinh}(a x)}+1\right )}{15 a c^3 \sqrt {a^2 c x^2+c}}+\frac {4 x \text {arcsinh}(a x)^2}{15 c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac {x \text {arcsinh}(a x)^2}{5 c \left (a^2 c x^2+c\right )^{5/2}}-\frac {x}{3 c^3 \sqrt {a^2 c x^2+c}}-\frac {x}{30 c^3 \left (a^2 x^2+1\right ) \sqrt {a^2 c x^2+c}} \]

[In]

Int[ArcSinh[a*x]^2/(c + a^2*c*x^2)^(7/2),x]

[Out]

-1/3*x/(c^3*Sqrt[c + a^2*c*x^2]) - x/(30*c^3*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]) + ArcSinh[a*x]/(10*a*c^3*(1 +
a^2*x^2)^(3/2)*Sqrt[c + a^2*c*x^2]) + (4*ArcSinh[a*x])/(15*a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) + (x*A
rcSinh[a*x]^2)/(5*c*(c + a^2*c*x^2)^(5/2)) + (4*x*ArcSinh[a*x]^2)/(15*c^2*(c + a^2*c*x^2)^(3/2)) + (8*x*ArcSin
h[a*x]^2)/(15*c^3*Sqrt[c + a^2*c*x^2]) + (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(15*a*c^3*Sqrt[c + a^2*c*x^2]) -
(16*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*Log[1 + E^(2*ArcSinh[a*x])])/(15*a*c^3*Sqrt[c + a^2*c*x^2]) - (8*Sqrt[1 +
a^2*x^2]*PolyLog[2, -E^(2*ArcSinh[a*x])])/(15*a*c^3*Sqrt[c + a^2*c*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

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, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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 3799

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

Rule 5787

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcSinh
[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Dist[b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Int[x*((a + b*ArcS
inh[c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 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 5797

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/e, Subst[Int[(
a + b*x)^n*Tanh[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]

Rubi steps \begin{align*} \text {integral}& = \frac {x \text {arcsinh}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 \int \frac {\text {arcsinh}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{5 c}-\frac {\left (2 a \sqrt {1+a^2 x^2}\right ) \int \frac {x \text {arcsinh}(a x)}{\left (1+a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}} \\ & = \frac {\text {arcsinh}(a x)}{10 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 \int \frac {\text {arcsinh}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{15 c^2}-\frac {\sqrt {1+a^2 x^2} \int \frac {1}{\left (1+a^2 x^2\right )^{5/2}} \, dx}{10 c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (8 a \sqrt {1+a^2 x^2}\right ) \int \frac {x \text {arcsinh}(a x)}{\left (1+a^2 x^2\right )^2} \, dx}{15 c^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {x}{30 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {\text {arcsinh}(a x)}{10 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {4 \text {arcsinh}(a x)}{15 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^2}{15 c^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \int \frac {1}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{15 c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{15 c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (16 a \sqrt {1+a^2 x^2}\right ) \int \frac {x \text {arcsinh}(a x)}{1+a^2 x^2} \, dx}{15 c^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {x}{3 c^3 \sqrt {c+a^2 c x^2}}-\frac {x}{30 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {\text {arcsinh}(a x)}{10 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {4 \text {arcsinh}(a x)}{15 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^2}{15 c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (16 \sqrt {1+a^2 x^2}\right ) \text {Subst}(\int x \tanh (x) \, dx,x,\text {arcsinh}(a x))}{15 a c^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {x}{3 c^3 \sqrt {c+a^2 c x^2}}-\frac {x}{30 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {\text {arcsinh}(a x)}{10 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {4 \text {arcsinh}(a x)}{15 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^2}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (32 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a c^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {x}{3 c^3 \sqrt {c+a^2 c x^2}}-\frac {x}{30 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {\text {arcsinh}(a x)}{10 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {4 \text {arcsinh}(a x)}{15 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^2}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {16 \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \log \left (1+e^{2 \text {arcsinh}(a x)}\right )}{15 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\left (16 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{15 a c^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {x}{3 c^3 \sqrt {c+a^2 c x^2}}-\frac {x}{30 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {\text {arcsinh}(a x)}{10 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {4 \text {arcsinh}(a x)}{15 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^2}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {16 \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \log \left (1+e^{2 \text {arcsinh}(a x)}\right )}{15 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(a x)}\right )}{15 a c^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {x}{3 c^3 \sqrt {c+a^2 c x^2}}-\frac {x}{30 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {\text {arcsinh}(a x)}{10 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {4 \text {arcsinh}(a x)}{15 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)^2}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {16 \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \log \left (1+e^{2 \text {arcsinh}(a x)}\right )}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(a x)}\right )}{15 a c^3 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.49 \[ \int \frac {\text {arcsinh}(a x)^2}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {a x \left (-10-\frac {1}{1+a^2 x^2}\right )+\left (-16 \sqrt {1+a^2 x^2}+\frac {2 a x \left (15+20 a^2 x^2+8 a^4 x^4\right )}{\left (1+a^2 x^2\right )^2}\right ) \text {arcsinh}(a x)^2+\frac {\text {arcsinh}(a x) \left (11+8 a^2 x^2-32 \left (1+a^2 x^2\right )^2 \log \left (1+e^{-2 \text {arcsinh}(a x)}\right )\right )}{\left (1+a^2 x^2\right )^{3/2}}+16 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(a x)}\right )}{30 a c^3 \sqrt {c+a^2 c x^2}} \]

[In]

Integrate[ArcSinh[a*x]^2/(c + a^2*c*x^2)^(7/2),x]

[Out]

(a*x*(-10 - (1 + a^2*x^2)^(-1)) + (-16*Sqrt[1 + a^2*x^2] + (2*a*x*(15 + 20*a^2*x^2 + 8*a^4*x^4))/(1 + a^2*x^2)
^2)*ArcSinh[a*x]^2 + (ArcSinh[a*x]*(11 + 8*a^2*x^2 - 32*(1 + a^2*x^2)^2*Log[1 + E^(-2*ArcSinh[a*x])]))/(1 + a^
2*x^2)^(3/2) + 16*Sqrt[1 + a^2*x^2]*PolyLog[2, -E^(-2*ArcSinh[a*x])])/(30*a*c^3*Sqrt[c + a^2*c*x^2])

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.56

method	result	size
default	\(\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (8 a^{5} x^{5}-8 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+20 a^{3} x^{3}-16 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+15 a x -8 \sqrt {a^{2} x^{2}+1}\right ) \left (-64 \,\operatorname {arcsinh}\left (a x \right ) a^{8} x^{8}-64 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{7} x^{7}-32 a^{8} x^{8}-32 \sqrt {a^{2} x^{2}+1}\, a^{7} x^{7}-280 \,\operatorname {arcsinh}\left (a x \right ) a^{6} x^{6}-248 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{5} x^{5}-142 a^{6} x^{6}-126 x^{5} a^{5} \sqrt {a^{2} x^{2}+1}+80 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{2}-456 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )-340 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}-265 a^{4} x^{4}-156 a^{3} x^{3} \sqrt {a^{2} x^{2}+1}+190 \operatorname {arcsinh}\left (a x \right )^{2} a^{2} x^{2}-328 a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )-165 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x -235 a^{2} x^{2}-62 a x \sqrt {a^{2} x^{2}+1}+128 \operatorname {arcsinh}\left (a x \right )^{2}-88 \,\operatorname {arcsinh}\left (a x \right )-80\right )}{30 \left (40 a^{10} x^{10}+215 a^{8} x^{8}+469 a^{6} x^{6}+517 a^{4} x^{4}+287 a^{2} x^{2}+64\right ) a \,c^{4}}+\frac {16 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (a x \right )^{2}}{15 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}}-\frac {16 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{15 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}}-\frac {8 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{15 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}}\)	\(570\)

[In]

int(arcsinh(a*x)^2/(a^2*c*x^2+c)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/30*(c*(a^2*x^2+1))^(1/2)*(8*a^5*x^5-8*a^4*x^4*(a^2*x^2+1)^(1/2)+20*a^3*x^3-16*a^2*x^2*(a^2*x^2+1)^(1/2)+15*a
*x-8*(a^2*x^2+1)^(1/2))*(-64*arcsinh(a*x)*a^8*x^8-64*arcsinh(a*x)*(a^2*x^2+1)^(1/2)*a^7*x^7-32*a^8*x^8-32*(a^2
*x^2+1)^(1/2)*a^7*x^7-280*arcsinh(a*x)*a^6*x^6-248*arcsinh(a*x)*(a^2*x^2+1)^(1/2)*a^5*x^5-142*a^6*x^6-126*x^5*
a^5*(a^2*x^2+1)^(1/2)+80*a^4*x^4*arcsinh(a*x)^2-456*a^4*x^4*arcsinh(a*x)-340*a^3*x^3*arcsinh(a*x)*(a^2*x^2+1)^
(1/2)-265*a^4*x^4-156*a^3*x^3*(a^2*x^2+1)^(1/2)+190*arcsinh(a*x)^2*a^2*x^2-328*a^2*x^2*arcsinh(a*x)-165*arcsin
h(a*x)*(a^2*x^2+1)^(1/2)*a*x-235*a^2*x^2-62*a*x*(a^2*x^2+1)^(1/2)+128*arcsinh(a*x)^2-88*arcsinh(a*x)-80)/(40*a
^10*x^10+215*a^8*x^8+469*a^6*x^6+517*a^4*x^4+287*a^2*x^2+64)/a/c^4+16/15/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/
2)/a/c^4*arcsinh(a*x)^2-16/15/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)/a/c^4*arcsinh(a*x)*ln(1+(a*x+(a^2*x^2+1)
^(1/2))^2)-8/15/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)/a/c^4*polylog(2,-(a*x+(a^2*x^2+1)^(1/2))^2)

Fricas [F]

\[ \int \frac {\text {arcsinh}(a x)^2}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(arcsinh(a*x)^2/(a^2*c*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*arcsinh(a*x)^2/(a^8*c^4*x^8 + 4*a^6*c^4*x^6 + 6*a^4*c^4*x^4 + 4*a^2*c^4*x^2 + c^4
), x)

Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)^2}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate(asinh(a*x)**2/(a**2*c*x**2+c)**(7/2),x)

[Out]

Integral(asinh(a*x)**2/(c*(a**2*x**2 + 1))**(7/2), x)

Maxima [F]

\[ \int \frac {\text {arcsinh}(a x)^2}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(arcsinh(a*x)^2/(a^2*c*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

integrate(arcsinh(a*x)^2/(a^2*c*x^2 + c)^(7/2), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\text {arcsinh}(a x)^2}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(arcsinh(a*x)^2/(a^2*c*x^2+c)^(7/2),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. \[ \int \frac {\text {arcsinh}(a x)^2}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{7/2}} \,d x \]

[In]

int(asinh(a*x)^2/(c + a^2*c*x^2)^(7/2),x)

[Out]

int(asinh(a*x)^2/(c + a^2*c*x^2)^(7/2), x)