3.4.40 \(\int \frac {\sinh ^{-1}(a x)^3}{(c+a^2 c x^2)^{7/2}} \, dx\) [340]

Optimal. Leaf size=515 \[ -\frac {1}{20 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}-\frac {x \sinh ^{-1}(a x)}{c^3 \sqrt {c+a^2 c x^2}}-\frac {x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {4 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}} \]

[Out]

1/5*x*arcsinh(a*x)^3/c/(a^2*c*x^2+c)^(5/2)+4/15*x*arcsinh(a*x)^3/c^2/(a^2*c*x^2+c)^(3/2)-x*arcsinh(a*x)/c^3/(a
^2*c*x^2+c)^(1/2)-1/10*x*arcsinh(a*x)/c^3/(a^2*x^2+1)/(a^2*c*x^2+c)^(1/2)+3/20*arcsinh(a*x)^2/a/c^3/(a^2*x^2+1
)^(3/2)/(a^2*c*x^2+c)^(1/2)+8/15*x*arcsinh(a*x)^3/c^3/(a^2*c*x^2+c)^(1/2)-1/20/a/c^3/(a^2*x^2+1)^(1/2)/(a^2*c*
x^2+c)^(1/2)+2/5*arcsinh(a*x)^2/a/c^3/(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+8/15*arcsinh(a*x)^3*(a^2*x^2+1)^(1
/2)/a/c^3/(a^2*c*x^2+c)^(1/2)-8/5*arcsinh(a*x)^2*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)+1/2*ln(a^2*x^2+1)*(a^2*x^2+1)^(1/2)/a/c^3/(a^2*c*x^2+c)^(1/2)-8/5*arcsinh(a*x)*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)+4/5*polylog(3,-(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]
time = 0.35, antiderivative size = 515, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {5788, 5787, 5797, 3799, 2221, 2611, 2320, 6724, 5798, 266, 267} \begin {gather*} -\frac {8 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x) \text {Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {a^2 c x^2+c}}+\frac {4 \sqrt {a^2 x^2+1} \text {Li}_3\left (-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {a^2 c x^2+c}}-\frac {1}{20 a c^3 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \log \left (a^2 x^2+1\right )}{2 a c^3 \sqrt {a^2 c x^2+c}}+\frac {8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt {a^2 c x^2+c}}+\frac {8 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt {a^2 c x^2+c}}+\frac {2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}+\frac {3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (a^2 x^2+1\right )^{3/2} \sqrt {a^2 c x^2+c}}-\frac {x \sinh ^{-1}(a x)}{c^3 \sqrt {a^2 c x^2+c}}-\frac {x \sinh ^{-1}(a x)}{10 c^3 \left (a^2 x^2+1\right ) \sqrt {a^2 c x^2+c}}-\frac {8 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2 \log \left (e^{2 \sinh ^{-1}(a x)}+1\right )}{5 a c^3 \sqrt {a^2 c x^2+c}}+\frac {4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac {x \sinh ^{-1}(a x)^3}{5 c \left (a^2 c x^2+c\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^3/(c + a^2*c*x^2)^(7/2),x]

[Out]

-1/20*1/(a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) - (x*ArcSinh[a*x])/(c^3*Sqrt[c + a^2*c*x^2]) - (x*ArcSin
h[a*x])/(10*c^3*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]) + (3*ArcSinh[a*x]^2)/(20*a*c^3*(1 + a^2*x^2)^(3/2)*Sqrt[c +
a^2*c*x^2]) + (2*ArcSinh[a*x]^2)/(5*a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) + (x*ArcSinh[a*x]^3)/(5*c*(c
+ a^2*c*x^2)^(5/2)) + (4*x*ArcSinh[a*x]^3)/(15*c^2*(c + a^2*c*x^2)^(3/2)) + (8*x*ArcSinh[a*x]^3)/(15*c^3*Sqrt
[c + a^2*c*x^2]) + (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(15*a*c^3*Sqrt[c + a^2*c*x^2]) - (8*Sqrt[1 + a^2*x^2]*
ArcSinh[a*x]^2*Log[1 + E^(2*ArcSinh[a*x])])/(5*a*c^3*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*Log[1 + a^2*x^2
])/(2*a*c^3*Sqrt[c + a^2*c*x^2]) - (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*PolyLog[2, -E^(2*ArcSinh[a*x])])/(5*a*c^3
*Sqrt[c + a^2*c*x^2]) + (4*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(2*ArcSinh[a*x])])/(5*a*c^3*Sqrt[c + a^2*c*x^2])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 267

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

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*} \int \frac {\sinh ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{7/2}} \, dx &=\frac {x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 \int \frac {\sinh ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{5 c}-\frac {\left (3 a \sqrt {1+a^2 x^2}\right ) \int \frac {x \sinh ^{-1}(a x)^2}{\left (1+a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 \int \frac {\sinh ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{15 c^2}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {\sinh ^{-1}(a x)}{\left (1+a^2 x^2\right )^{5/2}} \, dx}{10 c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (4 a \sqrt {1+a^2 x^2}\right ) \int \frac {x \sinh ^{-1}(a x)^2}{\left (1+a^2 x^2\right )^2} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \int \frac {\sinh ^{-1}(a x)}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \int \frac {\sinh ^{-1}(a x)}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}}+\frac {\left (a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^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 \sinh ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{20 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}-\frac {x \sinh ^{-1}(a x)}{c^3 \sqrt {c+a^2 c x^2}}-\frac {x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \tanh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\left (a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{1+a^2 x^2} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}}+\frac {\left (4 a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{1+a^2 x^2} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{20 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}-\frac {x \sinh ^{-1}(a x)}{c^3 \sqrt {c+a^2 c x^2}}-\frac {x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (16 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} x^2}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{20 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}-\frac {x \sinh ^{-1}(a x)}{c^3 \sqrt {c+a^2 c x^2}}-\frac {x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\left (16 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{20 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}-\frac {x \sinh ^{-1}(a x)}{c^3 \sqrt {c+a^2 c x^2}}-\frac {x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \text {Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{20 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}-\frac {x \sinh ^{-1}(a x)}{c^3 \sqrt {c+a^2 c x^2}}-\frac {x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \text {Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{20 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}-\frac {x \sinh ^{-1}(a x)}{c^3 \sqrt {c+a^2 c x^2}}-\frac {x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}+\frac {3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt {c+a^2 c x^2}}+\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt {c+a^2 c x^2}}-\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \text {Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}+\frac {4 \sqrt {1+a^2 x^2} \text {Li}_3\left (-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt {c+a^2 c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.45, size = 297, normalized size = 0.58 \begin {gather*} \frac {-\frac {3}{\sqrt {1+a^2 x^2}}-60 a x \sinh ^{-1}(a x)-\frac {6 a x \sinh ^{-1}(a x)}{1+a^2 x^2}+\frac {9 \sinh ^{-1}(a x)^2}{\left (1+a^2 x^2\right )^{3/2}}+\frac {24 \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}}+32 a x \sinh ^{-1}(a x)^3+\frac {12 a x \sinh ^{-1}(a x)^3}{\left (1+a^2 x^2\right )^2}+\frac {16 a x \sinh ^{-1}(a x)^3}{1+a^2 x^2}-32 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3-96 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{-2 \sinh ^{-1}(a x)}\right )+30 \sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )+96 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \text {PolyLog}\left (2,-e^{-2 \sinh ^{-1}(a x)}\right )+48 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-e^{-2 \sinh ^{-1}(a x)}\right )}{60 a c^3 \sqrt {c+a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]^3/(c + a^2*c*x^2)^(7/2),x]

[Out]

(-3/Sqrt[1 + a^2*x^2] - 60*a*x*ArcSinh[a*x] - (6*a*x*ArcSinh[a*x])/(1 + a^2*x^2) + (9*ArcSinh[a*x]^2)/(1 + a^2
*x^2)^(3/2) + (24*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2] + 32*a*x*ArcSinh[a*x]^3 + (12*a*x*ArcSinh[a*x]^3)/(1 + a^2
*x^2)^2 + (16*a*x*ArcSinh[a*x]^3)/(1 + a^2*x^2) - 32*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3 - 96*Sqrt[1 + a^2*x^2]*A
rcSinh[a*x]^2*Log[1 + E^(-2*ArcSinh[a*x])] + 30*Sqrt[1 + a^2*x^2]*Log[1 + a^2*x^2] + 96*Sqrt[1 + a^2*x^2]*ArcS
inh[a*x]*PolyLog[2, -E^(-2*ArcSinh[a*x])] + 48*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(-2*ArcSinh[a*x])])/(60*a*c^3*S
qrt[c + a^2*c*x^2])

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Maple [A]
time = 2.85, size = 888, normalized size = 1.72

method	result	size
default	\(\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (8 a^{5} x^{5}-8 a^{4} \sqrt {a^{2} x^{2}+1}\, x^{4}+20 a^{3} x^{3}-16 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+15 a x -8 \sqrt {a^{2} x^{2}+1}\right ) \left (24+24 a^{8} x^{8}+96 a^{6} x^{6}+45 \sqrt {a^{2} x^{2}+1}\, a x +256 \arcsinh \left (a x \right )^{3}-192 \sqrt {a^{2} x^{2}+1}\, \arcsinh \left (a x \right ) a^{7} x^{7}-756 \sqrt {a^{2} x^{2}+1}\, \arcsinh \left (a x \right ) a^{5} x^{5}-936 \sqrt {a^{2} x^{2}+1}\, \arcsinh \left (a x \right ) a^{3} x^{3}-372 \sqrt {a^{2} x^{2}+1}\, \arcsinh \left (a x \right ) a x -1410 x^{2} \arcsinh \left (a x \right ) a^{2}-480 \arcsinh \left (a x \right )+84 \sqrt {a^{2} x^{2}+1}\, a^{5} x^{5}+105 \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+144 a^{4} x^{4}+96 a^{2} x^{2}-264 \arcsinh \left (a x \right )^{2}-192 \arcsinh \left (a x \right )^{2} a^{8} x^{8}-840 \arcsinh \left (a x \right )^{2} a^{6} x^{6}+160 \arcsinh \left (a x \right )^{3} a^{4} x^{4}-1368 \arcsinh \left (a x \right )^{2} a^{4} x^{4}+380 \arcsinh \left (a x \right )^{3} a^{2} x^{2}-192 \arcsinh \left (a x \right ) a^{8} x^{8}-852 \arcsinh \left (a x \right ) a^{6} x^{6}-1590 \arcsinh \left (a x \right ) a^{4} x^{4}-984 \arcsinh \left (a x \right )^{2} a^{2} x^{2}+24 \sqrt {a^{2} x^{2}+1}\, a^{7} x^{7}-192 \sqrt {a^{2} x^{2}+1}\, \arcsinh \left (a x \right )^{2} a^{7} x^{7}-744 \sqrt {a^{2} x^{2}+1}\, \arcsinh \left (a x \right )^{2} a^{5} x^{5}-1020 \sqrt {a^{2} x^{2}+1}\, \arcsinh \left (a x \right )^{2} a^{3} x^{3}-495 \sqrt {a^{2} x^{2}+1}\, \arcsinh \left (a x \right )^{2} a x \right )}{60 \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 {2 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \ln \left (a x +\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{4}}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \ln \left (1+\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{4}}+\frac {16 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right )^{3}}{15 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}}-\frac {8 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{5 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}}-\frac {8 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right ) \polylog \left (2, -\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{5 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}}+\frac {4 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \polylog \left (3, -\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{5 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}}\)	\(888\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^3/(a^2*c*x^2+c)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/60*(c*(a^2*x^2+1))^(1/2)*(8*a^5*x^5-8*a^4*(a^2*x^2+1)^(1/2)*x^4+20*a^3*x^3-16*(a^2*x^2+1)^(1/2)*a^2*x^2+15*a
*x-8*(a^2*x^2+1)^(1/2))*(24+24*a^8*x^8+96*a^6*x^6+45*(a^2*x^2+1)^(1/2)*a*x-264*arcsinh(a*x)^2+256*arcsinh(a*x)
^3-372*(a^2*x^2+1)^(1/2)*arcsinh(a*x)*a*x+84*(a^2*x^2+1)^(1/2)*a^5*x^5+105*(a^2*x^2+1)^(1/2)*a^3*x^3+144*a^4*x
^4-480*arcsinh(a*x)+96*a^2*x^2-192*arcsinh(a*x)^2*a^8*x^8-192*arcsinh(a*x)*a^8*x^8-840*arcsinh(a*x)^2*a^6*x^6-
852*arcsinh(a*x)*a^6*x^6+160*arcsinh(a*x)^3*a^4*x^4-1368*arcsinh(a*x)^2*a^4*x^4+380*arcsinh(a*x)^3*a^2*x^2-159
0*arcsinh(a*x)*a^4*x^4-984*arcsinh(a*x)^2*a^2*x^2-1410*arcsinh(a*x)*a^2*x^2+24*(a^2*x^2+1)^(1/2)*a^7*x^7-936*(
a^2*x^2+1)^(1/2)*arcsinh(a*x)*a^3*x^3-192*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^2*a^7*x^7-192*(a^2*x^2+1)^(1/2)*arcsi
nh(a*x)*a^7*x^7-744*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^2*a^5*x^5-756*(a^2*x^2+1)^(1/2)*arcsinh(a*x)*a^5*x^5-1020*(
a^2*x^2+1)^(1/2)*arcsinh(a*x)^2*a^3*x^3-495*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^2*a*x)/(40*a^10*x^10+215*a^8*x^8+46
9*a^6*x^6+517*a^4*x^4+287*a^2*x^2+64)/a/c^4-2*(c*(a^2*x^2+1))^(1/2)/(a^2*x^2+1)^(1/2)/a/c^4*ln(a*x+(a^2*x^2+1)
^(1/2))+(c*(a^2*x^2+1))^(1/2)/(a^2*x^2+1)^(1/2)/a/c^4*ln(1+(a*x+(a^2*x^2+1)^(1/2))^2)+16/15*(c*(a^2*x^2+1))^(1
/2)/(a^2*x^2+1)^(1/2)/a/c^4*arcsinh(a*x)^3-8/5*(c*(a^2*x^2+1))^(1/2)/(a^2*x^2+1)^(1/2)/a/c^4*arcsinh(a*x)^2*ln
(1+(a*x+(a^2*x^2+1)^(1/2))^2)-8/5*(c*(a^2*x^2+1))^(1/2)/(a^2*x^2+1)^(1/2)/a/c^4*arcsinh(a*x)*polylog(2,-(a*x+(
a^2*x^2+1)^(1/2))^2)+4/5*(c*(a^2*x^2+1))^(1/2)/(a^2*x^2+1)^(1/2)/a/c^4*polylog(3,-(a*x+(a^2*x^2+1)^(1/2))^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/(a^2*c*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

integrate(arcsinh(a*x)^3/(a^2*c*x^2 + c)^(7/2), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/(a^2*c*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*arcsinh(a*x)^3/(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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**3/(a**2*c*x**2+c)**(7/2),x)

[Out]

Integral(asinh(a*x)**3/(c*(a**2*x**2 + 1))**(7/2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/(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:Warning, choosing root of [1,0,%%%{-2,[2,1,2]%%%}+%%%{-2,[2,0,2]%%%}+%%%{-2,[0,1,0]%%%}+%%%{-2,[0,0,0]%%%},
0,%%%{1,[4,

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a*x)^3/(c + a^2*c*x^2)^(7/2),x)

[Out]

int(asinh(a*x)^3/(c + a^2*c*x^2)^(7/2), x)

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