3.4.88 \(\int (c-a^2 c x^2)^{3/2} \cosh ^{-1}(a x)^{5/2} \, dx\) [388]

Optimal. Leaf size=580 \[ \frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}} \]

[Out]

1/4*x*(-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(5/2)+3/8*c*x*arccosh(a*x)^(5/2)*(-a^2*c*x^2+c)^(1/2)+45/256*c*arccosh
(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-15/32*a*c*x^2*arccosh(a*x)^(3/2)*(-a^2*c*x^2+c)
^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)+5/32*c*(-a^2*x^2+1)^2*arccosh(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/
2)/(a*x+1)^(1/2)-3/28*c*arccosh(a*x)^(7/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+15/512*c*erf(2^(
1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-15/512*c*erfi(2^(
1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-15/16384*c*erf(2*
arccosh(a*x)^(1/2))*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+15/16384*c*erfi(2*arccosh(a*x)
^(1/2))*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+225/512*c*x*(-a^2*c*x^2+c)^(1/2)*arccosh(a
*x)^(1/2)+15/256*c*x*(-a*x+1)*(a*x+1)*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.94, antiderivative size = 580, normalized size of antiderivative = 1.00, number of steps used = 41, number of rules used = 17, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {5897, 5895, 5893, 5884, 5939, 5887, 5556, 12, 3389, 2211, 2235, 2236, 5912, 5914, 5898, 5896, 5952} \begin {gather*} -\frac {15 \sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15 \sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{5/2}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {a x-1} \sqrt {a x+1}}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (a x+1) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2),x]

[Out]

(225*c*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/512 + (15*c*x*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]*Sqrt[Ar
cCosh[a*x]])/256 + (45*c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*a*
c*x^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(32*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (5*c*(1 - a^2*x^2)^2*Sqrt[c
- a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*c*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]
^(5/2))/8 + (x*(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2))/4 - (3*c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(7/2))/(28*
a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(16384*a*Sqrt[
-1 + a*x]*Sqrt[1 + a*x]) + (15*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(256*a*Sqrt[-
1 + a*x]*Sqrt[1 + a*x]) + (15*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(16384*a*Sqrt[-1 + a*
x]*Sqrt[1 + a*x]) - (15*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(256*a*Sqrt[-1 + a*
x]*Sqrt[1 + a*x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5884

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCosh[c*x])^n/(
m + 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])
), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 5887

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Cosh
[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5895

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

Rule 5896

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)], x_Symbol] :
> Simp[x*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*((a + b*ArcCosh[c*x])^n/2), x] + (-Dist[(1/2)*Simp[Sqrt[d1 + e1*x]/Sq
rt[1 + c*x]]*Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]], Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]),
x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d1 + e1*x]/Sqrt[1 + c*x]]*Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]], Int[x*(a + b
*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] &&
GtQ[n, 0]

Rule 5897

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

Rule 5898

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Simp[x*(d1 + e1*x)^p*(d2 + e2*x)^p*((a + b*ArcCosh[c*x])^n/(2*p + 1)), x] + (Dist[2*d1*d2*(p/(2*p + 1)),
Int[(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(2*p + 1))*Simp[(d1
+ e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[x*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b
*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] &&
GtQ[n, 0] && GtQ[p, 0]

Rule 5912

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(
x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]

Rule 5914

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

Rule 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1
*e2*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)
^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 +
e2*x)^p/(-1 + c*x)^p], Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IG
tQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rule 5952

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*
c^(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^
(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2
, 0] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{5/2} \, dx &=-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^{5/2} \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{5/2} \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (5 a c \sqrt {c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x)^{3/2} \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \sqrt {\cosh ^{-1}(a x)} \, dx}{64 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\cosh ^{-1}(a x)^{5/2}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 a c \sqrt {c-a^2 c x^2}\right ) \int x \cosh ^{-1}(a x)^{3/2} \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \int \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)} \, dx}{256 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x \left (-1+a^2 x^2\right )}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{512 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 a^2 c \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \sqrt {\cosh ^{-1}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{64 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\cosh ^{-1}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{512 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\cosh ^{-1}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{128 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^3(x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{512 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{1024 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{256 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \left (-\frac {\sinh (2 x)}{4 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{512 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{1024 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2048 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{1024 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8192 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8192 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2048 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{512 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{2048 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{2048 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{1024 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{1024 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{2048 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{2048 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{512 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{512 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 213, normalized size = 0.37 \begin {gather*} \frac {c \sqrt {c-a^2 c x^2} \left (-1536 \cosh ^{-1}(a x)^4-4480 \cosh ^{-1}(a x)^2 \cosh \left (2 \cosh ^{-1}(a x)\right )+420 \sqrt {2 \pi } \sqrt {\cosh ^{-1}(a x)} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )-420 \sqrt {2 \pi } \sqrt {\cosh ^{-1}(a x)} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )+7 \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {7}{2},-4 \cosh ^{-1}(a x)\right )+7 \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {7}{2},4 \cosh ^{-1}(a x)\right )+3360 \cosh ^{-1}(a x) \sinh \left (2 \cosh ^{-1}(a x)\right )+3584 \cosh ^{-1}(a x)^3 \sinh \left (2 \cosh ^{-1}(a x)\right )\right )}{14336 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\cosh ^{-1}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2),x]

[Out]

(c*Sqrt[c - a^2*c*x^2]*(-1536*ArcCosh[a*x]^4 - 4480*ArcCosh[a*x]^2*Cosh[2*ArcCosh[a*x]] + 420*Sqrt[2*Pi]*Sqrt[
ArcCosh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] - 420*Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*
x]]] + 7*Sqrt[-ArcCosh[a*x]]*Gamma[7/2, -4*ArcCosh[a*x]] + 7*Sqrt[ArcCosh[a*x]]*Gamma[7/2, 4*ArcCosh[a*x]] + 3
360*ArcCosh[a*x]*Sinh[2*ArcCosh[a*x]] + 3584*ArcCosh[a*x]^3*Sinh[2*ArcCosh[a*x]]))/(14336*a*Sqrt[(-1 + a*x)/(1
+ a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(5/2),x)

[Out]

int((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(5/2),x)

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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((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((-a^2*c*x^2 + c)^(3/2)*arccosh(a*x)^(5/2), x)

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Fricas [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((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**(3/2)*acosh(a*x)**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8568 deep

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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((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(5/2),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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {acosh}\left (a\,x\right )}^{5/2}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^(5/2)*(c - a^2*c*x^2)^(3/2),x)

[Out]

int(acosh(a*x)^(5/2)*(c - a^2*c*x^2)^(3/2), x)

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