3.2.10 \(\int \frac {x^3}{\cosh ^{-1}(a x)^{7/2}} \, dx\) [110]

Optimal. Leaf size=244 \[ -\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}+\frac {16 \sqrt {\pi } \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4} \]

[Out]

4/5*x^2/a^2/arccosh(a*x)^(3/2)-16/15*x^4/arccosh(a*x)^(3/2)+16/15*erf(2*arccosh(a*x)^(1/2))*Pi^(1/2)/a^4+16/15
*erfi(2*arccosh(a*x)^(1/2))*Pi^(1/2)/a^4+4/15*erf(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4+4/15*erfi(2
^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4-2/5*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(5/2)+16/5*
x*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3/arccosh(a*x)^(1/2)-128/15*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(1/
2)

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Rubi [A]
time = 0.49, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5886, 5951, 5885, 3388, 2211, 2235, 2236} \begin {gather*} \frac {16 \sqrt {\pi } \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {16 x \sqrt {a x-1} \sqrt {a x+1}}{5 a^3 \sqrt {\cosh ^{-1}(a x)}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}-\frac {128 x^3 \sqrt {a x-1} \sqrt {a x+1}}{15 a \sqrt {\cosh ^{-1}(a x)}}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/ArcCosh[a*x]^(7/2),x]

[Out]

(-2*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2)) + (4*x^2)/(5*a^2*ArcCosh[a*x]^(3/2)) - (16*x^4)
/(15*ArcCosh[a*x]^(3/2)) + (16*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a^3*Sqrt[ArcCosh[a*x]]) - (128*x^3*Sqrt[-1 +
a*x]*Sqrt[1 + a*x])/(15*a*Sqrt[ArcCosh[a*x]]) + (16*Sqrt[Pi]*Erf[2*Sqrt[ArcCosh[a*x]]])/(15*a^4) + (4*Sqrt[2*
Pi]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(15*a^4) + (16*Sqrt[Pi]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(15*a^4) + (4*Sqrt[2*
Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(15*a^4)

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 3388

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

Rule 5885

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((
a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^
(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; Free
Q[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 5886

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((
a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Dist[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcCosh
[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Dist[m/(b*c*(n + 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] && LtQ[n, -2]

Rule 5951

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

Rubi steps

\begin {align*} \int \frac {x^3}{\cosh ^{-1}(a x)^{7/2}} \, dx &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac {6 \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (8 a) \int \frac {x^4}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac {64}{15} \int \frac {x^3}{\cosh ^{-1}(a x)^{3/2}} \, dx-\frac {8 \int \frac {x}{\cosh ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}-\frac {16 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^4}-\frac {128 \text {Subst}\left (\int \left (-\frac {\cosh (2 x)}{2 \sqrt {x}}-\frac {\cosh (4 x)}{2 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}\\ &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}-\frac {8 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^4}-\frac {8 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a^4}+\frac {64 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}+\frac {64 \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}\\ &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}+\frac {32 \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}+\frac {32 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}+\frac {32 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}+\frac {32 \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^4}-\frac {16 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{5 a^4}-\frac {16 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{5 a^4}\\ &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}-\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{5 a^4}-\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{5 a^4}+\frac {64 \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {64 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {64 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {64 \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}\\ &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \cosh ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\cosh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}+\frac {16 \sqrt {\pi } \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{15 a^4}\\ \end {align*}

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Mathematica [A]
time = 0.41, size = 291, normalized size = 1.19 \begin {gather*} \frac {e^{-4 \cosh ^{-1}(a x)} \left (3-3 e^{8 \cosh ^{-1}(a x)}-8 \cosh ^{-1}(a x)-8 e^{8 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+64 \cosh ^{-1}(a x)^2-64 e^{8 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2+128 e^{4 \cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{5/2} \Gamma \left (\frac {1}{2},-4 \cosh ^{-1}(a x)\right )-8 e^{2 \cosh ^{-1}(a x)} \left (3 a e^{2 \cosh ^{-1}(a x)} x \sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\cosh ^{-1}(a x)+e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)-4 \cosh ^{-1}(a x)^2+4 e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2-4 \sqrt {2} e^{2 \cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{5/2} \Gamma \left (\frac {1}{2},-2 \cosh ^{-1}(a x)\right )+4 \sqrt {2} e^{2 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{5/2} \Gamma \left (\frac {1}{2},2 \cosh ^{-1}(a x)\right )\right )-128 e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{5/2} \Gamma \left (\frac {1}{2},4 \cosh ^{-1}(a x)\right )\right )}{120 a^4 \cosh ^{-1}(a x)^{5/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3/ArcCosh[a*x]^(7/2),x]

[Out]

(3 - 3*E^(8*ArcCosh[a*x]) - 8*ArcCosh[a*x] - 8*E^(8*ArcCosh[a*x])*ArcCosh[a*x] + 64*ArcCosh[a*x]^2 - 64*E^(8*A
rcCosh[a*x])*ArcCosh[a*x]^2 + 128*E^(4*ArcCosh[a*x])*(-ArcCosh[a*x])^(5/2)*Gamma[1/2, -4*ArcCosh[a*x]] - 8*E^(
2*ArcCosh[a*x])*(3*a*E^(2*ArcCosh[a*x])*x*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + ArcCosh[a*x] + E^(4*ArcCosh[a
*x])*ArcCosh[a*x] - 4*ArcCosh[a*x]^2 + 4*E^(4*ArcCosh[a*x])*ArcCosh[a*x]^2 - 4*Sqrt[2]*E^(2*ArcCosh[a*x])*(-Ar
cCosh[a*x])^(5/2)*Gamma[1/2, -2*ArcCosh[a*x]] + 4*Sqrt[2]*E^(2*ArcCosh[a*x])*ArcCosh[a*x]^(5/2)*Gamma[1/2, 2*A
rcCosh[a*x]]) - 128*E^(4*ArcCosh[a*x])*ArcCosh[a*x]^(5/2)*Gamma[1/2, 4*ArcCosh[a*x]])/(120*a^4*E^(4*ArcCosh[a*
x])*ArcCosh[a*x]^(5/2))

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Maple [A]
time = 9.39, size = 366, normalized size = 1.50

method	result	size
default	\(\frac {\sqrt {2}\, \left (-16 \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x -4 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}-3 \sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +8 \mathrm {arccosh}\left (a x \right )^{3} \pi \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )+8 \mathrm {arccosh}\left (a x \right )^{3} \pi \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )+2 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\right )}{30 \sqrt {\pi }\, a^{4} \mathrm {arccosh}\left (a x \right )^{3}}+\frac {-128 \sqrt {a x -1}\, \sqrt {a x +1}\, \sqrt {\pi }\, \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} a^{3} x^{3}-16 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{4} x^{4}-6 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+64 \sqrt {a x -1}\, \sqrt {a x +1}\, \sqrt {\pi }\, \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} a x +16 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{2} x^{2}+3 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +16 \mathrm {arccosh}\left (a x \right )^{3} \pi \erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )+16 \mathrm {arccosh}\left (a x \right )^{3} \pi \erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-2 \sqrt {\pi }\, \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}}{15 \sqrt {\pi }\, a^{4} \mathrm {arccosh}\left (a x \right )^{3}}\)	\(366\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/arccosh(a*x)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/30*2^(1/2)*(-16*arccosh(a*x)^(5/2)*2^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x-4*arccosh(a*x)^(3/2)*2^(
1/2)*Pi^(1/2)*a^2*x^2-3*2^(1/2)*arccosh(a*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x+8*arccosh(a*x)^3*P
i*erf(2^(1/2)*arccosh(a*x)^(1/2))+8*arccosh(a*x)^3*Pi*erfi(2^(1/2)*arccosh(a*x)^(1/2))+2*arccosh(a*x)^(3/2)*2^
(1/2)*Pi^(1/2))/Pi^(1/2)/a^4/arccosh(a*x)^3+1/15*(-128*(a*x-1)^(1/2)*(a*x+1)^(1/2)*Pi^(1/2)*arccosh(a*x)^(5/2)
*a^3*x^3-16*arccosh(a*x)^(3/2)*Pi^(1/2)*a^4*x^4-6*arccosh(a*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a^3*
x^3+64*(a*x-1)^(1/2)*(a*x+1)^(1/2)*Pi^(1/2)*arccosh(a*x)^(5/2)*a*x+16*arccosh(a*x)^(3/2)*Pi^(1/2)*a^2*x^2+3*ar
ccosh(a*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x+16*arccosh(a*x)^3*Pi*erf(2*arccosh(a*x)^(1/2))+16*ar
ccosh(a*x)^3*Pi*erfi(2*arccosh(a*x)^(1/2))-2*Pi^(1/2)*arccosh(a*x)^(3/2))/Pi^(1/2)/a^4/arccosh(a*x)^3

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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(x^3/arccosh(a*x)^(7/2),x, algorithm="maxima")

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/acosh(a*x)**(7/2),x)

[Out]

Integral(x**3/acosh(a*x)**(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(x^3/arccosh(a*x)^(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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/acosh(a*x)^(7/2),x)

[Out]

int(x^3/acosh(a*x)^(7/2), x)

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