\(\int \frac {x}{\text {arccosh}(a x)^{7/2}} \, dx\) [112]

Optimal result

Integrand size = 10, antiderivative size = 157 \[ \int \frac {x}{\text {arccosh}(a x)^{7/2}} \, dx=-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {4}{15 a^2 \text {arccosh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arccosh}(a x)^{3/2}}-\frac {32 x \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}+\frac {8 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{15 a^2}+\frac {8 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{15 a^2} \]

[Out]

4/15/a^2/arccosh(a*x)^(3/2)-8/15*x^2/arccosh(a*x)^(3/2)+8/15*erf(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/
a^2+8/15*erfi(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^2-2/5*x*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x
)^(5/2)-32/15*x*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5886, 5951, 5885, 3388, 2211, 2235, 2236, 5893} \[ \int \frac {x}{\text {arccosh}(a x)^{7/2}} \, dx=\frac {8 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{15 a^2}+\frac {8 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{15 a^2}+\frac {4}{15 a^2 \text {arccosh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arccosh}(a x)^{3/2}}-\frac {32 x \sqrt {a x-1} \sqrt {a x+1}}{15 a \sqrt {\text {arccosh}(a x)}}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}} \]

[In]

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

[Out]

(-2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2)) + 4/(15*a^2*ArcCosh[a*x]^(3/2)) - (8*x^2)/(15*Arc
Cosh[a*x]^(3/2)) - (32*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(15*a*Sqrt[ArcCosh[a*x]]) + (8*Sqrt[2*Pi]*Erf[Sqrt[2]*S
qrt[ArcCosh[a*x]]])/(15*a^2) + (8*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(15*a^2)

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.58 \[ \int \frac {x}{\text {arccosh}(a x)^{7/2}} \, dx=-\frac {\frac {4 \cosh (2 \text {arccosh}(a x))}{\text {arccosh}(a x)^{3/2}}-8 \sqrt {2 \pi } \left (\text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )+\frac {\left (3+16 \text {arccosh}(a x)^2\right ) \sinh (2 \text {arccosh}(a x))}{\text {arccosh}(a x)^{5/2}}}{15 a^2} \]

[In]

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

[Out]

-1/15*((4*Cosh[2*ArcCosh[a*x]])/ArcCosh[a*x]^(3/2) - 8*Sqrt[2*Pi]*(Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + Erfi[Sqrt
[2]*Sqrt[ArcCosh[a*x]]]) + ((3 + 16*ArcCosh[a*x]^2)*Sinh[2*ArcCosh[a*x]])/ArcCosh[a*x]^(5/2))/a^2

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {\sqrt {2}\, \left (16 \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +4 \sqrt {2}\, \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{2} x^{2}+3 \sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x -8 \operatorname {arccosh}\left (a x \right )^{3} \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-8 \operatorname {arccosh}\left (a x \right )^{3} \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-2 \sqrt {2}\, \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\right )}{15 \sqrt {\pi }\, a^{2} \operatorname {arccosh}\left (a x \right )^{3}}\)	\(153\)

[In]

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

[Out]

-1/15*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*2^(1/2)*arccosh(a*x)^(
3/2)*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*2^(1/2)*arccosh(a*x)^
(3/2)*Pi^(1/2))/Pi^(1/2)/a^2/arccosh(a*x)^3

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {x}{\text {arccosh}(a x)^{7/2}} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]

[In]

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

[Out]

integrate(x/arccosh(a*x)^(7/2), x)

Giac [F]

\[ \int \frac {x}{\text {arccosh}(a x)^{7/2}} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]

[In]

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

[Out]

integrate(x/arccosh(a*x)^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\text {arccosh}(a x)^{7/2}} \, dx=\int \frac {x}{{\mathrm {acosh}\left (a\,x\right )}^{7/2}} \,d x \]

[In]

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

[Out]

int(x/acosh(a*x)^(7/2), x)