3.2.13 \(\int \frac {1}{\cosh ^{-1}(a x)^{7/2}} \, dx\) [113]

Optimal. Leaf size=122 \[ -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac {4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\cosh ^{-1}(a x)}}+\frac {4 \sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{15 a}+\frac {4 \sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{15 a} \]

[Out]

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

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Rubi [A]
time = 0.26, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5880, 5951, 5953, 3388, 2211, 2235, 2236} \begin {gather*} \frac {4 \sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{15 a}+\frac {4 \sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{15 a}-\frac {4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac {8 \sqrt {a x-1} \sqrt {a x+1}}{15 a \sqrt {\cosh ^{-1}(a x)}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 5880

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

Rule 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[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, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]

Rubi steps

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

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Mathematica [A]
time = 0.14, size = 147, normalized size = 1.20 \begin {gather*} -\frac {2 e^{-\cosh ^{-1}(a x)} \left (3 e^{\cosh ^{-1}(a x)} \sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\cosh ^{-1}(a x)+e^{2 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)-2 \cosh ^{-1}(a x)^2+2 e^{2 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2-2 e^{\cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{5/2} \Gamma \left (\frac {1}{2},-\cosh ^{-1}(a x)\right )+2 e^{\cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{5/2} \Gamma \left (\frac {1}{2},\cosh ^{-1}(a x)\right )\right )}{15 a \cosh ^{-1}(a x)^{5/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*(3*E^ArcCosh[a*x]*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + ArcCosh[a*x] + E^(2*ArcCosh[a*x])*ArcCosh[a*x] -
2*ArcCosh[a*x]^2 + 2*E^(2*ArcCosh[a*x])*ArcCosh[a*x]^2 - 2*E^ArcCosh[a*x]*(-ArcCosh[a*x])^(5/2)*Gamma[1/2, -Ar
cCosh[a*x]] + 2*E^ArcCosh[a*x]*ArcCosh[a*x]^(5/2)*Gamma[1/2, ArcCosh[a*x]]))/(15*a*E^ArcCosh[a*x]*ArcCosh[a*x]
^(5/2))

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Maple [A]
time = 3.67, size = 111, normalized size = 0.91

method	result	size
default	\(\frac {-\frac {8 \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {\pi }}{15}-\frac {4 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a x}{15}+\frac {4 \mathrm {arccosh}\left (a x \right )^{3} \pi \erf \left (\sqrt {\mathrm {arccosh}\left (a x \right )}\right )}{15}+\frac {4 \mathrm {arccosh}\left (a x \right )^{3} \pi \erfi \left (\sqrt {\mathrm {arccosh}\left (a x \right )}\right )}{15}-\frac {2 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}}{5}}{\sqrt {\pi }\, a \mathrm {arccosh}\left (a x \right )^{3}}\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccosh(a*x)^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/15*(-4*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)^(5/2)*Pi^(1/2)-2*arccosh(a*x)^(3/2)*Pi^(1/2)*a*x+2*arccosh(a
*x)^3*Pi*erf(arccosh(a*x)^(1/2))+2*arccosh(a*x)^3*Pi*erfi(arccosh(a*x)^(1/2))-3*arccosh(a*x)^(1/2)*Pi^(1/2)*(a
*x+1)^(1/2)*(a*x-1)^(1/2))/Pi^(1/2)/a/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(1/arccosh(a*x)^(7/2),x, algorithm="maxima")

[Out]

integrate(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(1/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 {1}{\operatorname {acosh}^{\frac {7}{2}}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/acosh(a*x)**(7/2),x)

[Out]

Integral(acosh(a*x)**(-7/2), x)

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Giac [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(1/arccosh(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^(-7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/acosh(a*x)^(7/2),x)

[Out]

int(1/acosh(a*x)^(7/2), x)

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