3.2.7 \(\int \frac {1}{\cosh ^{-1}(a x)^{5/2}} \, dx\) [107]

Optimal. Leaf size=89 \[ -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}-\frac {4 x}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {2 \sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{3 a}+\frac {2 \sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{3 a} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5880, 5951, 5881, 3389, 2211, 2235, 2236} \begin {gather*} -\frac {2 \sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{3 a}+\frac {2 \sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{3 a}-\frac {4 x}{3 \sqrt {\cosh ^{-1}(a x)}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2)) - (4*x)/(3*Sqrt[ArcCosh[a*x]]) - (2*Sqrt[Pi]*Erf[Sq
rt[ArcCosh[a*x]]])/(3*a) + (2*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(3*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 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 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 5881

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

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

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Mathematica [A]
time = 0.12, size = 121, normalized size = 1.36 \begin {gather*} -\frac {2 e^{-\cosh ^{-1}(a x)} \left (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)+e^{\cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-\cosh ^{-1}(a x)\right )-e^{\cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},\cosh ^{-1}(a x)\right )\right )}{3 a \cosh ^{-1}(a x)^{3/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]
time = 3.49, size = 84, normalized size = 0.94

method	result	size
default	\(-\frac {2 \left (2 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a x +\mathrm {arccosh}\left (a x \right )^{2} \pi \erf \left (\sqrt {\mathrm {arccosh}\left (a x \right )}\right )-\mathrm {arccosh}\left (a x \right )^{2} \pi \erfi \left (\sqrt {\mathrm {arccosh}\left (a x \right )}\right )+\sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\right )}{3 \sqrt {\pi }\, a \mathrm {arccosh}\left (a x \right )^{2}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccosh(a*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

[Out]

integrate(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(1/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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\operatorname {acosh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/acosh(a*x)**(5/2),x)

[Out]

Integral(acosh(a*x)**(-5/2), 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)^(5/2),x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^(-5/2), 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 )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/acosh(a*x)^(5/2),x)

[Out]

int(1/acosh(a*x)^(5/2), x)

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