3.2.1 \(\int \frac {1}{\cosh ^{-1}(a x)^{3/2}} \, dx\) [101]

Optimal. Leaf size=68 \[ -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}+\frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{a}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{a} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5880, 5953, 3388, 2211, 2235, 2236} \begin {gather*} \frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{a}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\cosh ^{-1}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]]) + (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/a + (Sqrt[Pi]*Er
fi[Sqrt[ArcCosh[a*x]]])/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 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)^{3/2}} \, dx &=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}+(2 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}}{a \sqrt {\cosh ^{-1}(a x)}}+\frac {2 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}+\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a}+\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}+\frac {2 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{a}+\frac {2 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{a}\\ &=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{a}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 76, normalized size = 1.12 \begin {gather*} \frac {-2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-\cosh ^{-1}(a x)\right )-\sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},\cosh ^{-1}(a x)\right )}{a \sqrt {\cosh ^{-1}(a x)}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -ArcCosh[a*x]] - Sqrt[ArcCosh[a*x]]*
Gamma[1/2, ArcCosh[a*x]])/(a*Sqrt[ArcCosh[a*x]])

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Maple [A]
time = 3.67, size = 66, normalized size = 0.97

method	result	size
default	\(\frac {-2 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}+\mathrm {arccosh}\left (a x \right ) \pi \erf \left (\sqrt {\mathrm {arccosh}\left (a x \right )}\right )+\mathrm {arccosh}\left (a x \right ) \pi \erfi \left (\sqrt {\mathrm {arccosh}\left (a x \right )}\right )}{\sqrt {\pi }\, a \,\mathrm {arccosh}\left (a x \right )}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccosh(a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

[Out]

integrate(arccosh(a*x)^(-3/2), 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)^(3/2),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 {3}{2}}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/acosh(a*x)**(3/2),x)

[Out]

Integral(acosh(a*x)**(-3/2), 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)^(3/2),x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^(-3/2), 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 )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/acosh(a*x)^(3/2),x)

[Out]

int(1/acosh(a*x)^(3/2), x)

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