3.415 \(\int \frac{\sqrt{c-a^2 c x^2}}{\cosh ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=201 \[ \frac{2 \sqrt{2 \pi } \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{3 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2 \sqrt{2 \pi } \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{3 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{8 x \sqrt{c-a^2 c x^2}}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \sqrt{c-a^2 c x^2}}{3 a \cosh ^{-1}(a x)^{3/2}} \]

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[c - a^2*c*x^2])/(3*a*ArcCosh[a*x]^(3/2)) - (8*x*Sqrt[c - a^2*c*x^2])/(3*
Sqrt[ArcCosh[a*x]]) + (2*Sqrt[2*Pi]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(3*a*Sqrt[-1 + a*x]*S
qrt[1 + a*x]) + (2*Sqrt[2*Pi]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(3*a*Sqrt[-1 + a*x]*Sqrt[1
+ a*x])

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Rubi [A]  time = 0.223367, antiderivative size = 207, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {5713, 5697, 5666, 3307, 2180, 2204, 2205} \[ \frac{2 \sqrt{2 \pi } \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{3 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2 \sqrt{2 \pi } \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{3 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{8 x \sqrt{c-a^2 c x^2}}{3 \sqrt{\cosh ^{-1}(a x)}}+\frac{2 (1-a x) \sqrt{a x+1} \sqrt{c-a^2 c x^2}}{3 a \sqrt{a x-1} \cosh ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 - a*x)*Sqrt[1 + a*x]*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[-1 + a*x]*ArcCosh[a*x]^(3/2)) - (8*x*Sqrt[c - a^2*c*
x^2])/(3*Sqrt[ArcCosh[a*x]]) + (2*Sqrt[2*Pi]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(3*a*Sqrt[-1
+ a*x]*Sqrt[1 + a*x]) + (2*Sqrt[2*Pi]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(3*a*Sqrt[-1 + a*
x]*Sqrt[1 + a*x])

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((-d)^IntPart[p]*(
d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5697

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol
] :> Simp[(Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)
), x] - Dist[(c*(2*p + 1)*(-(d1*d2))^(p - 1/2)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(b*(n + 1)*Sqrt[1 + c*x]*Sqrt[
-1 + c*x]), Int[x*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2,
e2, p}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && LtQ[n, -1] && IntegerQ[p - 1/2]

Rule 5666

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*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a +
b*x)^(n + 1)*Cosh[x]^(m - 1)*(m - (m + 1)*Cosh[x]^2), x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 3307

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 2180

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] &&  !$UseGamma === True

Rule 2204

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 2205

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]

Rubi steps

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

Mathematica [A]  time = 0.300379, size = 141, normalized size = 0.7 \[ -\frac{2 \sqrt{c-a^2 c x^2} \left (\sqrt{2} \left (-\cosh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-2 \cosh ^{-1}(a x)\right )+\sqrt{2} \cosh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},2 \cosh ^{-1}(a x)\right )+(a x+1) \left (a x+4 a x \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)-1\right )\right )}{3 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*Sqrt[c - a^2*c*x^2]*((1 + a*x)*(-1 + a*x + 4*a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]) + Sqrt[2]*(-ArcC
osh[a*x])^(3/2)*Gamma[1/2, -2*ArcCosh[a*x]] + Sqrt[2]*ArcCosh[a*x]^(3/2)*Gamma[1/2, 2*ArcCosh[a*x]]))/(3*a*Sqr
t[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x]^(3/2))

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Maple [F]  time = 0.499, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{-{a}^{2}c{x}^{2}+c} \left ({\rm arccosh} \left (ax\right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(5/2),x)

[Out]

int((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} c x^{2} + c}}{\operatorname{arcosh}\left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*c*x^2 + c)/arccosh(a*x)^(5/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(5/2),x, algorithm="giac")

[Out]

sage0*x