3.5.3 \(\int \frac {(c-a^2 c x^2)^{3/2}}{\sqrt {\cosh ^{-1}(a x)}} \, dx\) [403]

Optimal. Leaf size=294 \[ -\frac {3 c \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a \sqrt {-1+a x} \sqrt {1+a x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5906, 3393, 3388, 2211, 2235, 2236} \begin {gather*} -\frac {\sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{32 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {\sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{32 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {\sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {3 c \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{4 a \sqrt {a x-1} \sqrt {a x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*c*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*
x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sq
rt[2]*Sqrt[ArcCosh[a*x]]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[Ar
cCosh[a*x]]])/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCos
h[a*x]]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 5906

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.19, size = 153, normalized size = 0.52 \begin {gather*} -\frac {c \sqrt {c-a^2 c x^2} \left (\sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 \cosh ^{-1}(a x)\right )-4 \sqrt {2} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 \cosh ^{-1}(a x)\right )+\sqrt {\cosh ^{-1}(a x)} \left (24 \sqrt {\cosh ^{-1}(a x)}+4 \sqrt {2} \Gamma \left (\frac {1}{2},2 \cosh ^{-1}(a x)\right )-\Gamma \left (\frac {1}{2},4 \cosh ^{-1}(a x)\right )\right )\right )}{32 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\cosh ^{-1}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/32*(c*Sqrt[c - a^2*c*x^2]*(Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -4*ArcCosh[a*x]] - 4*Sqrt[2]*Sqrt[-ArcCosh[a*x]]*
Gamma[1/2, -2*ArcCosh[a*x]] + Sqrt[ArcCosh[a*x]]*(24*Sqrt[ArcCosh[a*x]] + 4*Sqrt[2]*Gamma[1/2, 2*ArcCosh[a*x]]
- Gamma[1/2, 4*ArcCosh[a*x]])))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])

________________________________________________________________________________________

Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\sqrt {\mathrm {arccosh}\left (a x \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(1/2),x)

[Out]

int((-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(1/2),x)

________________________________________________________________________________________

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

[Out]

integrate((-a^2*c*x^2 + c)^(3/2)/sqrt(arccosh(a*x)), x)

________________________________________________________________________________________

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((-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\sqrt {\operatorname {acosh}{\left (a x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-c*(a*x - 1)*(a*x + 1))**(3/2)/sqrt(acosh(a*x)), x)

________________________________________________________________________________________

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((-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(1/2),x, algorithm="giac")

[Out]

integrate((-a^2*c*x^2 + c)^(3/2)/sqrt(arccosh(a*x)), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}}{\sqrt {\mathrm {acosh}\left (a\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________