3.5.9 \(\int \frac {(c-a^2 c x^2)^{3/2}}{\cosh ^{-1}(a x)^{3/2}} \, dx\) [409]

Optimal. Leaf size=286 \[ -\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{3/2}}{a \sqrt {\cosh ^{-1}(a x)}}+\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{a \sqrt {-1+a x} \sqrt {1+a x}} \]

[Out]

-1/2*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)+1/2
*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)+1/4*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)-1/4*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)-2*(-a^2*c*x^2+c)^(3/2)*(a*x-1)^(1/2)*(a*x
+1)^(1/2)/a/arccosh(a*x)^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5904, 5912, 5952, 5556, 3389, 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 )}{4 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 )}{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 )}{4 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 )}{a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{a \sqrt {\cosh ^{-1}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c - a^2*c*x^2)^(3/2))/(a*Sqrt[ArcCosh[a*x]]) + (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^
2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[
2]*Sqrt[ArcCosh[a*x]]])/(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh
[a*x]]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]
]])/(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 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 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5904

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

Rule 5912

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(
x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]

Rule 5952

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

Rubi steps

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

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Mathematica [A]
time = 0.33, size = 239, normalized size = 0.84 \begin {gather*} -\frac {c e^{-4 \cosh ^{-1}(a x)} \sqrt {c-a^2 c x^2} \left (-1-14 e^{4 \cosh ^{-1}(a x)}-e^{8 \cosh ^{-1}(a x)}+16 a^2 e^{4 \cosh ^{-1}(a x)} x^2+4 e^{4 \cosh ^{-1}(a x)} \sqrt {2 \pi } \sqrt {\cosh ^{-1}(a x)} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )-4 e^{4 \cosh ^{-1}(a x)} \sqrt {2 \pi } \sqrt {\cosh ^{-1}(a x)} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )+2 e^{4 \cosh ^{-1}(a x)} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 \cosh ^{-1}(a x)\right )+2 e^{4 \cosh ^{-1}(a x)} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 \cosh ^{-1}(a x)\right )\right )}{8 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)/ArcCosh[a*x]^(3/2),x]

[Out]

-1/8*(c*Sqrt[c - a^2*c*x^2]*(-1 - 14*E^(4*ArcCosh[a*x]) - E^(8*ArcCosh[a*x]) + 16*a^2*E^(4*ArcCosh[a*x])*x^2 +
4*E^(4*ArcCosh[a*x])*Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] - 4*E^(4*ArcCosh[a*x])*Sqr
t[2*Pi]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 2*E^(4*ArcCosh[a*x])*Sqrt[-ArcCosh[a*x]]*Gamma[1
/2, -4*ArcCosh[a*x]] + 2*E^(4*ArcCosh[a*x])*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 4*ArcCosh[a*x]]))/(a*E^(4*ArcCosh[a*
x])*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((-a^2*c*x^2 + c)^(3/2)/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((-a^2*c*x^2+c)^(3/2)/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 {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\operatorname {acosh}^{\frac {3}{2}}{\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)**(3/2),x)

[Out]

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

[Out]

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

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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}}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}} \,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)^(3/2),x)

[Out]

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

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