\(\int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx\) [415]

Optimal result

Integrand size = 24, antiderivative size = 201 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c-a^2 c x^2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {8 x \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {arccosh}(a x)}}+\frac {2 \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2 \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}} \]

[Out]

2/3*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)+2/3*er
fi(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)-2/3*(a*x-1)
^(1/2)*(a*x+1)^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/arccosh(a*x)^(3/2)-8/3*x*(-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5904, 5885, 3388, 2211, 2235, 2236} \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=\frac {2 \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {8 x \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {arccosh}(a x)}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{3 a \text {arccosh}(a x)^{3/2}} \]

[In]

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

[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])

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 5885

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

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]

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

Mathematica [A] (warning: unable to verify)

Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {c-a^2 c x^2} \left ((1+a x) \left (-1+a x+4 a x \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)\right )+\sqrt {2} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-2 \text {arccosh}(a x)\right )+\sqrt {2} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \text {arccosh}(a x)\right )\right )}{3 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)^{3/2}} \]

[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))

Maple [F]

\[\int \frac {\sqrt {-a^{2} c \,x^{2}+c}}{\operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\operatorname {acosh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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