\(\int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx\) [567]

Optimal result

Integrand size = 12, antiderivative size = 120 \[ \int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=-\frac {2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c} \]

[Out]

exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c+erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1
/2)/b^(3/2)/c/exp(a/b)-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))^(1/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5880, 5953, 3388, 2211, 2236, 2235} \[ \int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {\sqrt {\pi } e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}} \]

[In]

Int[(a + b*ArcCosh[c*x])^(-3/2),x]

[Out]

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

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

Mathematica [F]

\[ \int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {1}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \]

[In]

int(1/(a + b*acosh(c*x))^(3/2),x)

[Out]

int(1/(a + b*acosh(c*x))^(3/2), x)