Integrand size = 8, antiderivative size = 68 \[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\text {arccosh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{a} \]
erf(arccosh(a*x)^(1/2))*Pi^(1/2)/a+erfi(arccosh(a*x)^(1/2))*Pi^(1/2)/a-2*( a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(1/2)
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\frac {-2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )-\sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )}{a \sqrt {\text {arccosh}(a x)}} \]
(-2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -ArcCosh[a*x]] - Sqrt[ArcCosh[a*x]]*Gamma[1/2, ArcCosh[a*x]])/(a*Sqrt[ArcC osh[a*x]])
Time = 0.57 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6295, 6368, 3042, 3788, 26, 2611, 2633, 2634}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 6295 |
\(\displaystyle 2 a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}dx-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 6368 |
\(\displaystyle \frac {2 \int \frac {a x}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a}\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \left (\frac {1}{2} i \int -\frac {i e^{\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \int \frac {i e^{-\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {e^{-\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)+\frac {1}{2} \int \frac {e^{\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {2 \left (\int e^{-\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}+\int e^{\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {2 \left (\int e^{-\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {2 \left (\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\) |
(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]]) + (2*((Sqrt[Pi]*E rf[Sqrt[ArcCosh[a*x]]])/2 + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/2))/a
3.2.1.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[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]
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]
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] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [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]
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] - Simp[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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x _))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p] Subst[In t[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]
Time = 0.35 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {-2 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}+\operatorname {arccosh}\left (a x \right ) \pi \,\operatorname {erf}\left (\sqrt {\operatorname {arccosh}\left (a x \right )}\right )+\operatorname {arccosh}\left (a x \right ) \pi \,\operatorname {erfi}\left (\sqrt {\operatorname {arccosh}\left (a x \right )}\right )}{\sqrt {\pi }\, a \,\operatorname {arccosh}\left (a x \right )}\) | \(66\) |
(-2*arccosh(a*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)+arccosh(a*x)*P i*erf(arccosh(a*x)^(1/2))+arccosh(a*x)*Pi*erfi(arccosh(a*x)^(1/2)))/Pi^(1/ 2)/a/arccosh(a*x)
Exception generated. \[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {1}{\operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
\[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}} \,d x \]