Integrand size = 24, antiderivative size = 170 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {arccosh}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{a \sqrt {-1+a x} \sqrt {1+a x}} \]
-1/2*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*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)-2*(a*x-1)^(1 /2)*(a*x+1)^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/arccosh(a*x)^(1/2)
Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (4-4 a^2 x^2-\sqrt {2 \pi } \sqrt {\text {arccosh}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\sqrt {2 \pi } \sqrt {\text {arccosh}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{2 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\text {arccosh}(a x)}} \]
(Sqrt[c - a^2*c*x^2]*(4 - 4*a^2*x^2 - Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erf[Sq rt[2]*Sqrt[ArcCosh[a*x]]] + Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqr t[ArcCosh[a*x]]]))/(2*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[ a*x]])
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.87, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6319, 6302, 5971, 27, 3042, 26, 3789, 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.
\(\displaystyle \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 6319 |
\(\displaystyle \frac {4 a \sqrt {c-a^2 c x^2} \int \frac {x}{\sqrt {\text {arccosh}(a x)}}dx}{\sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 6302 |
\(\displaystyle \frac {4 \sqrt {c-a^2 c x^2} \int \frac {a x \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \frac {4 \sqrt {c-a^2 c x^2} \int \frac {\sinh (2 \text {arccosh}(a x))}{2 \sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 \sqrt {c-a^2 c x^2} \int \frac {\sinh (2 \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \sqrt {c-a^2 c x^2} \int -\frac {i \sin (2 i \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \sqrt {c-a^2 c x^2} \int \frac {\sin (2 i \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \sqrt {c-a^2 c x^2} \left (\frac {1}{2} i \int \frac {e^{2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \int \frac {e^{-2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a \sqrt {a x-1} \sqrt {a x+1}}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \sqrt {c-a^2 c x^2} \left (i \int e^{2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}-i \int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a \sqrt {a x-1} \sqrt {a x+1}}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \sqrt {c-a^2 c x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-i \int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a \sqrt {a x-1} \sqrt {a x+1}}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \sqrt {c-a^2 c x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a \sqrt {a x-1} \sqrt {a x+1}}\) |
(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[c - a^2*c*x^2])/(a*Sqrt[ArcCosh[a*x] ]) - ((2*I)*Sqrt[c - a^2*c*x^2]*((-1/2*I)*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcC osh[a*x]]] + (I/2)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]]))/(a*Sqrt[- 1 + a*x]*Sqrt[1 + a*x])
3.5.10.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
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*A rcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Si mp[(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*ArcCosh[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]
\[\int \frac {\sqrt {-a^{2} c \,x^{2}+c}}{\operatorname {arccosh}\left (a x \right )^{\frac {3}{2}}}d x\]
Exception generated. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
\[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}} \,d x \]