Integrand size = 24, antiderivative size = 201 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c-a^2 c x^2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {8 x \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {arccosh}(a x)}}+\frac {2 \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2 \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}} \]
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*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)-2/3*(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)
Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {c-a^2 c x^2} \left ((1+a x) \left (-1+a x+4 a x \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)\right )+\sqrt {2} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-2 \text {arccosh}(a x)\right )+\sqrt {2} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \text {arccosh}(a x)\right )\right )}{3 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)^{3/2}} \]
(-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]*(-ArcCosh[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*Sqrt[ (-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x]^(3/2))
Time = 0.62 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6319, 6300, 25, 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.
\(\displaystyle \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 6319 |
\(\displaystyle \frac {4 a \sqrt {c-a^2 c x^2} \int \frac {x}{\text {arccosh}(a x)^{3/2}}dx}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 6300 |
\(\displaystyle \frac {4 a \sqrt {c-a^2 c x^2} \left (-\frac {2 \int -\frac {\cosh (2 \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {4 a \sqrt {c-a^2 c x^2} \left (\frac {2 \int \frac {\cosh (2 \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {4 a \sqrt {c-a^2 c x^2} \left (-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \int \frac {\sin \left (2 i \text {arccosh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {4 a \sqrt {c-a^2 c x^2} \left (-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 \left (\frac {1}{2} i \int \frac {i e^{-2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \int -\frac {i e^{2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a^2}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {4 a \sqrt {c-a^2 c x^2} \left (-\frac {2 \left (-\frac {1}{2} \int \frac {e^{-2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} \int \frac {e^{2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {4 a \sqrt {c-a^2 c x^2} \left (-\frac {2 \left (-\int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}-\int e^{2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {4 a \sqrt {c-a^2 c x^2} \left (-\frac {2 \left (-\int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}-\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {4 a \sqrt {c-a^2 c x^2} \left (-\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{3 a \text {arccosh}(a x)^{3/2}}\) |
(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[c - a^2*c*x^2])/(3*a*ArcCosh[a*x]^(3 /2)) + (4*a*Sqrt[c - a^2*c*x^2]*((-2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sq rt[ArcCosh[a*x]]) - (2*(-1/2*(Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]]) - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/2))/a^2))/(3*Sqrt[-1 + a*x ]*Sqrt[1 + a*x])
3.5.15.3.1 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_)^(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] + Simp[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] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
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]
\[\int \frac {\sqrt {-a^{2} c \,x^{2}+c}}{\operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\operatorname {acosh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
\[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}} \,d x \]