Integrand size = 10, antiderivative size = 89 \[ \int \frac {x}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\text {arccosh}(a x)}}+\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{a^2}+\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{a^2} \]
1/2*erf(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^2+1/2*erfi(2^(1/2)* arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^2-2*x*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a /arccosh(a*x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.71 \[ \int \frac {x}{\text {arccosh}(a x)^{3/2}} \, dx=\frac {\sqrt {\frac {\pi }{2}} \left (\text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )-\frac {\sinh (2 \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}}{a^2} \]
(Sqrt[Pi/2]*(Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + Erfi[Sqrt[2]*Sqrt[ArcCosh[a *x]]]) - Sinh[2*ArcCosh[a*x]]/Sqrt[ArcCosh[a*x]])/a^2
Time = 0.38 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {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.
\(\displaystyle \int \frac {x}{\text {arccosh}(a x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 6300 |
\(\displaystyle -\frac {2 \int -\frac {\cosh (2 \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 \int \frac {\cosh (2 \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \int \frac {\sin \left (2 i \text {arccosh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle -\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 \left (\frac {1}{2} i \int \frac {i e^{-2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \int -\frac {i e^{2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {2 \left (-\frac {1}{2} \int \frac {e^{-2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} \int \frac {e^{2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle -\frac {2 \left (-\int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}-\int e^{2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle -\frac {2 \left (-\int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}-\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle -\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\) |
(-2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]]) - (2*(-1/2*(Sqr t[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]]) - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[A rcCosh[a*x]]])/2))/a^2
3.1.100.3.1 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]
Time = 0.41 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (2 \sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x -\operatorname {arccosh}\left (a x \right ) \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-\operatorname {arccosh}\left (a x \right ) \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )\right )}{2 \sqrt {\pi }\, a^{2} \operatorname {arccosh}\left (a x \right )}\) | \(85\) |
-1/2*2^(1/2)*(2*2^(1/2)*arccosh(a*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^ (1/2)*a*x-arccosh(a*x)*Pi*erf(2^(1/2)*arccosh(a*x)^(1/2))-arccosh(a*x)*Pi* erfi(2^(1/2)*arccosh(a*x)^(1/2)))/Pi^(1/2)/a^2/arccosh(a*x)
Exception generated. \[ \int \frac {x}{\text {arccosh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {x}{\operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
\[ \int \frac {x}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {x}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}} \,d x \]