Integrand size = 12, antiderivative size = 166 \[ \int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {8 x}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arccosh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{6 a^3}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{2 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{2 a^3} \] Output:
-2/3*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(3/2)+8/3*x/a^2/arccos h(a*x)^(1/2)-4*x^3/arccosh(a*x)^(1/2)-1/6*Pi^(1/2)*erf(arccosh(a*x)^(1/2)) /a^3-1/2*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*arccosh(a*x)^(1/2))/a^3+1/6*Pi^(1/2) *erfi(arccosh(a*x)^(1/2))/a^3+1/2*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*arccosh(a* x)^(1/2))/a^3
Time = 0.44 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.17 \[ \int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx=\frac {-\sqrt {\frac {-1+a x}{1+a x}} (1+a x)-3 e^{-3 \text {arccosh}(a x)} \text {arccosh}(a x)-e^{-\text {arccosh}(a x)} \text {arccosh}(a x)-e^{\text {arccosh}(a x)} \text {arccosh}(a x)-3 e^{3 \text {arccosh}(a x)} \text {arccosh}(a x)-3 \sqrt {3} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-3 \text {arccosh}(a x)\right )-(-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )+\text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )+3 \sqrt {3} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},3 \text {arccosh}(a x)\right )-\sinh (3 \text {arccosh}(a x))}{6 a^3 \text {arccosh}(a x)^{3/2}} \] Input:
Integrate[x^2/ArcCosh[a*x]^(5/2),x]
Output:
(-(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)) - (3*ArcCosh[a*x])/E^(3*ArcCosh[a *x]) - ArcCosh[a*x]/E^ArcCosh[a*x] - E^ArcCosh[a*x]*ArcCosh[a*x] - 3*E^(3* ArcCosh[a*x])*ArcCosh[a*x] - 3*Sqrt[3]*(-ArcCosh[a*x])^(3/2)*Gamma[1/2, -3 *ArcCosh[a*x]] - (-ArcCosh[a*x])^(3/2)*Gamma[1/2, -ArcCosh[a*x]] + ArcCosh [a*x]^(3/2)*Gamma[1/2, ArcCosh[a*x]] + 3*Sqrt[3]*ArcCosh[a*x]^(3/2)*Gamma[ 1/2, 3*ArcCosh[a*x]] - Sinh[3*ArcCosh[a*x]])/(6*a^3*ArcCosh[a*x]^(3/2))
Result contains complex when optimal does not.
Time = 1.41 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.36, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6301, 6366, 6296, 3042, 26, 3789, 2611, 2633, 2634, 6302, 5971, 2009}
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.
\(\displaystyle \int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 6301 |
\(\displaystyle 2 a \int \frac {x^3}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}dx-\frac {4 \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}dx}{3 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 6366 |
\(\displaystyle 2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {4 \left (\frac {2 \int \frac {1}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 6296 |
\(\displaystyle -\frac {4 \left (\frac {2 \int \frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}+2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {4 \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \int -\frac {i \sin (i \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}\right )}{3 a}+2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {4 \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \int \frac {\sin (i \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}\right )}{3 a}+2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle -\frac {4 \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \int \frac {e^{\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \int \frac {e^{-\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a^2}\right )}{3 a}+2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle -\frac {4 \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (i \int e^{\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}-i \int e^{-\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a^2}\right )}{3 a}+2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle -\frac {4 \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-i \int e^{-\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a^2}\right )}{3 a}+2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle 2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {4 \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )\right )}{a^2}\right )}{3 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 6302 |
\(\displaystyle 2 a \left (\frac {6 \int \frac {a^2 x^2 \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^4}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {4 \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )\right )}{a^2}\right )}{3 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle 2 a \left (\frac {6 \int \left (\frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1)}{4 \sqrt {\text {arccosh}(a x)}}+\frac {\sinh (3 \text {arccosh}(a x))}{4 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^4}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {4 \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )\right )}{a^2}\right )}{3 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 a \left (\frac {6 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^4}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {4 \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )\right )}{a^2}\right )}{3 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\) |
Input:
Int[x^2/ArcCosh[a*x]^(5/2),x]
Output:
(-2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2)) - (4*((-2*x )/(a*Sqrt[ArcCosh[a*x]]) - ((2*I)*((-1/2*I)*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x] ]] + (I/2)*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]]))/a^2))/(3*a) + 2*a*((-2*x^3) /(a*Sqrt[ArcCosh[a*x]]) + (6*(-1/8*(Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]]) - (S qrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/8 + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh [a*x]]])/8 + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/8))/a^4)
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_.) + (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_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
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[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcCosh[c*x ])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]) ), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
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_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1 _) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x ]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Simp[f*(m/(b*c*(n + 1)))*Simp [Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Int[ (f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e 1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
\[\int \frac {x^{2}}{\operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}}d x\]
Input:
int(x^2/arccosh(a*x)^(5/2),x)
Output:
int(x^2/arccosh(a*x)^(5/2),x)
Exception generated. \[ \int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2/arccosh(a*x)^(5/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {x^{2}}{\operatorname {acosh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \] Input:
integrate(x**2/acosh(a*x)**(5/2),x)
Output:
Integral(x**2/acosh(a*x)**(5/2), x)
\[ \int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^2/arccosh(a*x)^(5/2),x, algorithm="maxima")
Output:
integrate(x^2/arccosh(a*x)^(5/2), x)
\[ \int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^2/arccosh(a*x)^(5/2),x, algorithm="giac")
Output:
integrate(x^2/arccosh(a*x)^(5/2), x)
Timed out. \[ \int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {x^2}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}} \,d x \] Input:
int(x^2/acosh(a*x)^(5/2),x)
Output:
int(x^2/acosh(a*x)^(5/2), x)
\[ \int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx=\frac {2 \mathit {acosh} \left (a x \right )^{2} \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{3}}{\mathit {acosh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{2}}d x \right ) a^{2}-\frac {4 \mathit {acosh} \left (a x \right )^{2} \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x}{\mathit {acosh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{2}}d x \right )}{3}-\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{2}}{3}}{\mathit {acosh} \left (a x \right )^{2} a} \] Input:
int(x^2/acosh(a*x)^(5/2),x)
Output:
(2*(3*acosh(a*x)**2*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x**3 )/(acosh(a*x)**2*a**2*x**2 - acosh(a*x)**2),x)*a**2 - 2*acosh(a*x)**2*int( (sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x)/(acosh(a*x)**2*a**2*x**2 - acosh(a*x)**2),x) - sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x**2))/ (3*acosh(a*x)**2*a)