Integrand size = 10, antiderivative size = 115 \[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=2 a^2 \text {arccosh}(a x)^3+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{x}-\frac {\text {arccosh}(a x)^4}{2 x^2}-6 a^2 \text {arccosh}(a x)^2 \log \left (1+e^{2 \text {arccosh}(a x)}\right )-6 a^2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right ) \]
2*a^2*arccosh(a*x)^3-1/2*arccosh(a*x)^4/x^2-6*a^2*arccosh(a*x)^2*ln(1+(a*x +(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)-6*a^2*arccosh(a*x)*polylog(2,-(a*x+(a*x-1 )^(1/2)*(a*x+1)^(1/2))^2)+3*a^2*polylog(3,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2 ))^2)+2*a*arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x
Time = 0.78 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=-\frac {\text {arccosh}(a x)^4}{2 x^2}+a^2 \left (2 \text {arccosh}(a x)^2 \left (-\text {arccosh}(a x)+\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)}{a x}-3 \log \left (1+e^{-2 \text {arccosh}(a x)}\right )\right )+6 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}(a x)}\right )\right ) \]
-1/2*ArcCosh[a*x]^4/x^2 + a^2*(2*ArcCosh[a*x]^2*(-ArcCosh[a*x] + (Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x])/(a*x) - 3*Log[1 + E^(-2*ArcCosh [a*x])]) + 6*ArcCosh[a*x]*PolyLog[2, -E^(-2*ArcCosh[a*x])] + 3*PolyLog[3, -E^(-2*ArcCosh[a*x])])
Result contains complex when optimal does not.
Time = 0.94 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6298, 6333, 6297, 3042, 26, 4201, 2620, 3011, 2720, 7143}
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.
| \(\displaystyle \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle 2 a \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {\text {arccosh}(a x)^4}{2 x^2}\) |
| \(\Big \downarrow \) 6333 |
| \(\displaystyle 2 a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{x}-3 a \int \frac {\text {arccosh}(a x)^2}{x}dx\right )-\frac {\text {arccosh}(a x)^4}{2 x^2}\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle 2 a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{x}-3 a \int \frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1) \text {arccosh}(a x)^2}{a x}d\text {arccosh}(a x)\right )-\frac {\text {arccosh}(a x)^4}{2 x^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^4}{2 x^2}+2 a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{x}-3 a \int -i \text {arccosh}(a x)^2 \tan (i \text {arccosh}(a x))d\text {arccosh}(a x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^4}{2 x^2}+2 a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{x}+3 i a \int \text {arccosh}(a x)^2 \tan (i \text {arccosh}(a x))d\text {arccosh}(a x)\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^4}{2 x^2}+2 a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{x}+3 i a \left (2 i \int \frac {e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)^2}{1+e^{2 \text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{3} i \text {arccosh}(a x)^3\right )\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^4}{2 x^2}+2 a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{x}+3 i a \left (2 i \left (\frac {1}{2} \text {arccosh}(a x)^2 \log \left (e^{2 \text {arccosh}(a x)}+1\right )-\int \text {arccosh}(a x) \log \left (1+e^{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^4}{2 x^2}+2 a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{x}+3 i a \left (2 i \left (-\frac {1}{2} \int \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)+\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )+\frac {1}{2} \text {arccosh}(a x)^2 \log \left (e^{2 \text {arccosh}(a x)}+1\right )\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^4}{2 x^2}+2 a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{x}+3 i a \left (2 i \left (-\frac {1}{4} \int e^{-2 \text {arccosh}(a x)} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )de^{2 \text {arccosh}(a x)}+\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )+\frac {1}{2} \text {arccosh}(a x)^2 \log \left (e^{2 \text {arccosh}(a x)}+1\right )\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^4}{2 x^2}+2 a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{x}+3 i a \left (2 i \left (\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )-\frac {1}{4} \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )+\frac {1}{2} \text {arccosh}(a x)^2 \log \left (e^{2 \text {arccosh}(a x)}+1\right )\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )\right )\) |
-1/2*ArcCosh[a*x]^4/x^2 + 2*a*((Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^ 3)/x + (3*I)*a*((-1/3*I)*ArcCosh[a*x]^3 + (2*I)*((ArcCosh[a*x]^2*Log[1 + E ^(2*ArcCosh[a*x])])/2 + (ArcCosh[a*x]*PolyLog[2, -E^(2*ArcCosh[a*x])])/2 - PolyLog[3, -E^(2*ArcCosh[a*x])]/4)))
3.1.40.3.1 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_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(f*x)^(m + 1) *(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d1*d2*f*( m + 1))), x] + Simp[b*c*(n/(f*(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Sim p[(d2 + e2*x)^p/(-1 + c*x)^p] Int[(f*x)^(m + 1)*(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, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 0.16 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.30
| method | result | size |
| derivativedivides | \(a^{2} \left (-\frac {\operatorname {arccosh}\left (a x \right )^{3} \left (4 a^{2} x^{2}-4 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +\operatorname {arccosh}\left (a x \right )\right )}{2 a^{2} x^{2}}+4 \operatorname {arccosh}\left (a x \right )^{3}-6 \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-6 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+3 \operatorname {polylog}\left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )\right )\) | \(149\) |
| default | \(a^{2} \left (-\frac {\operatorname {arccosh}\left (a x \right )^{3} \left (4 a^{2} x^{2}-4 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +\operatorname {arccosh}\left (a x \right )\right )}{2 a^{2} x^{2}}+4 \operatorname {arccosh}\left (a x \right )^{3}-6 \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-6 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+3 \operatorname {polylog}\left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )\right )\) | \(149\) |
a^2*(-1/2*arccosh(a*x)^3*(4*a^2*x^2-4*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x+arcc osh(a*x))/a^2/x^2+4*arccosh(a*x)^3-6*arccosh(a*x)^2*ln(1+(a*x+(a*x-1)^(1/2 )*(a*x+1)^(1/2))^2)-6*arccosh(a*x)*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^( 1/2))^2)+3*polylog(3,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))
\[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{3}} \,d x } \]
\[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=\int \frac {\operatorname {acosh}^{4}{\left (a x \right )}}{x^{3}}\, dx \]
\[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{3}} \,d x } \]
-1/2*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^4/x^2 + integrate(2*(a^3*x^2 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x - a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/(a^3*x^5 - a*x^3 + (a^2*x^4 - x^2)*sqrt(a*x + 1)*sqrt(a*x - 1)), x)
Exception generated. \[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^4}{x^3} \,d x \]