Integrand size = 10, antiderivative size = 183 \[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\frac {a^2 \text {arccosh}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x^2}-\frac {\text {arccosh}(a x)^3}{3 x^3}+a^3 \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )-a^3 \arctan \left (\sqrt {-1+a x} \sqrt {1+a x}\right )-i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+i a^3 \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-i a^3 \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right ) \] Output:
a^2*arccosh(a*x)/x+1/2*a*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)^2/x^2-1/ 3*arccosh(a*x)^3/x^3+a^3*arccosh(a*x)^2*arctan(a*x+(a*x-1)^(1/2)*(a*x+1)^( 1/2))-a^3*arctan((a*x-1)^(1/2)*(a*x+1)^(1/2))-I*a^3*arccosh(a*x)*polylog(2 ,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+I*a^3*arccosh(a*x)*polylog(2,I*(a*x +(a*x-1)^(1/2)*(a*x+1)^(1/2)))+I*a^3*polylog(3,-I*(a*x+(a*x-1)^(1/2)*(a*x+ 1)^(1/2)))-I*a^3*polylog(3,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))
Time = 0.40 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.10 \[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\frac {1}{6} \left (\frac {6 a^2 \text {arccosh}(a x)}{x}+\frac {3 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)^2}{x^2}-\frac {2 \text {arccosh}(a x)^3}{x^3}-3 i a^3 \left (-4 i \arctan \left (\tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )\right )+\text {arccosh}(a x)^2 \log \left (1-i e^{-\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \log \left (1+i e^{-\text {arccosh}(a x)}\right )+2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )-2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{-\text {arccosh}(a x)}\right )\right )\right ) \] Input:
Integrate[ArcCosh[a*x]^3/x^4,x]
Output:
((6*a^2*ArcCosh[a*x])/x + (3*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCos h[a*x]^2)/x^2 - (2*ArcCosh[a*x]^3)/x^3 - (3*I)*a^3*((-4*I)*ArcTan[Tanh[Arc Cosh[a*x]/2]] + ArcCosh[a*x]^2*Log[1 - I/E^ArcCosh[a*x]] - ArcCosh[a*x]^2* Log[1 + I/E^ArcCosh[a*x]] + 2*ArcCosh[a*x]*PolyLog[2, (-I)/E^ArcCosh[a*x]] - 2*ArcCosh[a*x]*PolyLog[2, I/E^ArcCosh[a*x]] + 2*PolyLog[3, (-I)/E^ArcCo sh[a*x]] - 2*PolyLog[3, I/E^ArcCosh[a*x]]))/6
Time = 1.71 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.96, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6298, 6348, 6298, 103, 218, 6362, 3042, 4668, 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.
\(\displaystyle \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx\) |
\(\Big \downarrow \) 6298 |
\(\displaystyle a \int \frac {\text {arccosh}(a x)^2}{x^3 \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {\text {arccosh}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 6348 |
\(\displaystyle a \left (\frac {1}{2} a^2 \int \frac {\text {arccosh}(a x)^2}{x \sqrt {a x-1} \sqrt {a x+1}}dx-a \int \frac {\text {arccosh}(a x)}{x^2}dx+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 x^2}\right )-\frac {\text {arccosh}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 6298 |
\(\displaystyle a \left (\frac {1}{2} a^2 \int \frac {\text {arccosh}(a x)^2}{x \sqrt {a x-1} \sqrt {a x+1}}dx-a \left (a \int \frac {1}{x \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {\text {arccosh}(a x)}{x}\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 x^2}\right )-\frac {\text {arccosh}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle a \left (\frac {1}{2} a^2 \int \frac {\text {arccosh}(a x)^2}{x \sqrt {a x-1} \sqrt {a x+1}}dx-a \left (a^2 \int \frac {1}{(a x-1) (a x+1) a+a}d\left (\sqrt {a x-1} \sqrt {a x+1}\right )-\frac {\text {arccosh}(a x)}{x}\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 x^2}\right )-\frac {\text {arccosh}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle a \left (\frac {1}{2} a^2 \int \frac {\text {arccosh}(a x)^2}{x \sqrt {a x-1} \sqrt {a x+1}}dx-a \left (a \arctan \left (\sqrt {a x-1} \sqrt {a x+1}\right )-\frac {\text {arccosh}(a x)}{x}\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 x^2}\right )-\frac {\text {arccosh}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 6362 |
\(\displaystyle a \left (\frac {1}{2} a^2 \int \frac {\text {arccosh}(a x)^2}{a x}d\text {arccosh}(a x)-a \left (a \arctan \left (\sqrt {a x-1} \sqrt {a x+1}\right )-\frac {\text {arccosh}(a x)}{x}\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 x^2}\right )-\frac {\text {arccosh}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\text {arccosh}(a x)^3}{3 x^3}+a \left (\frac {1}{2} a^2 \int \text {arccosh}(a x)^2 \csc \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )d\text {arccosh}(a x)-a \left (a \arctan \left (\sqrt {a x-1} \sqrt {a x+1}\right )-\frac {\text {arccosh}(a x)}{x}\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 x^2}\right )\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle -\frac {\text {arccosh}(a x)^3}{3 x^3}+a \left (\frac {1}{2} a^2 \left (-2 i \int \text {arccosh}(a x) \log \left (1-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 i \int \text {arccosh}(a x) \log \left (1+i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )-a \left (a \arctan \left (\sqrt {a x-1} \sqrt {a x+1}\right )-\frac {\text {arccosh}(a x)}{x}\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 x^2}\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {\text {arccosh}(a x)^3}{3 x^3}+a \left (\frac {1}{2} a^2 \left (2 i \left (\int \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-2 i \left (\int \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )+2 \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )-a \left (a \arctan \left (\sqrt {a x-1} \sqrt {a x+1}\right )-\frac {\text {arccosh}(a x)}{x}\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 x^2}\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {\text {arccosh}(a x)^3}{3 x^3}+a \left (\frac {1}{2} a^2 \left (2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )+2 \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )-a \left (a \arctan \left (\sqrt {a x-1} \sqrt {a x+1}\right )-\frac {\text {arccosh}(a x)}{x}\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 x^2}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {\text {arccosh}(a x)^3}{3 x^3}+a \left (\frac {1}{2} a^2 \left (2 \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )+2 i \left (\operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-2 i \left (\operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )\right )-a \left (a \arctan \left (\sqrt {a x-1} \sqrt {a x+1}\right )-\frac {\text {arccosh}(a x)}{x}\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 x^2}\right )\) |
Input:
Int[ArcCosh[a*x]^3/x^4,x]
Output:
-1/3*ArcCosh[a*x]^3/x^3 + a*((Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2) /(2*x^2) - a*(-(ArcCosh[a*x]/x) + a*ArcTan[Sqrt[-1 + a*x]*Sqrt[1 + a*x]]) + (a^2*(2*ArcCosh[a*x]^2*ArcTan[E^ArcCosh[a*x]] + (2*I)*(-(ArcCosh[a*x]*Po lyLog[2, (-I)*E^ArcCosh[a*x]]) + PolyLog[3, (-I)*E^ArcCosh[a*x]]) - (2*I)* (-(ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]]) + PolyLog[3, I*E^ArcCosh[a*x ]])))/2)
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 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[c^2*((m + 2*p + 3)/(f^2*(m + 1))) Int[(f*x)^(m + 2)* (d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp[b*c*(n/(f *(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(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*ArcCos h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && Eq Q[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && ILtQ[m, -1]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1 _.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/c^(m + 1))*Simp[ Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Subst [Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && Inte gerQ[m]
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]
\[\int \frac {\operatorname {arccosh}\left (a x \right )^{3}}{x^{4}}d x\]
Input:
int(arccosh(a*x)^3/x^4,x)
Output:
int(arccosh(a*x)^3/x^4,x)
\[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{4}} \,d x } \] Input:
integrate(arccosh(a*x)^3/x^4,x, algorithm="fricas")
Output:
integral(arccosh(a*x)^3/x^4, x)
\[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{4}}\, dx \] Input:
integrate(acosh(a*x)**3/x**4,x)
Output:
Integral(acosh(a*x)**3/x**4, x)
\[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{4}} \,d x } \] Input:
integrate(arccosh(a*x)^3/x^4,x, algorithm="maxima")
Output:
-1/3*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/x^3 + integrate((a^3*x^2 + s qrt(a*x + 1)*sqrt(a*x - 1)*a^2*x - a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1 ))^2/(a^3*x^6 - a*x^4 + (a^2*x^5 - x^3)*sqrt(a*x + 1)*sqrt(a*x - 1)), x)
\[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{4}} \,d x } \] Input:
integrate(arccosh(a*x)^3/x^4,x, algorithm="giac")
Output:
integrate(arccosh(a*x)^3/x^4, x)
Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^4} \,d x \] Input:
int(acosh(a*x)^3/x^4,x)
Output:
int(acosh(a*x)^3/x^4, x)
\[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int \frac {\mathit {acosh} \left (a x \right )^{3}}{x^{4}}d x \] Input:
int(acosh(a*x)^3/x^4,x)
Output:
int(acosh(a*x)**3/x**4,x)