Integrand size = 10, antiderivative size = 268 \[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\frac {2 a^2 \text {arccosh}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 x^2}-\frac {\text {arccosh}(a x)^4}{3 x^3}-8 a^3 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )+\frac {4}{3} a^3 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+4 i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-4 i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(a x)}\right ) \] Output:
2*a^2*arccosh(a*x)^2/x+2/3*a*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)^3/x^ 2-1/3*arccosh(a*x)^4/x^3-8*a^3*arccosh(a*x)*arctan(a*x+(a*x-1)^(1/2)*(a*x+ 1)^(1/2))+4/3*a^3*arccosh(a*x)^3*arctan(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+4 *I*a^3*polylog(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))-2*I*a^3*arccosh(a*x )^2*polylog(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))-4*I*a^3*polylog(2,I*(a *x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+2*I*a^3*arccosh(a*x)^2*polylog(2,I*(a*x+( a*x-1)^(1/2)*(a*x+1)^(1/2)))+4*I*a^3*arccosh(a*x)*polylog(3,-I*(a*x+(a*x-1 )^(1/2)*(a*x+1)^(1/2)))-4*I*a^3*arccosh(a*x)*polylog(3,I*(a*x+(a*x-1)^(1/2 )*(a*x+1)^(1/2)))-4*I*a^3*polylog(4,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+ 4*I*a^3*polylog(4,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(595\) vs. \(2(268)=536\).
Time = 2.29 (sec) , antiderivative size = 595, normalized size of antiderivative = 2.22 \[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx =\text {Too large to display} \] Input:
Integrate[ArcCosh[a*x]^4/x^4,x]
Output:
a^3*((I/2)*(8 + Pi^2 - (4*I)*Pi*ArcCosh[a*x] - 4*ArcCosh[a*x]^2)*PolyLog[2 , (-I)/E^ArcCosh[a*x]] - (I/96)*(7*Pi^4 + (8*I)*Pi^3*ArcCosh[a*x] + 24*Pi^ 2*ArcCosh[a*x]^2 + ((192*I)*ArcCosh[a*x]^2)/(a*x) - (32*I)*Pi*ArcCosh[a*x] ^3 + ((64*I)*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x]^3)/(a^2*x^2 ) - 16*ArcCosh[a*x]^4 - ((32*I)*ArcCosh[a*x]^4)/(a^3*x^3) - 384*ArcCosh[a* x]*Log[1 - I/E^ArcCosh[a*x]] + (8*I)*Pi^3*Log[1 + I/E^ArcCosh[a*x]] + 384* ArcCosh[a*x]*Log[1 + I/E^ArcCosh[a*x]] + 48*Pi^2*ArcCosh[a*x]*Log[1 + I/E^ ArcCosh[a*x]] - (96*I)*Pi*ArcCosh[a*x]^2*Log[1 + I/E^ArcCosh[a*x]] - 64*Ar cCosh[a*x]^3*Log[1 + I/E^ArcCosh[a*x]] - 48*Pi^2*ArcCosh[a*x]*Log[1 - I*E^ ArcCosh[a*x]] + (96*I)*Pi*ArcCosh[a*x]^2*Log[1 - I*E^ArcCosh[a*x]] - (8*I) *Pi^3*Log[1 + I*E^ArcCosh[a*x]] + 64*ArcCosh[a*x]^3*Log[1 + I*E^ArcCosh[a* x]] + (8*I)*Pi^3*Log[Tan[(Pi + (2*I)*ArcCosh[a*x])/4]] + 384*PolyLog[2, I/ E^ArcCosh[a*x]] + 192*ArcCosh[a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]] - 48* Pi^2*PolyLog[2, I*E^ArcCosh[a*x]] + (192*I)*Pi*ArcCosh[a*x]*PolyLog[2, I*E ^ArcCosh[a*x]] + (192*I)*Pi*PolyLog[3, (-I)/E^ArcCosh[a*x]] + 384*ArcCosh[ a*x]*PolyLog[3, (-I)/E^ArcCosh[a*x]] - 384*ArcCosh[a*x]*PolyLog[3, (-I)*E^ ArcCosh[a*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcCosh[a*x]] + 384*PolyLog[4, ( -I)/E^ArcCosh[a*x]] + 384*PolyLog[4, (-I)*E^ArcCosh[a*x]]))
Time = 3.36 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6298, 6348, 6298, 6362, 3042, 4668, 2715, 2838, 3011, 7163, 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.
\(\displaystyle \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx\) |
\(\Big \downarrow \) 6298 |
\(\displaystyle \frac {4}{3} a \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {\text {arccosh}(a x)^4}{3 x^3}\) |
\(\Big \downarrow \) 6348 |
\(\displaystyle \frac {4}{3} a \left (\frac {1}{2} a^2 \int \frac {\text {arccosh}(a x)^3}{x \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {3}{2} a \int \frac {\text {arccosh}(a x)^2}{x^2}dx+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 x^2}\right )-\frac {\text {arccosh}(a x)^4}{3 x^3}\) |
\(\Big \downarrow \) 6298 |
\(\displaystyle \frac {4}{3} a \left (\frac {1}{2} a^2 \int \frac {\text {arccosh}(a x)^3}{x \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {3}{2} a \left (2 a \int \frac {\text {arccosh}(a x)}{x \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {\text {arccosh}(a x)^2}{x}\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 x^2}\right )-\frac {\text {arccosh}(a x)^4}{3 x^3}\) |
\(\Big \downarrow \) 6362 |
\(\displaystyle \frac {4}{3} a \left (\frac {1}{2} a^2 \int \frac {\text {arccosh}(a x)^3}{a x}d\text {arccosh}(a x)-\frac {3}{2} a \left (2 a \int \frac {\text {arccosh}(a x)}{a x}d\text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{x}\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 x^2}\right )-\frac {\text {arccosh}(a x)^4}{3 x^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\text {arccosh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \int \text {arccosh}(a x)^3 \csc \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )d\text {arccosh}(a x)-\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \int \text {arccosh}(a x) \csc \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )d\text {arccosh}(a x)\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 x^2}\right )\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle -\frac {\text {arccosh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \left (-3 i \int \text {arccosh}(a x)^2 \log \left (1-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+3 i \int \text {arccosh}(a x)^2 \log \left (1+i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )-\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \left (-i \int \log \left (1-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+i \int \log \left (1+i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )\right )\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 x^2}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {\text {arccosh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \left (-3 i \int \text {arccosh}(a x)^2 \log \left (1-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+3 i \int \text {arccosh}(a x)^2 \log \left (1+i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )-\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \left (-i \int e^{-\text {arccosh}(a x)} \log \left (1-i e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}+i \int e^{-\text {arccosh}(a x)} \log \left (1+i e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}+2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )\right )\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 x^2}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {\text {arccosh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \left (-3 i \int \text {arccosh}(a x)^2 \log \left (1-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+3 i \int \text {arccosh}(a x)^2 \log \left (1+i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )-\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \left (2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 x^2}\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {\text {arccosh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \left (3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )-\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \left (2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 x^2}\right )\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle -\frac {\text {arccosh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \left (3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )-\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \left (2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 x^2}\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {\text {arccosh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \left (3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )-\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )+2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )\right )-\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \left (2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 x^2}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {\text {arccosh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \left (2 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )+3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-\operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )\right )-3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )-\operatorname {PolyLog}\left (4,i e^{\text {arccosh}(a x)}\right )\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )\right )-\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{x}+2 a \left (2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 x^2}\right )\) |
Input:
Int[ArcCosh[a*x]^4/x^4,x]
Output:
-1/3*ArcCosh[a*x]^4/x^3 + (4*a*((Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] ^3)/(2*x^2) - (3*a*(-(ArcCosh[a*x]^2/x) + 2*a*(2*ArcCosh[a*x]*ArcTan[E^Arc Cosh[a*x]] - I*PolyLog[2, (-I)*E^ArcCosh[a*x]] + I*PolyLog[2, I*E^ArcCosh[ a*x]])))/2 + (a^2*(2*ArcCosh[a*x]^3*ArcTan[E^ArcCosh[a*x]] + (3*I)*(-(ArcC osh[a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]]) + 2*(ArcCosh[a*x]*PolyLog[3, ( -I)*E^ArcCosh[a*x]] - PolyLog[4, (-I)*E^ArcCosh[a*x]])) - (3*I)*(-(ArcCosh [a*x]^2*PolyLog[2, I*E^ArcCosh[a*x]]) + 2*(ArcCosh[a*x]*PolyLog[3, I*E^Arc Cosh[a*x]] - PolyLog[4, I*E^ArcCosh[a*x]]))))/2))/3
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
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[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
\[\int \frac {\operatorname {arccosh}\left (a x \right )^{4}}{x^{4}}d x\]
Input:
int(arccosh(a*x)^4/x^4,x)
Output:
int(arccosh(a*x)^4/x^4,x)
\[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{4}} \,d x } \] Input:
integrate(arccosh(a*x)^4/x^4,x, algorithm="fricas")
Output:
integral(arccosh(a*x)^4/x^4, x)
\[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int \frac {\operatorname {acosh}^{4}{\left (a x \right )}}{x^{4}}\, dx \] Input:
integrate(acosh(a*x)**4/x**4,x)
Output:
Integral(acosh(a*x)**4/x**4, x)
\[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{4}} \,d x } \] Input:
integrate(arccosh(a*x)^4/x^4,x, algorithm="maxima")
Output:
-1/3*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^4/x^3 + integrate(4/3*(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^6 - a*x^4 + (a^2*x^5 - x^3)*sqrt(a*x + 1)*sqrt(a*x - 1)), x)
\[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{4}} \,d x } \] Input:
integrate(arccosh(a*x)^4/x^4,x, algorithm="giac")
Output:
integrate(arccosh(a*x)^4/x^4, x)
Timed out. \[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^4}{x^4} \,d x \] Input:
int(acosh(a*x)^4/x^4,x)
Output:
int(acosh(a*x)^4/x^4, x)
\[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int \frac {\mathit {acosh} \left (a x \right )^{4}}{x^{4}}d x \] Input:
int(acosh(a*x)^4/x^4,x)
Output:
int(acosh(a*x)**4/x**4,x)