Integrand size = 24, antiderivative size = 166 \[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\frac {a \sqrt {-1+a x} \text {arccosh}(a x)^3}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}-\frac {3 a \sqrt {-1+a x} \text {arccosh}(a x)^2 \log \left (1+e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 a \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 a \sqrt {-1+a x} \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}} \]
a*arccosh(a*x)^3*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-3*a*arccosh(a*x)^2*ln(1+(a*x +(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-3*a*arccosh( a*x)*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*(a*x-1)^(1/2)/(-a*x+1 )^(1/2)+3/2*a*polylog(3,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*(a*x-1)^(1/2 )/(-a*x+1)^(1/2)-arccosh(a*x)^3*(-a^2*x^2+1)^(1/2)/x
Time = 0.44 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.83 \[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\frac {a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (2 \text {arccosh}(a x)^2 \left (-\text {arccosh}(a x)+\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)}{a x}-3 \log \left (1+e^{-2 \text {arccosh}(a x)}\right )\right )+6 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}(a x)}\right )\right )}{2 \sqrt {-((-1+a x) (1+a x))}} \]
(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(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*P olyLog[3, -E^(-2*ArcCosh[a*x])]))/(2*Sqrt[-((-1 + a*x)*(1 + a*x))])
Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.75, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6332, 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.
\(\displaystyle \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 6332 |
\(\displaystyle -\frac {3 a \sqrt {a x-1} \int \frac {\text {arccosh}(a x)^2}{x}dx}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}\) |
\(\Big \downarrow \) 6297 |
\(\displaystyle -\frac {3 a \sqrt {a x-1} \int \frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1) \text {arccosh}(a x)^2}{a x}d\text {arccosh}(a x)}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}-\frac {3 a \sqrt {a x-1} \int -i \text {arccosh}(a x)^2 \tan (i \text {arccosh}(a x))d\text {arccosh}(a x)}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}+\frac {3 i a \sqrt {a x-1} \int \text {arccosh}(a x)^2 \tan (i \text {arccosh}(a x))d\text {arccosh}(a x)}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}+\frac {3 i a \sqrt {a x-1} \left (2 i \int \frac {e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)^2}{1+e^{2 \text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}+\frac {3 i a \sqrt {a x-1} \left (2 i \left (\frac {1}{2} \text {arccosh}(a x)^2 \log \left (e^{2 \text {arccosh}(a x)}+1\right )-\int \text {arccosh}(a x) \log \left (1+e^{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}+\frac {3 i a \sqrt {a x-1} \left (2 i \left (-\frac {1}{2} \int \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)+\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )+\frac {1}{2} \text {arccosh}(a x)^2 \log \left (e^{2 \text {arccosh}(a x)}+1\right )\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}+\frac {3 i a \sqrt {a x-1} \left (2 i \left (-\frac {1}{4} \int e^{-2 \text {arccosh}(a x)} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )de^{2 \text {arccosh}(a x)}+\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )+\frac {1}{2} \text {arccosh}(a x)^2 \log \left (e^{2 \text {arccosh}(a x)}+1\right )\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}+\frac {3 i a \sqrt {a x-1} \left (2 i \left (\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )-\frac {1}{4} \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )+\frac {1}{2} \text {arccosh}(a x)^2 \log \left (e^{2 \text {arccosh}(a x)}+1\right )\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{\sqrt {1-a x}}\) |
-((Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/x) + ((3*I)*a*Sqrt[-1 + a*x]*((-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*ArcCo sh[a*x])]/4)))/Sqrt[1 - a*x]
3.3.59.3.1 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_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + Simp[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*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, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3 , 0] && NeQ[m, -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 = 1.05 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.89
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x -1\right ) \operatorname {arccosh}\left (a x \right )^{3}}{x \left (a^{2} x^{2}-1\right )}-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{3} a}{a^{2} x^{2}-1}+\frac {3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1}+\frac {3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {polylog}\left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{2 \left (a^{2} x^{2}-1\right )}\) | \(313\) |
-(-a^2*x^2+1)^(1/2)*(a^2*x^2-(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x-1)*arccosh(a* x)^3/x/(a^2*x^2-1)-2*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/(a^2*x ^2-1)*arccosh(a*x)^3*a+3*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/(a ^2*x^2-1)*arccosh(a*x)^2*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*a+3*(-a ^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/(a^2*x^2-1)*arccosh(a*x)*polyl og(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*a-3/2*(-a^2*x^2+1)^(1/2)*(a*x-1 )^(1/2)*(a*x+1)^(1/2)/(a^2*x^2-1)*polylog(3,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1 /2))^2)*a
\[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \]
\[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
\[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \]
(a^2*x^2 - 1)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/(sqrt(a*x + 1)*sqrt (-a*x + 1)*x) - integrate(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))^2/((sqrt(a*x + 1)*a*x^2 + (a*x + 1)*sqrt(a*x - 1)*x)*sqrt(-a*x + 1)), x)
\[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \]