Integrand size = 24, antiderivative size = 124 \[ \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx=-\frac {a \sqrt {1-a x} \text {arccosh}(a x)^2}{\sqrt {-1+a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{x}+\frac {2 a \sqrt {1-a x} \text {arccosh}(a x) \log \left (1+e^{2 \text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}+\frac {a \sqrt {1-a x} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )}{\sqrt {-1+a x}} \] Output:
-a*(-a*x+1)^(1/2)*arccosh(a*x)^2/(a*x-1)^(1/2)-(-a^2*x^2+1)^(1/2)*arccosh( a*x)^2/x+2*a*(-a*x+1)^(1/2)*arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^( 1/2))^2)/(a*x-1)^(1/2)+a*(-a*x+1)^(1/2)*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x +1)^(1/2))^2)/(a*x-1)^(1/2)
Time = 0.37 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx=\frac {a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (\text {arccosh}(a x) \left (-\text {arccosh}(a x)+\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)}{a x}-2 \log \left (1+e^{-2 \text {arccosh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a x)}\right )\right )}{\sqrt {-((-1+a x) (1+a x))}} \] Input:
Integrate[ArcCosh[a*x]^2/(x^2*Sqrt[1 - a^2*x^2]),x]
Output:
(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(ArcCosh[a*x]*(-ArcCosh[a*x] + (Sq rt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x])/(a*x) - 2*Log[1 + E^(-2*A rcCosh[a*x])]) + PolyLog[2, -E^(-2*ArcCosh[a*x])]))/Sqrt[-((-1 + a*x)*(1 + a*x))]
Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.83, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6332, 6297, 3042, 26, 4201, 2620, 2715, 2838}
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.
\(\displaystyle \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 6332 |
\(\displaystyle -\frac {2 a \sqrt {a x-1} \int \frac {\text {arccosh}(a x)}{x}dx}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{x}\) |
\(\Big \downarrow \) 6297 |
\(\displaystyle -\frac {2 a \sqrt {a x-1} \int \frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1) \text {arccosh}(a x)}{a x}d\text {arccosh}(a x)}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{x}-\frac {2 a \sqrt {a x-1} \int -i \text {arccosh}(a x) \tan (i \text {arccosh}(a x))d\text {arccosh}(a x)}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{x}+\frac {2 i a \sqrt {a x-1} \int \text {arccosh}(a x) \tan (i \text {arccosh}(a x))d\text {arccosh}(a x)}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{x}+\frac {2 i a \sqrt {a x-1} \left (2 i \int \frac {e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)}{1+e^{2 \text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \text {arccosh}(a x)^2\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{x}+\frac {2 i a \sqrt {a x-1} \left (2 i \left (\frac {1}{2} \text {arccosh}(a x) \log \left (e^{2 \text {arccosh}(a x)}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{x}+\frac {2 i a \sqrt {a x-1} \left (2 i \left (\frac {1}{2} \text {arccosh}(a x) \log \left (e^{2 \text {arccosh}(a x)}+1\right )-\frac {1}{4} \int e^{-2 \text {arccosh}(a x)} \log \left (1+e^{2 \text {arccosh}(a x)}\right )de^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{x}+\frac {2 i a \sqrt {a x-1} \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )+\frac {1}{2} \text {arccosh}(a x) \log \left (e^{2 \text {arccosh}(a x)}+1\right )\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )}{\sqrt {1-a x}}\) |
Input:
Int[ArcCosh[a*x]^2/(x^2*Sqrt[1 - a^2*x^2]),x]
Output:
-((Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/x) + ((2*I)*a*Sqrt[-1 + a*x]*((-1/2*I )*ArcCosh[a*x]^2 + (2*I)*((ArcCosh[a*x]*Log[1 + E^(2*ArcCosh[a*x])])/2 + P olyLog[2, -E^(2*ArcCosh[a*x])]/4)))/Sqrt[1 - a*x]
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[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[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[((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]
Time = 0.45 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.94
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x -1\right ) \operatorname {arccosh}\left (a x \right )^{2}}{x \left (a^{2} x^{2}-1\right )}-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{2} a}{a^{2} x^{2}-1}+\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1}+\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1}\) | \(241\) |
Input:
int(arccosh(a*x)^2/x^2/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-(-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)^2/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)^2*a+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)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*a+(-a^2*x ^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/(a^2*x^2-1)*polylog(2,-(a*x+(a*x-1 )^(1/2)*(a*x+1)^(1/2))^2)*a
\[ \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \] Input:
integrate(arccosh(a*x)^2/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-a^2*x^2 + 1)*arccosh(a*x)^2/(a^2*x^4 - x^2), x)
\[ \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:
integrate(acosh(a*x)**2/x**2/(-a**2*x**2+1)**(1/2),x)
Output:
Integral(acosh(a*x)**2/(x**2*sqrt(-(a*x - 1)*(a*x + 1))), x)
\[ \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \] Input:
integrate(arccosh(a*x)^2/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
(a^2*x^2 - 1)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/(sqrt(a*x + 1)*sqrt (-a*x + 1)*x) - 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))/((sqrt(a*x + 1)*a*x^2 + (a*x + 1)*sqrt(a*x - 1)*x)*sqrt(-a*x + 1)), x)
\[ \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \] Input:
integrate(arccosh(a*x)^2/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
integrate(arccosh(a*x)^2/(sqrt(-a^2*x^2 + 1)*x^2), x)
Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \] Input:
int(acosh(a*x)^2/(x^2*(1 - a^2*x^2)^(1/2)),x)
Output:
int(acosh(a*x)^2/(x^2*(1 - a^2*x^2)^(1/2)), x)
\[ \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acosh} \left (a x \right )^{2}}{\sqrt {-a^{2} x^{2}+1}\, x^{2}}d x \] Input:
int(acosh(a*x)^2/x^2/(-a^2*x^2+1)^(1/2),x)
Output:
int(acosh(a*x)**2/(sqrt( - a**2*x**2 + 1)*x**2),x)