Integrand size = 22, antiderivative size = 74 \[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {2 \sqrt {1-a^2 x^2}}{a^2}+\frac {2 x \sqrt {1-a x} \text {arccosh}(a x)}{a \sqrt {-1+a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{a^2} \] Output:
-2*(-a^2*x^2+1)^(1/2)/a^2+2*x*(-a*x+1)^(1/2)*arccosh(a*x)/a/(a*x-1)^(1/2)- (-a^2*x^2+1)^(1/2)*arccosh(a*x)^2/a^2
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.73 \[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (-2+\frac {2 a x \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}}-\text {arccosh}(a x)^2\right )}{a^2} \] Input:
Integrate[(x*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]
Output:
(Sqrt[1 - a^2*x^2]*(-2 + (2*a*x*ArcCosh[a*x])/(Sqrt[-1 + a*x]*Sqrt[1 + a*x ]) - ArcCosh[a*x]^2))/a^2
Time = 0.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6329, 6294, 83}
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.
\(\displaystyle \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 6329 |
\(\displaystyle -\frac {2 \sqrt {a x-1} \int \text {arccosh}(a x)dx}{a \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{a^2}\) |
\(\Big \downarrow \) 6294 |
\(\displaystyle -\frac {2 \sqrt {a x-1} \left (x \text {arccosh}(a x)-a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{a \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{a^2}\) |
\(\Big \downarrow \) 83 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{a^2}-\frac {2 \sqrt {a x-1} \left (x \text {arccosh}(a x)-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a}\right )}{a \sqrt {1-a x}}\) |
Input:
Int[(x*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]
Output:
-((Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/a^2) - (2*Sqrt[-1 + a*x]*(-((Sqrt[-1 + a*x]*Sqrt[1 + a*x])/a) + x*ArcCosh[a*x]))/(a*Sqrt[1 - a*x])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcCosh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt [1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Time = 0.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.11
method | result | size |
default | \(-\frac {\left (\operatorname {arccosh}\left (a x \right )^{2} a^{2} x^{2}-2 \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a x +2 a^{2} x^{2}-\operatorname {arccosh}\left (a x \right )^{2}-2\right ) \sqrt {-a^{2} x^{2}+1}}{\left (a^{2} x^{2}-1\right ) a^{2}}\) | \(82\) |
orering | \(\frac {\left (a^{4} x^{4}-4 a^{2} x^{2}+2\right ) \operatorname {arccosh}\left (a x \right )^{2}}{a^{4} x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (\frac {\operatorname {arccosh}\left (a x \right )^{2}}{\sqrt {-a^{2} x^{2}+1}}+\frac {2 x \,\operatorname {arccosh}\left (a x \right ) a}{\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}}+\frac {x^{2} \operatorname {arccosh}\left (a x \right )^{2} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{a^{4} x^{2}}+\frac {\left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (\frac {4 \,\operatorname {arccosh}\left (a x \right ) a}{\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}}+\frac {3 \operatorname {arccosh}\left (a x \right )^{2} x \,a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {2 x \,a^{2}}{\left (a x -1\right ) \left (a x +1\right ) \sqrt {-a^{2} x^{2}+1}}+\frac {4 x^{2} \operatorname {arccosh}\left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \sqrt {a x -1}\, \sqrt {a x +1}}-\frac {x \,\operatorname {arccosh}\left (a x \right ) a^{2}}{\sqrt {-a^{2} x^{2}+1}\, \left (a x -1\right )^{\frac {3}{2}} \sqrt {a x +1}}-\frac {x \,\operatorname {arccosh}\left (a x \right ) a^{2}}{\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \left (a x +1\right )^{\frac {3}{2}}}+\frac {3 x^{3} \operatorname {arccosh}\left (a x \right )^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{a^{4} x}\) | \(388\) |
Input:
int(x*arccosh(a*x)^2/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-(arccosh(a*x)^2*a^2*x^2-2*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x+2* a^2*x^2-arccosh(a*x)^2-2)*(-a^2*x^2+1)^(1/2)/(a^2*x^2-1)/a^2
Time = 0.10 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.54 \[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {2 \, \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-a^{2} x^{2} + 1}}{a^{4} x^{2} - a^{2}} \] Input:
integrate(x*arccosh(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
(2*sqrt(a^2*x^2 - 1)*sqrt(-a^2*x^2 + 1)*a*x*log(a*x + sqrt(a^2*x^2 - 1)) + (-a^2*x^2 + 1)^(3/2)*log(a*x + sqrt(a^2*x^2 - 1))^2 - 2*(a^2*x^2 - 1)*sqr t(-a^2*x^2 + 1))/(a^4*x^2 - a^2)
\[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x \operatorname {acosh}^{2}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:
integrate(x*acosh(a*x)**2/(-a**2*x**2+1)**(1/2),x)
Output:
Integral(x*acosh(a*x)**2/sqrt(-(a*x - 1)*(a*x + 1)), x)
Result contains complex when optimal does not.
Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.68 \[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {2 i \, x \operatorname {arcosh}\left (a x\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )^{2}}{a^{2}} - \frac {2 i \, \sqrt {a^{2} x^{2} - 1}}{a^{2}} \] Input:
integrate(x*arccosh(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
2*I*x*arccosh(a*x)/a - sqrt(-a^2*x^2 + 1)*arccosh(a*x)^2/a^2 - 2*I*sqrt(a^ 2*x^2 - 1)/a^2
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.08 \[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}}{a^{2}} + \frac {2 i \, {\left (x \log \left (a x + i \, \sqrt {-a^{2} x^{2} + 1}\right ) - \frac {i \, \sqrt {-a^{2} x^{2} + 1}}{a}\right )}}{a} \] Input:
integrate(x*arccosh(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
-sqrt(-a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 - 1))^2/a^2 + 2*I*(x*log(a*x + I*sqrt(-a^2*x^2 + 1)) - I*sqrt(-a^2*x^2 + 1)/a)/a
Timed out. \[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x\,{\mathrm {acosh}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \] Input:
int((x*acosh(a*x)^2)/(1 - a^2*x^2)^(1/2),x)
Output:
int((x*acosh(a*x)^2)/(1 - a^2*x^2)^(1/2), x)
\[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acosh} \left (a x \right )^{2} x}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:
int(x*acosh(a*x)^2/(-a^2*x^2+1)^(1/2),x)
Output:
int((acosh(a*x)**2*x)/sqrt( - a**2*x**2 + 1),x)