Integrand size = 22, antiderivative size = 105 \[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {6 x \sqrt {1-a x}}{a \sqrt {-1+a x}}-\frac {6 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{a^2}+\frac {3 x \sqrt {1-a x} \text {arccosh}(a x)^2}{a \sqrt {-1+a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{a^2} \] Output:
6*x*(-a*x+1)^(1/2)/a/(a*x-1)^(1/2)-6*(-a^2*x^2+1)^(1/2)*arccosh(a*x)/a^2+3 *x*(-a*x+1)^(1/2)*arccosh(a*x)^2/a/(a*x-1)^(1/2)-(-a^2*x^2+1)^(1/2)*arccos h(a*x)^3/a^2
Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96 \[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (6 a x-6 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)+3 a x \text {arccosh}(a x)^2-\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3\right )}{a^2 \sqrt {-1+a x} \sqrt {1+a x}} \] Input:
Integrate[(x*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2],x]
Output:
(Sqrt[1 - a^2*x^2]*(6*a*x - 6*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] + 3*a*x*ArcCosh[a*x]^2 - Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3))/(a^2* Sqrt[-1 + a*x]*Sqrt[1 + a*x])
Time = 0.79 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6329, 6294, 6330, 24}
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.
\(\displaystyle \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 6329 |
\(\displaystyle -\frac {3 \sqrt {a x-1} \int \text {arccosh}(a x)^2dx}{a \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{a^2}\) |
\(\Big \downarrow \) 6294 |
\(\displaystyle -\frac {3 \sqrt {a x-1} \left (x \text {arccosh}(a x)^2-2 a \int \frac {x \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{a \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{a^2}\) |
\(\Big \downarrow \) 6330 |
\(\displaystyle -\frac {3 \sqrt {a x-1} \left (x \text {arccosh}(a x)^2-2 a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{a^2}-\frac {\int 1dx}{a}\right )\right )}{a \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{a^2}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{a^2}-\frac {3 \sqrt {a x-1} \left (x \text {arccosh}(a x)^2-2 a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{a^2}-\frac {x}{a}\right )\right )}{a \sqrt {1-a x}}\) |
Input:
Int[(x*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2],x]
Output:
-((Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/a^2) - (3*Sqrt[-1 + a*x]*(x*ArcCosh[a *x]^2 - 2*a*(-(x/a) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/a^2)))/( a*Sqrt[1 - a*x])
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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p _)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 *c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*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, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
Time = 0.44 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.06
method | result | size |
default | \(-\frac {\left (x^{2} \operatorname {arccosh}\left (a x \right )^{3} a^{2}-3 a x \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+6 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )-6 \sqrt {a x -1}\, \sqrt {a x +1}\, a x -\operatorname {arccosh}\left (a x \right )^{3}-6 \,\operatorname {arccosh}\left (a x \right )\right ) \sqrt {-a^{2} x^{2}+1}}{\left (a^{2} x^{2}-1\right ) a^{2}}\) | \(111\) |
orering | \(\text {Expression too large to display}\) | \(1033\) |
Input:
int(x*arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-(x^2*arccosh(a*x)^3*a^2-3*a*x*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+ 6*a^2*x^2*arccosh(a*x)-6*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x-arccosh(a*x)^3-6* arccosh(a*x))*(-a^2*x^2+1)^(1/2)/(a^2*x^2-1)/a^2
Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.51 \[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {3 \, \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} + 6 \, \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} a x - 6 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{a^{4} x^{2} - a^{2}} \] Input:
integrate(x*arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
(3*sqrt(a^2*x^2 - 1)*sqrt(-a^2*x^2 + 1)*a*x*log(a*x + sqrt(a^2*x^2 - 1))^2 + (-a^2*x^2 + 1)^(3/2)*log(a*x + sqrt(a^2*x^2 - 1))^3 + 6*sqrt(a^2*x^2 - 1)*sqrt(-a^2*x^2 + 1)*a*x - 6*(a^2*x^2 - 1)*sqrt(-a^2*x^2 + 1)*log(a*x + s qrt(a^2*x^2 - 1)))/(a^4*x^2 - a^2)
\[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x \operatorname {acosh}^{3}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:
integrate(x*acosh(a*x)**3/(-a**2*x**2+1)**(1/2),x)
Output:
Integral(x*acosh(a*x)**3/sqrt(-(a*x - 1)*(a*x + 1)), x)
Result contains complex when optimal does not.
Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.62 \[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {3 i \, x \operatorname {arcosh}\left (a x\right )^{2}}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )^{3}}{a^{2}} + \frac {6 \, {\left (i \, x - \frac {i \, \sqrt {a^{2} x^{2} - 1} \operatorname {arcosh}\left (a x\right )}{a}\right )}}{a} \] Input:
integrate(x*arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
3*I*x*arccosh(a*x)^2/a - sqrt(-a^2*x^2 + 1)*arccosh(a*x)^3/a^2 + 6*(I*x - I*sqrt(a^2*x^2 - 1)*arccosh(a*x)/a)/a
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3}}{a^{2}} + \frac {3 i \, {\left (x \log \left (a x + i \, \sqrt {-a^{2} x^{2} + 1}\right )^{2} + 2 \, x - \frac {2 i \, \sqrt {-a^{2} x^{2} + 1} \log \left (a x + i \, \sqrt {-a^{2} x^{2} + 1}\right )}{a}\right )}}{a} \] Input:
integrate(x*arccosh(a*x)^3/(-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))^3/a^2 + 3*I*(x*log(a*x + I*sqrt(-a^2*x^2 + 1))^2 + 2*x - 2*I*sqrt(-a^2*x^2 + 1)*log(a*x + I*sqrt(-a ^2*x^2 + 1))/a)/a
Timed out. \[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x\,{\mathrm {acosh}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \] Input:
int((x*acosh(a*x)^3)/(1 - a^2*x^2)^(1/2),x)
Output:
int((x*acosh(a*x)^3)/(1 - a^2*x^2)^(1/2), x)
\[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acosh} \left (a x \right )^{3} x}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:
int(x*acosh(a*x)^3/(-a^2*x^2+1)^(1/2),x)
Output:
int((acosh(a*x)**3*x)/sqrt( - a**2*x**2 + 1),x)