Integrand size = 22, antiderivative size = 88 \[ \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {x^2 \sqrt {1-a x}}{4 a \sqrt {-1+a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 a^2}-\frac {\sqrt {1-a x} \text {arccosh}(a x)^2}{4 a^3 \sqrt {-1+a x}} \] Output:
1/4*x^2*(-a*x+1)^(1/2)/a/(a*x-1)^(1/2)-1/2*x*(-a^2*x^2+1)^(1/2)*arccosh(a* x)/a^2-1/4*(-a*x+1)^(1/2)*arccosh(a*x)^2/a^3/(a*x-1)^(1/2)
Time = 0.15 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.85 \[ \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {-((-1+a x) (1+a x))} (-\cosh (2 \text {arccosh}(a x))+2 \text {arccosh}(a x) (\text {arccosh}(a x)+\sinh (2 \text {arccosh}(a x))))}{8 a^3 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \] Input:
Integrate[(x^2*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]
Output:
-1/8*(Sqrt[-((-1 + a*x)*(1 + a*x))]*(-Cosh[2*ArcCosh[a*x]] + 2*ArcCosh[a*x ]*(ArcCosh[a*x] + Sinh[2*ArcCosh[a*x]])))/(a^3*Sqrt[(-1 + a*x)/(1 + a*x)]* (1 + a*x))
Time = 0.36 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6353, 15, 6307}
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.
\(\displaystyle \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 6353 |
\(\displaystyle \frac {\int \frac {\text {arccosh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {\sqrt {a x-1} \int xdx}{2 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 a^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\int \frac {\text {arccosh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 a^2}-\frac {x^2 \sqrt {a x-1}}{4 a \sqrt {1-a x}}\) |
\(\Big \downarrow \) 6307 |
\(\displaystyle \frac {\sqrt {a x-1} \text {arccosh}(a x)^2}{4 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 a^2}-\frac {x^2 \sqrt {a x-1}}{4 a \sqrt {1-a x}}\) |
Input:
Int[(x^2*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]
Output:
-1/4*(x^2*Sqrt[-1 + a*x])/(a*Sqrt[1 - a*x]) - (x*Sqrt[1 - a^2*x^2]*ArcCosh [a*x])/(2*a^2) + (Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(4*a^3*Sqrt[1 - a*x])
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x ] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2 *p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 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*Ar cCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(222\) vs. \(2(72)=144\).
Time = 0.28 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.53
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{2}}{4 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (a x \right )\right )}{16 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (-2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 a^{3} x^{3}+\sqrt {a x -1}\, \sqrt {a x +1}-2 a x \right ) \left (1+2 \,\operatorname {arccosh}\left (a x \right )\right )}{16 a^{3} \left (a^{2} x^{2}-1\right )}\) | \(223\) |
Input:
int(x^2*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/4*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3/(a^2*x^2-1)*arccos h(a*x)^2-1/16*(-a^2*x^2+1)^(1/2)*(2*a^3*x^3-2*a*x+2*a^2*x^2*(a*x-1)^(1/2)* (a*x+1)^(1/2)-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(-1+2*arccosh(a*x))/a^3/(a^2*x^ 2-1)-1/16*(-a^2*x^2+1)^(1/2)*(-2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2*a^3 *x^3+(a*x-1)^(1/2)*(a*x+1)^(1/2)-2*a*x)*(1+2*arccosh(a*x))/a^3/(a^2*x^2-1)
\[ \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{2} \operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate(x^2*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-a^2*x^2 + 1)*x^2*arccosh(a*x)/(a^2*x^2 - 1), x)
\[ \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{2} \operatorname {acosh}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:
integrate(x**2*acosh(a*x)/(-a**2*x**2+1)**(1/2),x)
Output:
Integral(x**2*acosh(a*x)/sqrt(-(a*x - 1)*(a*x + 1)), x)
Exception generated. \[ \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^2*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
\[ \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{2} \operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate(x^2*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
integrate(x^2*arccosh(a*x)/sqrt(-a^2*x^2 + 1), x)
Timed out. \[ \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^2\,\mathrm {acosh}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \] Input:
int((x^2*acosh(a*x))/(1 - a^2*x^2)^(1/2),x)
Output:
int((x^2*acosh(a*x))/(1 - a^2*x^2)^(1/2), x)
\[ \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acosh} \left (a x \right ) x^{2}}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:
int(x^2*acosh(a*x)/(-a^2*x^2+1)^(1/2),x)
Output:
int((acosh(a*x)*x**2)/sqrt( - a**2*x**2 + 1),x)