Integrand size = 24, antiderivative size = 65 \[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\frac {\sqrt {-1+a x} \text {Chi}(2 \text {arccosh}(a x))}{2 a^3 \sqrt {1-a x}}+\frac {\sqrt {-1+a x} \log (\text {arccosh}(a x))}{2 a^3 \sqrt {1-a x}} \] Output:
1/2*(a*x-1)^(1/2)*Chi(2*arccosh(a*x))/a^3/(-a*x+1)^(1/2)+1/2*(a*x-1)^(1/2) *ln(arccosh(a*x))/a^3/(-a*x+1)^(1/2)
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=-\frac {\sqrt {-((-1+a x) (1+a x))} (\text {Chi}(2 \text {arccosh}(a x))+\log (\text {arccosh}(a x)))}{2 a^3 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \] Input:
Integrate[x^2/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]
Output:
-1/2*(Sqrt[-((-1 + a*x)*(1 + a*x))]*(CoshIntegral[2*ArcCosh[a*x]] + Log[Ar cCosh[a*x]]))/(a^3*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))
Time = 0.49 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.68, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6367, 3042, 3793, 2009}
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.
\(\displaystyle \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx\) |
\(\Big \downarrow \) 6367 |
\(\displaystyle \frac {\sqrt {a x-1} \int \frac {a^2 x^2}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^3 \sqrt {1-a x}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a x-1} \int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )^2}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^3 \sqrt {1-a x}}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {\sqrt {a x-1} \int \left (\frac {\cosh (2 \text {arccosh}(a x))}{2 \text {arccosh}(a x)}+\frac {1}{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)}{a^3 \sqrt {1-a x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {a x-1} \left (\frac {1}{2} \text {Chi}(2 \text {arccosh}(a x))+\frac {1}{2} \log (\text {arccosh}(a x))\right )}{a^3 \sqrt {1-a x}}\) |
Input:
Int[x^2/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]
Output:
(Sqrt[-1 + a*x]*(CoshIntegral[2*ArcCosh[a*x]]/2 + Log[ArcCosh[a*x]]/2))/(a ^3*Sqrt[1 - a*x])
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x )^p*(-1 + c*x)^p)] Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && Eq Q[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 0.18 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \left (\operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )+\operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (a x \right )\right )-2 \ln \left (\operatorname {arccosh}\left (a x \right )\right )\right )}{4 a^{3} \left (a^{2} x^{2}-1\right )}\) | \(67\) |
Input:
int(x^2/(-a^2*x^2+1)^(1/2)/arccosh(a*x),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)*(Ei(1,2 *arccosh(a*x))+Ei(1,-2*arccosh(a*x))-2*ln(arccosh(a*x)))
\[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int { \frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )} \,d x } \] Input:
integrate(x^2/(-a^2*x^2+1)^(1/2)/arccosh(a*x),x, algorithm="fricas")
Output:
integral(-sqrt(-a^2*x^2 + 1)*x^2/((a^2*x^2 - 1)*arccosh(a*x)), x)
\[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int \frac {x^{2}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}{\left (a x \right )}}\, dx \] Input:
integrate(x**2/(-a**2*x**2+1)**(1/2)/acosh(a*x),x)
Output:
Integral(x**2/(sqrt(-(a*x - 1)*(a*x + 1))*acosh(a*x)), x)
\[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int { \frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )} \,d x } \] Input:
integrate(x^2/(-a^2*x^2+1)^(1/2)/arccosh(a*x),x, algorithm="maxima")
Output:
integrate(x^2/(sqrt(-a^2*x^2 + 1)*arccosh(a*x)), x)
\[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int { \frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )} \,d x } \] Input:
integrate(x^2/(-a^2*x^2+1)^(1/2)/arccosh(a*x),x, algorithm="giac")
Output:
integrate(x^2/(sqrt(-a^2*x^2 + 1)*arccosh(a*x)), x)
Timed out. \[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int \frac {x^2}{\mathrm {acosh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \] 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)
\[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int \frac {x^{2}}{\sqrt {-a^{2} x^{2}+1}\, \mathit {acosh} \left (a x \right )}d x \] Input:
int(x^2/(-a^2*x^2+1)^(1/2)/acosh(a*x),x)
Output:
int(x**2/(sqrt( - a**2*x**2 + 1)*acosh(a*x)),x)