Integrand size = 22, antiderivative size = 28 \[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\frac {\sqrt {-1+a x} \text {Chi}(\text {arccosh}(a x))}{a^2 \sqrt {1-a x}} \] Output:
(a*x-1)^(1/2)*Chi(arccosh(a*x))/a^2/(-a*x+1)^(1/2)
Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=-\frac {\sqrt {-((-1+a x) (1+a x))} \text {Chi}(\text {arccosh}(a x))}{a^2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \] Input:
Integrate[x/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]
Output:
-((Sqrt[-((-1 + a*x)*(1 + a*x))]*CoshIntegral[ArcCosh[a*x]])/(a^2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)))
Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6367, 3042, 3782}
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.
\(\displaystyle \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx\) |
\(\Big \downarrow \) 6367 |
\(\displaystyle \frac {\sqrt {a x-1} \int \frac {a x}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2 \sqrt {1-a x}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a x-1} \int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2 \sqrt {1-a x}}\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle \frac {\sqrt {a x-1} \text {Chi}(\text {arccosh}(a x))}{a^2 \sqrt {1-a x}}\) |
Input:
Int[x/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]
Output:
(Sqrt[-1 + a*x]*CoshIntegral[ArcCosh[a*x]])/(a^2*Sqrt[1 - a*x])
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
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]
\[\int \frac {x}{\sqrt {-a^{2} x^{2}+1}\, \operatorname {arccosh}\left (a x \right )}d x\]
Input:
int(x/(-a^2*x^2+1)^(1/2)/arccosh(a*x),x)
Output:
int(x/(-a^2*x^2+1)^(1/2)/arccosh(a*x),x)
\[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int { \frac {x}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )} \,d x } \] Input:
integrate(x/(-a^2*x^2+1)^(1/2)/arccosh(a*x),x, algorithm="fricas")
Output:
integral(-sqrt(-a^2*x^2 + 1)*x/((a^2*x^2 - 1)*arccosh(a*x)), x)
\[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int \frac {x}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}{\left (a x \right )}}\, dx \] Input:
integrate(x/(-a**2*x**2+1)**(1/2)/acosh(a*x),x)
Output:
Integral(x/(sqrt(-(a*x - 1)*(a*x + 1))*acosh(a*x)), x)
\[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int { \frac {x}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )} \,d x } \] Input:
integrate(x/(-a^2*x^2+1)^(1/2)/arccosh(a*x),x, algorithm="maxima")
Output:
integrate(x/(sqrt(-a^2*x^2 + 1)*arccosh(a*x)), x)
\[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int { \frac {x}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )} \,d x } \] Input:
integrate(x/(-a^2*x^2+1)^(1/2)/arccosh(a*x),x, algorithm="giac")
Output:
integrate(x/(sqrt(-a^2*x^2 + 1)*arccosh(a*x)), x)
Timed out. \[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int \frac {x}{\mathrm {acosh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \] Input:
int(x/(acosh(a*x)*(1 - a^2*x^2)^(1/2)),x)
Output:
int(x/(acosh(a*x)*(1 - a^2*x^2)^(1/2)), x)
\[ \int \frac {x}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int \frac {x}{\sqrt {-a^{2} x^{2}+1}\, \mathit {acosh} \left (a x \right )}d x \] Input:
int(x/(-a^2*x^2+1)^(1/2)/acosh(a*x),x)
Output:
int(x/(sqrt( - a**2*x**2 + 1)*acosh(a*x)),x)