\(\int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) [256]

Optimal result

Integrand size = 22, antiderivative size = 110 \[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {6 x \sqrt {-1+a x}}{a \sqrt {1-a x}}-\frac {6 \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{a^2}-\frac {3 x \sqrt {-1+a x} \text {arccosh}(a x)^2}{a \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{a^2} \]

[Out]

-6*x*(a*x-1)^(1/2)/a/(-a*x+1)^(1/2)-3*x*arccosh(a*x)^2*(a*x-1)^(1/2)/a/(-a*x+1)^(1/2)-6*arccosh(a*x)*(-a*x+1)^
(1/2)*(a*x+1)^(1/2)/a^2-arccosh(a*x)^3*(-a^2*x^2+1)^(1/2)/a^2

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5914, 5879, 5915, 8} \[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{a^2}-\frac {6 \sqrt {1-a x} \sqrt {a x+1} \text {arccosh}(a x)}{a^2}-\frac {3 x \sqrt {a x-1} \text {arccosh}(a x)^2}{a \sqrt {1-a x}}-\frac {6 x \sqrt {a x-1}}{a \sqrt {1-a x}} \]

[In]

Int[(x*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2],x]

[Out]

(-6*x*Sqrt[-1 + a*x])/(a*Sqrt[1 - a*x]) - (6*Sqrt[1 - a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/a^2 - (3*x*Sqrt[-1 + a*
x]*ArcCosh[a*x]^2)/(a*Sqrt[1 - a*x]) - (Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/a^2

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 5879

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[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]

Rule 5914

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] - Dist[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]

Rule 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Dist[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] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{a^2}-\frac {\left (3 \sqrt {-1+a x}\right ) \int \text {arccosh}(a x)^2 \, dx}{a \sqrt {1-a x}} \\ & = -\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}+\frac {\left (6 \sqrt {-1+a x}\right ) \int \frac {x \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x}} \\ & = -\frac {6 \sqrt {1-a x} \sqrt {1+a x} \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}-\frac {\left (6 \sqrt {-1+a x}\right ) \int 1 \, dx}{a \sqrt {1-a x}} \\ & = -\frac {6 x \sqrt {-1+a x}}{a \sqrt {1-a x}}-\frac {6 \sqrt {1-a x} \sqrt {1+a x} \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.92 \[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (6 a x-6 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)+3 a x \text {arccosh}(a x)^2-\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3\right )}{a^2 \sqrt {-1+a x} \sqrt {1+a x}} \]

[In]

Integrate[(x*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2],x]

[Out]

(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])

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.41

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x -1}\, \sqrt {a x +1}\, a x +a^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (a x \right )^{3}-3 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )-6\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x -1\right ) \left (\operatorname {arccosh}\left (a x \right )^{3}+3 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+6\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}\)	\(155\)

[In]

int(x*arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(-a^2*x^2+1)^(1/2)*((a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x+a^2*x^2-1)*(arccosh(a*x)^3-3*arccosh(a*x)^2+6*arccosh
(a*x)-6)/a^2/(a^2*x^2-1)-1/2*(-a^2*x^2+1)^(1/2)*(a^2*x^2-(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x-1)*(arccosh(a*x)^3+3*
arccosh(a*x)^2+6*arccosh(a*x)+6)/a^2/(a^2*x^2-1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.45 \[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {3 \, \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} + 6 \, \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} a x - 6 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{a^{4} x^{2} - a^{2}} \]

[In]

integrate(x*arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

(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 + sq
rt(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 +
sqrt(a^2*x^2 - 1)))/(a^4*x^2 - a^2)

Sympy [F]

\[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x \operatorname {acosh}^{3}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

[In]

integrate(x*acosh(a*x)**3/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(x*acosh(a*x)**3/sqrt(-(a*x - 1)*(a*x + 1)), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.59 \[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {3 i \, x \operatorname {arcosh}\left (a x\right )^{2}}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )^{3}}{a^{2}} + \frac {6 \, {\left (i \, x - \frac {i \, \sqrt {a^{2} x^{2} - 1} \operatorname {arcosh}\left (a x\right )}{a}\right )}}{a} \]

[In]

integrate(x*arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94 \[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3}}{a^{2}} - \frac {3 i \, {\left (x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 2 \, a {\left (\frac {x}{a} - \frac {\sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{a^{2}}\right )}\right )}}{a} \]

[In]

integrate(x*arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

-sqrt(-a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 - 1))^3/a^2 - 3*I*(x*log(a*x + sqrt(a^2*x^2 - 1))^2 + 2*a*(x/a - sq
rt(a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1))/a^2))/a

Mupad [F(-1)]

Timed out. \[ \int \frac {x \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x\,{\mathrm {acosh}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \]

[In]

int((x*acosh(a*x)^3)/(1 - a^2*x^2)^(1/2),x)

[Out]

int((x*acosh(a*x)^3)/(1 - a^2*x^2)^(1/2), x)