\(\int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx\) [136]

Optimal result

Integrand size = 22, antiderivative size = 110 \[ \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {2 x \sqrt {-1+a x}}{3 a^3 \sqrt {1-a x}}-\frac {x^3 \sqrt {-1+a x}}{9 a \sqrt {1-a x}}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{3 a^2} \]

[Out]

-2/3*x*(a*x-1)^(1/2)/a^3/(-a*x+1)^(1/2)-1/9*x^3*(a*x-1)^(1/2)/a/(-a*x+1)^(1/2)-2/3*arccosh(a*x)*(-a^2*x^2+1)^(
1/2)/a^4-1/3*x^2*arccosh(a*x)*(-a^2*x^2+1)^(1/2)/a^2

Rubi [A] (verified)

Time = 0.08 (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 = {5938, 5914, 8, 30} \[ \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {2 x \sqrt {a x-1}}{3 a^3 \sqrt {1-a x}}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{3 a^2}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{3 a^4}-\frac {x^3 \sqrt {a x-1}}{9 a \sqrt {1-a x}} \]

[In]

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

[Out]

(-2*x*Sqrt[-1 + a*x])/(3*a^3*Sqrt[1 - a*x]) - (x^3*Sqrt[-1 + a*x])/(9*a*Sqrt[1 - a*x]) - (2*Sqrt[1 - a^2*x^2]*
ArcCosh[a*x])/(3*a^4) - (x^2*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 5938

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] + (Dist[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] - Dist[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*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && I
GtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{3 a^2}+\frac {2 \int \frac {x \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{3 a^2}-\frac {\sqrt {-1+a x} \int x^2 \, dx}{3 a \sqrt {1-a x}} \\ & = -\frac {x^3 \sqrt {-1+a x}}{9 a \sqrt {1-a x}}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{3 a^2}-\frac {\left (2 \sqrt {-1+a x}\right ) \int 1 \, dx}{3 a^3 \sqrt {1-a x}} \\ & = -\frac {2 x \sqrt {-1+a x}}{3 a^3 \sqrt {1-a x}}-\frac {x^3 \sqrt {-1+a x}}{9 a \sqrt {1-a x}}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{3 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.67 \[ \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {a x \sqrt {-1+a x} \sqrt {1+a x} \left (6+a^2 x^2\right )-3 \left (-2+a^2 x^2+a^4 x^4\right ) \text {arccosh}(a x)}{9 a^4 \sqrt {1-a^2 x^2}} \]

[In]

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

[Out]

-1/9*(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(6 + a^2*x^2) - 3*(-2 + a^2*x^2 + a^4*x^4)*ArcCosh[a*x])/(a^4*Sqrt[1 -
a^2*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(310\) vs. \(2(90)=180\).

Time = 1.18 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.83

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 a^{4} x^{4}-5 a^{2} x^{2}+4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (a x \right )\right )}{72 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x -1}\, \sqrt {a x +1}\, a x +a^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (a x \right )\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x -1\right ) \left (1+\operatorname {arccosh}\left (a x \right )\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 a^{4} x^{4}-5 a^{2} x^{2}-4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}+3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +1\right ) \left (1+3 \,\operatorname {arccosh}\left (a x \right )\right )}{72 a^{4} \left (a^{2} x^{2}-1\right )}\)	\(311\)

[In]

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

[Out]

-1/72*(-a^2*x^2+1)^(1/2)*(4*a^4*x^4-5*a^2*x^2+4*a^3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-3*(a*x-1)^(1/2)*(a*x+1)^(1
/2)*a*x+1)*(-1+3*arccosh(a*x))/a^4/(a^2*x^2-1)-3/8*(-a^2*x^2+1)^(1/2)*((a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x+a^2*x^2
-1)*(-1+arccosh(a*x))/a^4/(a^2*x^2-1)-3/8*(-a^2*x^2+1)^(1/2)*(a^2*x^2-(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x-1)*(1+ar
ccosh(a*x))/a^4/(a^2*x^2-1)-1/72*(-a^2*x^2+1)^(1/2)*(4*a^4*x^4-5*a^2*x^2-4*a^3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)
+3*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x+1)*(1+3*arccosh(a*x))/a^4/(a^2*x^2-1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {3 \, {\left (a^{4} x^{4} + a^{2} x^{2} - 2\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (a^{3} x^{3} + 6 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1}}{9 \, {\left (a^{6} x^{2} - a^{4}\right )}} \]

[In]

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

[Out]

-1/9*(3*(a^4*x^4 + a^2*x^2 - 2)*sqrt(-a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 - 1)) - (a^3*x^3 + 6*a*x)*sqrt(a^2*x
^2 - 1)*sqrt(-a^2*x^2 + 1))/(a^6*x^2 - a^4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.93 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.56 \[ \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {1}{9} \, a {\left (\frac {i \, x^{3}}{a^{2}} + \frac {6 i \, x}{a^{4}}\right )} - \frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arcosh}\left (a x\right ) \]

[In]

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

[Out]

1/9*a*(I*x^3/a^2 + 6*I*x/a^4) - 1/3*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arccosh(a*x)

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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