\(\int x^2 \text {arccosh}(a x)^3 \, dx\) [24]

Optimal result

Integrand size = 10, antiderivative size = 155 \[ \int x^2 \text {arccosh}(a x)^3 \, dx=-\frac {40 \sqrt {-1+a x} \sqrt {1+a x}}{27 a^3}-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {4 x \text {arccosh}(a x)}{3 a^2}+\frac {2}{9} x^3 \text {arccosh}(a x)-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{3 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^3 \]

[Out]

4/3*x*arccosh(a*x)/a^2+2/9*x^3*arccosh(a*x)+1/3*x^3*arccosh(a*x)^3-40/27*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-2/27*
x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a-2/3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-1/3*x^2*arccosh(a*x)^2*(a
*x-1)^(1/2)*(a*x+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5883, 5939, 5915, 5879, 75, 102, 12} \[ \int x^2 \text {arccosh}(a x)^3 \, dx=-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 a^3}-\frac {40 \sqrt {a x-1} \sqrt {a x+1}}{27 a^3}+\frac {4 x \text {arccosh}(a x)}{3 a^2}+\frac {1}{3} x^3 \text {arccosh}(a x)^3+\frac {2}{9} x^3 \text {arccosh}(a x)-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{27 a} \]

[In]

Int[x^2*ArcCosh[a*x]^3,x]

[Out]

(-40*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a^3) - (2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a) + (4*x*ArcCosh[a*x])
/(3*a^2) + (2*x^3*ArcCosh[a*x])/9 - (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(3*a^3) - (x^2*Sqrt[-1 + a
*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(3*a) + (x^3*ArcCosh[a*x]^3)/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

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 5883

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcC
osh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -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]

Rule 5939

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \text {arccosh}(a x)^3-a \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{3 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^3+\frac {2}{3} \int x^2 \text {arccosh}(a x) \, dx-\frac {2 \int \frac {x \text {arccosh}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{3 a} \\ & = \frac {2}{9} x^3 \text {arccosh}(a x)-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{3 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^3+\frac {4 \int \text {arccosh}(a x) \, dx}{3 a^2}-\frac {1}{9} (2 a) \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {4 x \text {arccosh}(a x)}{3 a^2}+\frac {2}{9} x^3 \text {arccosh}(a x)-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{3 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^3-\frac {2 \int \frac {2 x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{27 a}-\frac {4 \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{3 a} \\ & = -\frac {4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a^3}-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {4 x \text {arccosh}(a x)}{3 a^2}+\frac {2}{9} x^3 \text {arccosh}(a x)-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{3 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^3-\frac {4 \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{27 a} \\ & = -\frac {40 \sqrt {-1+a x} \sqrt {1+a x}}{27 a^3}-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {4 x \text {arccosh}(a x)}{3 a^2}+\frac {2}{9} x^3 \text {arccosh}(a x)-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{3 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.66 \[ \int x^2 \text {arccosh}(a x)^3 \, dx=\frac {-2 \sqrt {-1+a x} \sqrt {1+a x} \left (20+a^2 x^2\right )+6 a x \left (6+a^2 x^2\right ) \text {arccosh}(a x)-9 \sqrt {-1+a x} \sqrt {1+a x} \left (2+a^2 x^2\right ) \text {arccosh}(a x)^2+9 a^3 x^3 \text {arccosh}(a x)^3}{27 a^3} \]

[In]

Integrate[x^2*ArcCosh[a*x]^3,x]

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(20 + a^2*x^2) + 6*a*x*(6 + a^2*x^2)*ArcCosh[a*x] - 9*Sqrt[-1 + a*x]*Sqrt[1 +
a*x]*(2 + a^2*x^2)*ArcCosh[a*x]^2 + 9*a^3*x^3*ArcCosh[a*x]^3)/(27*a^3)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\frac {a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{3}}{3}-\frac {2 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{3}-\frac {a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{3}+\frac {4 a x \,\operatorname {arccosh}\left (a x \right )}{3}-\frac {40 \sqrt {a x -1}\, \sqrt {a x +1}}{27}+\frac {2 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )}{9}-\frac {2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{27}}{a^{3}}\)	\(128\)
default	\(\frac {\frac {a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{3}}{3}-\frac {2 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{3}-\frac {a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{3}+\frac {4 a x \,\operatorname {arccosh}\left (a x \right )}{3}-\frac {40 \sqrt {a x -1}\, \sqrt {a x +1}}{27}+\frac {2 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )}{9}-\frac {2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{27}}{a^{3}}\)	\(128\)

[In]

int(x^2*arccosh(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a^3*(1/3*a^3*x^3*arccosh(a*x)^3-2/3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/3*a^2*x^2*arccosh(a*x)^2*(a
*x-1)^(1/2)*(a*x+1)^(1/2)+4/3*a*x*arccosh(a*x)-40/27*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2/9*a^3*x^3*arccosh(a*x)-2/27
*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.80 \[ \int x^2 \text {arccosh}(a x)^3 \, dx=\frac {9 \, a^{3} x^{3} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 9 \, {\left (a^{2} x^{2} + 2\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 6 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (a^{2} x^{2} + 20\right )} \sqrt {a^{2} x^{2} - 1}}{27 \, a^{3}} \]

[In]

integrate(x^2*arccosh(a*x)^3,x, algorithm="fricas")

[Out]

1/27*(9*a^3*x^3*log(a*x + sqrt(a^2*x^2 - 1))^3 - 9*(a^2*x^2 + 2)*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1)
)^2 + 6*(a^3*x^3 + 6*a*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 2*(a^2*x^2 + 20)*sqrt(a^2*x^2 - 1))/a^3

Sympy [F]

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

[In]

integrate(x**2*acosh(a*x)**3,x)

[Out]

Integral(x**2*acosh(a*x)**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.75 \[ \int x^2 \text {arccosh}(a x)^3 \, dx=\frac {1}{3} \, x^{3} \operatorname {arcosh}\left (a x\right )^{3} - \frac {1}{3} \, a {\left (\frac {\sqrt {a^{2} x^{2} - 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {a^{2} x^{2} - 1}}{a^{4}}\right )} \operatorname {arcosh}\left (a x\right )^{2} - \frac {2}{27} \, a {\left (\frac {\sqrt {a^{2} x^{2} - 1} x^{2} + \frac {20 \, \sqrt {a^{2} x^{2} - 1}}{a^{2}}}{a^{2}} - \frac {3 \, {\left (a^{2} x^{3} + 6 \, x\right )} \operatorname {arcosh}\left (a x\right )}{a^{3}}\right )} \]

[In]

integrate(x^2*arccosh(a*x)^3,x, algorithm="maxima")

[Out]

1/3*x^3*arccosh(a*x)^3 - 1/3*a*(sqrt(a^2*x^2 - 1)*x^2/a^2 + 2*sqrt(a^2*x^2 - 1)/a^4)*arccosh(a*x)^2 - 2/27*a*(
(sqrt(a^2*x^2 - 1)*x^2 + 20*sqrt(a^2*x^2 - 1)/a^2)/a^2 - 3*(a^2*x^3 + 6*x)*arccosh(a*x)/a^3)

Giac [F(-2)]

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

[In]

integrate(x^2*arccosh(a*x)^3,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. \[ \int x^2 \text {arccosh}(a x)^3 \, dx=\int x^2\,{\mathrm {acosh}\left (a\,x\right )}^3 \,d x \]

[In]

int(x^2*acosh(a*x)^3,x)

[Out]

int(x^2*acosh(a*x)^3, x)