3.24 \(\int x^2 \cosh ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=155 \[ -\frac {40 \sqrt {a x-1} \sqrt {a x+1}}{27 a^3}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{3 a^3}+\frac {4 x \cosh ^{-1}(a x)}{3 a^2}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^3+\frac {2}{9} x^3 \cosh ^{-1}(a x)-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{27 a}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{3 a} \]

[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

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Rubi [A]  time = 0.47, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5662, 5759, 5718, 5654, 74, 100, 12} \[ -\frac {40 \sqrt {a x-1} \sqrt {a x+1}}{27 a^3}+\frac {4 x \cosh ^{-1}(a x)}{3 a^2}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{3 a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{27 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^3+\frac {2}{9} x^3 \cosh ^{-1}(a x)-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{3 a} \]

Antiderivative was successfully verified.

[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 74

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 100

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 5654

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 5662

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))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5718

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

Rule 5759

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

Rubi steps

\begin {align*} \int x^2 \cosh ^{-1}(a x)^3 \, dx &=\frac {1}{3} x^3 \cosh ^{-1}(a x)^3-a \int \frac {x^3 \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^3+\frac {2}{3} \int x^2 \cosh ^{-1}(a x) \, dx-\frac {2 \int \frac {x \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{3 a}\\ &=\frac {2}{9} x^3 \cosh ^{-1}(a x)-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^3+\frac {4 \int \cosh ^{-1}(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 \cosh ^{-1}(a x)}{3 a^2}+\frac {2}{9} x^3 \cosh ^{-1}(a x)-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \cosh ^{-1}(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 \cosh ^{-1}(a x)}{3 a^2}+\frac {2}{9} x^3 \cosh ^{-1}(a x)-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \cosh ^{-1}(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 \cosh ^{-1}(a x)}{3 a^2}+\frac {2}{9} x^3 \cosh ^{-1}(a x)-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^3\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 103, normalized size = 0.66 \[ \frac {9 a^3 x^3 \cosh ^{-1}(a x)^3-2 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 x^2+20\right )-9 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 x^2+2\right ) \cosh ^{-1}(a x)^2+6 a x \left (a^2 x^2+6\right ) \cosh ^{-1}(a x)}{27 a^3} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.63, size = 124, normalized size = 0.80 \[ \frac {9 \, a^{3} x^{3} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 9 \, {\left (a^{2} x^{2} + 2\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 6 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (a^{2} x^{2} + 20\right )} \sqrt {a^{2} x^{2} - 1}}{27 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.04, size = 128, normalized size = 0.83 \[ \frac {\frac {a^{3} x^{3} \mathrm {arccosh}\left (a x \right )^{3}}{3}-\frac {2 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{3}-\frac {\mathrm {arccosh}\left (a x \right )^{2} a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{3}+\frac {4 a x \,\mathrm {arccosh}\left (a x \right )}{3}-\frac {40 \sqrt {a x -1}\, \sqrt {a x +1}}{27}+\frac {2 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )}{9}-\frac {2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{27}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*arccosh(a*x)^2*a^2*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))

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maxima [A]  time = 0.67, size = 116, normalized size = 0.75 \[ \frac {1}{3} \, x^{3} \operatorname {arcosh}\left (a x\right )^{3} - \frac {1}{3} \, a {\left (\frac {\sqrt {a^{2} x^{2} - 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {a^{2} x^{2} - 1}}{a^{4}}\right )} \operatorname {arcosh}\left (a x\right )^{2} - \frac {2}{27} \, a {\left (\frac {\sqrt {a^{2} x^{2} - 1} x^{2} + \frac {20 \, \sqrt {a^{2} x^{2} - 1}}{a^{2}}}{a^{2}} - \frac {3 \, {\left (a^{2} x^{3} + 6 \, x\right )} \operatorname {arcosh}\left (a x\right )}{a^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {acosh}\left (a\,x\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*acosh(a*x)^3, x)

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sympy [A]  time = 2.04, size = 138, normalized size = 0.89 \[ \begin {cases} \frac {x^{3} \operatorname {acosh}^{3}{\left (a x \right )}}{3} + \frac {2 x^{3} \operatorname {acosh}{\left (a x \right )}}{9} - \frac {x^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{3 a} - \frac {2 x^{2} \sqrt {a^{2} x^{2} - 1}}{27 a} + \frac {4 x \operatorname {acosh}{\left (a x \right )}}{3 a^{2}} - \frac {2 \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{3 a^{3}} - \frac {40 \sqrt {a^{2} x^{2} - 1}}{27 a^{3}} & \text {for}\: a \neq 0 \\- \frac {i \pi ^{3} x^{3}}{24} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**3*acosh(a*x)**3/3 + 2*x**3*acosh(a*x)/9 - x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)**2/(3*a) - 2*x**2*
sqrt(a**2*x**2 - 1)/(27*a) + 4*x*acosh(a*x)/(3*a**2) - 2*sqrt(a**2*x**2 - 1)*acosh(a*x)**2/(3*a**3) - 40*sqrt(
a**2*x**2 - 1)/(27*a**3), Ne(a, 0)), (-I*pi**3*x**3/24, True))

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