3.1.35 \(\int x^2 \cosh ^{-1}(a x)^4 \, dx\) [35]

Optimal. Leaf size=182 \[ \frac {160 x}{27 a^2}+\frac {8 x^3}{81}-\frac {160 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{27 a^3}-\frac {8 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{27 a}+\frac {8 x \cosh ^{-1}(a x)^2}{3 a^2}+\frac {4}{9} x^3 \cosh ^{-1}(a x)^2-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{9 a^3}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{9 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^4 \]

[Out]

160/27*x/a^2+8/81*x^3+8/3*x*arccosh(a*x)^2/a^2+4/9*x^3*arccosh(a*x)^2+1/3*x^3*arccosh(a*x)^4-160/27*arccosh(a*
x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-8/27*x^2*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a-8/9*arccosh(a*x)^3*(a*x
-1)^(1/2)*(a*x+1)^(1/2)/a^3-4/9*x^2*arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

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Rubi [A]
time = 0.57, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5883, 5939, 5915, 5879, 8, 30} \begin {gather*} -\frac {8 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{9 a^3}-\frac {160 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{27 a^3}+\frac {160 x}{27 a^2}+\frac {8 x \cosh ^{-1}(a x)^2}{3 a^2}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^4+\frac {4}{9} x^3 \cosh ^{-1}(a x)^2-\frac {4 x^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{9 a}-\frac {8 x^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{27 a}+\frac {8 x^3}{81} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*ArcCosh[a*x]^4,x]

[Out]

(160*x)/(27*a^2) + (8*x^3)/81 - (160*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(27*a^3) - (8*x^2*Sqrt[-1 + a*
x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(27*a) + (8*x*ArcCosh[a*x]^2)/(3*a^2) + (4*x^3*ArcCosh[a*x]^2)/9 - (8*Sqrt[-1 +
a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(9*a^3) - (4*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(9*a) + (x^3
*ArcCosh[a*x]^4)/3

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 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*} \int x^2 \cosh ^{-1}(a x)^4 \, dx &=\frac {1}{3} x^3 \cosh ^{-1}(a x)^4-\frac {1}{3} (4 a) \int \frac {x^3 \cosh ^{-1}(a x)^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{9 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^4+\frac {4}{3} \int x^2 \cosh ^{-1}(a x)^2 \, dx-\frac {8 \int \frac {x \cosh ^{-1}(a x)^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{9 a}\\ &=\frac {4}{9} x^3 \cosh ^{-1}(a x)^2-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{9 a^3}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{9 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^4+\frac {8 \int \cosh ^{-1}(a x)^2 \, dx}{3 a^2}-\frac {1}{9} (8 a) \int \frac {x^3 \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {8 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{27 a}+\frac {8 x \cosh ^{-1}(a x)^2}{3 a^2}+\frac {4}{9} x^3 \cosh ^{-1}(a x)^2-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{9 a^3}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{9 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^4+\frac {8 \int x^2 \, dx}{27}-\frac {16 \int \frac {x \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{27 a}-\frac {16 \int \frac {x \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{3 a}\\ &=\frac {8 x^3}{81}-\frac {160 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{27 a^3}-\frac {8 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{27 a}+\frac {8 x \cosh ^{-1}(a x)^2}{3 a^2}+\frac {4}{9} x^3 \cosh ^{-1}(a x)^2-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{9 a^3}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{9 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^4+\frac {16 \int 1 \, dx}{27 a^2}+\frac {16 \int 1 \, dx}{3 a^2}\\ &=\frac {160 x}{27 a^2}+\frac {8 x^3}{81}-\frac {160 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{27 a^3}-\frac {8 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{27 a}+\frac {8 x \cosh ^{-1}(a x)^2}{3 a^2}+\frac {4}{9} x^3 \cosh ^{-1}(a x)^2-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{9 a^3}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{9 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^4\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 122, normalized size = 0.67 \begin {gather*} \frac {8 a x \left (60+a^2 x^2\right )-24 \sqrt {-1+a x} \sqrt {1+a x} \left (20+a^2 x^2\right ) \cosh ^{-1}(a x)+36 a x \left (6+a^2 x^2\right ) \cosh ^{-1}(a x)^2-36 \sqrt {-1+a x} \sqrt {1+a x} \left (2+a^2 x^2\right ) \cosh ^{-1}(a x)^3+27 a^3 x^3 \cosh ^{-1}(a x)^4}{81 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcCosh[a*x]^4,x]

[Out]

(8*a*x*(60 + a^2*x^2) - 24*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(20 + a^2*x^2)*ArcCosh[a*x] + 36*a*x*(6 + a^2*x^2)*Arc
Cosh[a*x]^2 - 36*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(2 + a^2*x^2)*ArcCosh[a*x]^3 + 27*a^3*x^3*ArcCosh[a*x]^4)/(81*a^
3)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{2} \mathrm {arccosh}\left (a x \right )^{4}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccosh(a*x)^4,x)

[Out]

int(x^2*arccosh(a*x)^4,x)

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Maxima [A]
time = 0.27, size = 143, normalized size = 0.79 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {arcosh}\left (a x\right )^{4} - \frac {4}{9} \, 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 )^{3} - \frac {4}{81} \, {\left (2 \, a {\left (\frac {3 \, {\left (\sqrt {a^{2} x^{2} - 1} x^{2} + \frac {20 \, \sqrt {a^{2} x^{2} - 1}}{a^{2}}\right )} \operatorname {arcosh}\left (a x\right )}{a^{3}} - \frac {a^{2} x^{3} + 60 \, x}{a^{4}}\right )} - \frac {9 \, {\left (a^{2} x^{3} + 6 \, x\right )} \operatorname {arcosh}\left (a x\right )^{2}}{a^{3}}\right )} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccosh(a*x)^4,x, algorithm="maxima")

[Out]

1/3*x^3*arccosh(a*x)^4 - 4/9*a*(sqrt(a^2*x^2 - 1)*x^2/a^2 + 2*sqrt(a^2*x^2 - 1)/a^4)*arccosh(a*x)^3 - 4/81*(2*
a*(3*(sqrt(a^2*x^2 - 1)*x^2 + 20*sqrt(a^2*x^2 - 1)/a^2)*arccosh(a*x)/a^3 - (a^2*x^3 + 60*x)/a^4) - 9*(a^2*x^3
+ 6*x)*arccosh(a*x)^2/a^3)*a

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Fricas [A]
time = 0.37, size = 154, normalized size = 0.85 \begin {gather*} \frac {27 \, a^{3} x^{3} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{4} + 8 \, a^{3} x^{3} - 36 \, {\left (a^{2} x^{2} + 2\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} + 36 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 24 \, {\left (a^{2} x^{2} + 20\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + 480 \, a x}{81 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccosh(a*x)^4,x, algorithm="fricas")

[Out]

1/81*(27*a^3*x^3*log(a*x + sqrt(a^2*x^2 - 1))^4 + 8*a^3*x^3 - 36*(a^2*x^2 + 2)*sqrt(a^2*x^2 - 1)*log(a*x + sqr
t(a^2*x^2 - 1))^3 + 36*(a^3*x^3 + 6*a*x)*log(a*x + sqrt(a^2*x^2 - 1))^2 - 24*(a^2*x^2 + 20)*sqrt(a^2*x^2 - 1)*
log(a*x + sqrt(a^2*x^2 - 1)) + 480*a*x)/a^3

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Sympy [A]
time = 0.47, size = 165, normalized size = 0.91 \begin {gather*} \begin {cases} \frac {x^{3} \operatorname {acosh}^{4}{\left (a x \right )}}{3} + \frac {4 x^{3} \operatorname {acosh}^{2}{\left (a x \right )}}{9} + \frac {8 x^{3}}{81} - \frac {4 x^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{3}{\left (a x \right )}}{9 a} - \frac {8 x^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{27 a} + \frac {8 x \operatorname {acosh}^{2}{\left (a x \right )}}{3 a^{2}} + \frac {160 x}{27 a^{2}} - \frac {8 \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{3}{\left (a x \right )}}{9 a^{3}} - \frac {160 \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{27 a^{3}} & \text {for}\: a \neq 0 \\\frac {\pi ^{4} x^{3}}{48} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*acosh(a*x)**4,x)

[Out]

Piecewise((x**3*acosh(a*x)**4/3 + 4*x**3*acosh(a*x)**2/9 + 8*x**3/81 - 4*x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)**
3/(9*a) - 8*x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)/(27*a) + 8*x*acosh(a*x)**2/(3*a**2) + 160*x/(27*a**2) - 8*sqrt
(a**2*x**2 - 1)*acosh(a*x)**3/(9*a**3) - 160*sqrt(a**2*x**2 - 1)*acosh(a*x)/(27*a**3), Ne(a, 0)), (pi**4*x**3/
48, True))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccosh(a*x)^4,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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\mathrm {acosh}\left (a\,x\right )}^4 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*acosh(a*x)^4,x)

[Out]

int(x^2*acosh(a*x)^4, x)

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