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

Optimal. Leaf size=106 \[ -\frac {3 \cosh ^{-1}(a x)^2}{32 a^4}-\frac {3 x \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{16 a^3}+\frac {3 x^2}{32 a^2}+\frac {1}{4} x^4 \cosh ^{-1}(a x)^2-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{8 a}+\frac {x^4}{32} \]

[Out]

3/32*x^2/a^2+1/32*x^4-3/32*arccosh(a*x)^2/a^4+1/4*x^4*arccosh(a*x)^2-3/16*x*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)
^(1/2)/a^3-1/8*x^3*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

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Rubi [A]  time = 0.44, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5662, 5759, 5676, 30} \[ \frac {3 x^2}{32 a^2}-\frac {3 x \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{16 a^3}-\frac {3 \cosh ^{-1}(a x)^2}{32 a^4}+\frac {1}{4} x^4 \cosh ^{-1}(a x)^2-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{8 a}+\frac {x^4}{32} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x^2)/(32*a^2) + x^4/32 - (3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(16*a^3) - (x^3*Sqrt[-1 + a*x]*Sqr
t[1 + a*x]*ArcCosh[a*x])/(8*a) - (3*ArcCosh[a*x]^2)/(32*a^4) + (x^4*ArcCosh[a*x]^2)/4

Rule 30

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

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 5676

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

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^3 \cosh ^{-1}(a x)^2 \, dx &=\frac {1}{4} x^4 \cosh ^{-1}(a x)^2-\frac {1}{2} a \int \frac {x^4 \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{8 a}+\frac {1}{4} x^4 \cosh ^{-1}(a x)^2+\frac {\int x^3 \, dx}{8}-\frac {3 \int \frac {x^2 \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 a}\\ &=\frac {x^4}{32}-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{16 a^3}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{8 a}+\frac {1}{4} x^4 \cosh ^{-1}(a x)^2-\frac {3 \int \frac {\cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 a^3}+\frac {3 \int x \, dx}{16 a^2}\\ &=\frac {3 x^2}{32 a^2}+\frac {x^4}{32}-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{16 a^3}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{8 a}-\frac {3 \cosh ^{-1}(a x)^2}{32 a^4}+\frac {1}{4} x^4 \cosh ^{-1}(a x)^2\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 77, normalized size = 0.73 \[ \frac {\left (8 a^4 x^4-3\right ) \cosh ^{-1}(a x)^2+a^2 x^2 \left (a^2 x^2+3\right )-2 a x \sqrt {a x-1} \sqrt {a x+1} \left (2 a^2 x^2+3\right ) \cosh ^{-1}(a x)}{32 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x^2*(3 + a^2*x^2) - 2*a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(3 + 2*a^2*x^2)*ArcCosh[a*x] + (-3 + 8*a^4*x^4)*Ar
cCosh[a*x]^2)/(32*a^4)

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fricas [A]  time = 0.57, size = 92, normalized size = 0.87 \[ \frac {a^{4} x^{4} + 3 \, a^{2} x^{2} + {\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 2 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{32 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(a^4*x^4 + 3*a^2*x^2 + (8*a^4*x^4 - 3)*log(a*x + sqrt(a^2*x^2 - 1))^2 - 2*(2*a^3*x^3 + 3*a*x)*sqrt(a^2*x^
2 - 1)*log(a*x + sqrt(a^2*x^2 - 1)))/a^4

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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^3*arccosh(a*x)^2,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 = 92, normalized size = 0.87 \[ \frac {\frac {a^{4} x^{4} \mathrm {arccosh}\left (a x \right )^{2}}{4}-\frac {\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{3} x^{3}}{8}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) a x \sqrt {a x -1}\, \sqrt {a x +1}}{16}-\frac {3 \mathrm {arccosh}\left (a x \right )^{2}}{32}+\frac {x^{4} a^{4}}{32}+\frac {3 a^{2} x^{2}}{32}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^4*(1/4*a^4*x^4*arccosh(a*x)^2-1/8*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^3*x^3-3/16*arccosh(a*x)*a*x*(
a*x-1)^(1/2)*(a*x+1)^(1/2)-3/32*arccosh(a*x)^2+1/32*x^4*a^4+3/32*a^2*x^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, x^{4} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2} - \int \frac {{\left (a^{3} x^{6} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x^{5} - a x^{4}\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}{2 \, {\left (a^{3} x^{3} + {\left (a^{2} x^{2} - 1\right )} \sqrt {a x + 1} \sqrt {a x - 1} - a x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*acosh(a*x)^2, x)

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sympy [A]  time = 2.06, size = 99, normalized size = 0.93 \[ \begin {cases} \frac {x^{4} \operatorname {acosh}^{2}{\left (a x \right )}}{4} + \frac {x^{4}}{32} - \frac {x^{3} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{8 a} + \frac {3 x^{2}}{32 a^{2}} - \frac {3 x \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{16 a^{3}} - \frac {3 \operatorname {acosh}^{2}{\left (a x \right )}}{32 a^{4}} & \text {for}\: a \neq 0 \\- \frac {\pi ^{2} x^{4}}{16} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**4*acosh(a*x)**2/4 + x**4/32 - x**3*sqrt(a**2*x**2 - 1)*acosh(a*x)/(8*a) + 3*x**2/(32*a**2) - 3*x
*sqrt(a**2*x**2 - 1)*acosh(a*x)/(16*a**3) - 3*acosh(a*x)**2/(32*a**4), Ne(a, 0)), (-pi**2*x**4/16, True))

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