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

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

[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

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Rubi [A]  time = 0.440192, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, 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 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 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]

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 30

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

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*}

Mathematica [A]  time = 0.0715097, size = 77, normalized size = 0.73 \[ \frac{a^2 x^2 \left (a^2 x^2+3\right )-2 a x \sqrt{a x-1} \sqrt{a x+1} \left (2 a^2 x^2+3\right ) \cosh ^{-1}(a x)+\left (8 a^4 x^4-3\right ) \cosh ^{-1}(a x)^2}{32 a^4} \]

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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Maple [A]  time = 0.035, size = 126, normalized size = 1.2 \begin{align*}{\frac{1}{{a}^{4}} \left ({\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2} \left ( ax-1 \right ) \left ( ax+1 \right ){a}^{2}{x}^{2}}{4}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{2}{x}^{2}}{4}}-{\frac{{a}^{3}{x}^{3}{\rm arccosh} \left (ax\right )}{8}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{3\,ax{\rm arccosh} \left (ax\right )}{16}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{32}}+{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ){a}^{2}{x}^{2}}{32}}+{\frac{{a}^{2}{x}^{2}}{8}} \right ) } \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 2.36711, size = 205, normalized size = 1.93 \begin{align*} \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}} \end{align*}

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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Sympy [A]  time = 2.87612, size = 99, normalized size = 0.93 \begin{align*} \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} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{arcosh}\left (a x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*arccosh(a*x)^2, x)