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

Optimal. Leaf size=77 \[ -\frac{3 x \sqrt{a x-1} \sqrt{a x+1}}{32 a^3}-\frac{3 \cosh ^{-1}(a x)}{32 a^4}-\frac{x^3 \sqrt{a x-1} \sqrt{a x+1}}{16 a}+\frac{1}{4} x^4 \cosh ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.0299139, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5662, 100, 12, 90, 52} \[ -\frac{3 x \sqrt{a x-1} \sqrt{a x+1}}{32 a^3}-\frac{3 \cosh ^{-1}(a x)}{32 a^4}-\frac{x^3 \sqrt{a x-1} \sqrt{a x+1}}{16 a}+\frac{1}{4} x^4 \cosh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(32*a^3) - (x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(16*a) - (3*ArcCosh[a*x])/(3
2*a^4) + (x^4*ArcCosh[a*x])/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 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 12

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

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

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

Mathematica [A]  time = 0.0622652, size = 71, normalized size = 0.92 \[ -\frac{a x \sqrt{a x-1} \sqrt{a x+1} \left (2 a^2 x^2+3\right )-8 a^4 x^4 \cosh ^{-1}(a x)+6 \tanh ^{-1}\left (\sqrt{\frac{a x-1}{a x+1}}\right )}{32 a^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(3 + 2*a^2*x^2) - 8*a^4*x^4*ArcCosh[a*x] + 6*ArcTanh[Sqrt[(-1 + a*x)/(1 + a
*x)]])/(32*a^4)

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Maple [A]  time = 0.017, size = 99, normalized size = 1.3 \begin{align*}{\frac{{x}^{4}{\rm arccosh} \left (ax\right )}{4}}-{\frac{{x}^{3}}{16\,a}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{3\,x}{32\,{a}^{3}}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{3}{32\,{a}^{4}}\sqrt{ax-1}\sqrt{ax+1}\ln \left ( ax+\sqrt{{a}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14385, size = 116, normalized size = 1.51 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{arcosh}\left (a x\right ) - \frac{1}{32} \,{\left (\frac{2 \, \sqrt{a^{2} x^{2} - 1} x^{3}}{a^{2}} + \frac{3 \, \sqrt{a^{2} x^{2} - 1} x}{a^{4}} + \frac{3 \, \log \left (2 \, a^{2} x + 2 \, \sqrt{a^{2} x^{2} - 1} \sqrt{a^{2}}\right )}{\sqrt{a^{2}} a^{4}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.27403, size = 131, normalized size = 1.7 \begin{align*} \frac{{\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) -{\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt{a^{2} x^{2} - 1}}{32 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.34646, size = 68, normalized size = 0.88 \begin{align*} \begin{cases} \frac{x^{4} \operatorname{acosh}{\left (a x \right )}}{4} - \frac{x^{3} \sqrt{a^{2} x^{2} - 1}}{16 a} - \frac{3 x \sqrt{a^{2} x^{2} - 1}}{32 a^{3}} - \frac{3 \operatorname{acosh}{\left (a x \right )}}{32 a^{4}} & \text{for}\: a \neq 0 \\\frac{i \pi x^{4}}{8} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.35273, size = 109, normalized size = 1.42 \begin{align*} \frac{1}{4} \, x^{4} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{1}{32} \,{\left (\sqrt{a^{2} x^{2} - 1} x{\left (\frac{2 \, x^{2}}{a^{2}} + \frac{3}{a^{4}}\right )} - \frac{3 \, \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right )}{a^{4}{\left | a \right |}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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