3.34 \(\int x^3 \cosh ^{-1}(a x)^4 \, dx\)

Optimal. Leaf size=214 \[ \frac{45 x^2}{128 a^2}+\frac{9 x^2 \cosh ^{-1}(a x)^2}{16 a^2}-\frac{3 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{8 a^3}-\frac{45 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{64 a^3}-\frac{3 \cosh ^{-1}(a x)^4}{32 a^4}-\frac{45 \cosh ^{-1}(a x)^2}{128 a^4}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^4+\frac{3}{16} x^4 \cosh ^{-1}(a x)^2-\frac{x^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{4 a}-\frac{3 x^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{32 a}+\frac{3 x^4}{128} \]

[Out]

(45*x^2)/(128*a^2) + (3*x^4)/128 - (45*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(64*a^3) - (3*x^3*Sqrt[-1
+ a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(32*a) - (45*ArcCosh[a*x]^2)/(128*a^4) + (9*x^2*ArcCosh[a*x]^2)/(16*a^2) +
(3*x^4*ArcCosh[a*x]^2)/16 - (3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(8*a^3) - (x^3*Sqrt[-1 + a*x]*Sq
rt[1 + a*x]*ArcCosh[a*x]^3)/(4*a) - (3*ArcCosh[a*x]^4)/(32*a^4) + (x^4*ArcCosh[a*x]^4)/4

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Rubi [A]  time = 1.31439, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5662, 5759, 5676, 30} \[ \frac{45 x^2}{128 a^2}+\frac{9 x^2 \cosh ^{-1}(a x)^2}{16 a^2}-\frac{3 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{8 a^3}-\frac{45 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{64 a^3}-\frac{3 \cosh ^{-1}(a x)^4}{32 a^4}-\frac{45 \cosh ^{-1}(a x)^2}{128 a^4}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^4+\frac{3}{16} x^4 \cosh ^{-1}(a x)^2-\frac{x^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{4 a}-\frac{3 x^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{32 a}+\frac{3 x^4}{128} \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcCosh[a*x]^4,x]

[Out]

(45*x^2)/(128*a^2) + (3*x^4)/128 - (45*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(64*a^3) - (3*x^3*Sqrt[-1
+ a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(32*a) - (45*ArcCosh[a*x]^2)/(128*a^4) + (9*x^2*ArcCosh[a*x]^2)/(16*a^2) +
(3*x^4*ArcCosh[a*x]^2)/16 - (3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(8*a^3) - (x^3*Sqrt[-1 + a*x]*Sq
rt[1 + a*x]*ArcCosh[a*x]^3)/(4*a) - (3*ArcCosh[a*x]^4)/(32*a^4) + (x^4*ArcCosh[a*x]^4)/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)^4 \, dx &=\frac{1}{4} x^4 \cosh ^{-1}(a x)^4-a \int \frac{x^4 \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{4 a}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^4+\frac{3}{4} \int x^3 \cosh ^{-1}(a x)^2 \, dx-\frac{3 \int \frac{x^2 \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{4 a}\\ &=\frac{3}{16} x^4 \cosh ^{-1}(a x)^2-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{8 a^3}-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{4 a}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^4-\frac{3 \int \frac{\cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{8 a^3}+\frac{9 \int x \cosh ^{-1}(a x)^2 \, dx}{8 a^2}-\frac{1}{8} (3 a) \int \frac{x^4 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{3 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{32 a}+\frac{9 x^2 \cosh ^{-1}(a x)^2}{16 a^2}+\frac{3}{16} x^4 \cosh ^{-1}(a x)^2-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{8 a^3}-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{4 a}-\frac{3 \cosh ^{-1}(a x)^4}{32 a^4}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^4+\frac{3 \int x^3 \, dx}{32}-\frac{9 \int \frac{x^2 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{32 a}-\frac{9 \int \frac{x^2 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{8 a}\\ &=\frac{3 x^4}{128}-\frac{45 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{64 a^3}-\frac{3 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{32 a}+\frac{9 x^2 \cosh ^{-1}(a x)^2}{16 a^2}+\frac{3}{16} x^4 \cosh ^{-1}(a x)^2-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{8 a^3}-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{4 a}-\frac{3 \cosh ^{-1}(a x)^4}{32 a^4}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^4-\frac{9 \int \frac{\cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{64 a^3}-\frac{9 \int \frac{\cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{16 a^3}+\frac{9 \int x \, dx}{64 a^2}+\frac{9 \int x \, dx}{16 a^2}\\ &=\frac{45 x^2}{128 a^2}+\frac{3 x^4}{128}-\frac{45 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{64 a^3}-\frac{3 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{32 a}-\frac{45 \cosh ^{-1}(a x)^2}{128 a^4}+\frac{9 x^2 \cosh ^{-1}(a x)^2}{16 a^2}+\frac{3}{16} x^4 \cosh ^{-1}(a x)^2-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{8 a^3}-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{4 a}-\frac{3 \cosh ^{-1}(a x)^4}{32 a^4}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^4\\ \end{align*}

Mathematica [A]  time = 0.109415, size = 143, normalized size = 0.67 \[ \frac{3 a^2 x^2 \left (a^2 x^2+15\right )+4 \left (8 a^4 x^4-3\right ) \cosh ^{-1}(a x)^4-16 a x \sqrt{a x-1} \sqrt{a x+1} \left (2 a^2 x^2+3\right ) \cosh ^{-1}(a x)^3+3 \left (8 a^4 x^4+24 a^2 x^2-15\right ) \cosh ^{-1}(a x)^2-6 a x \sqrt{a x-1} \sqrt{a x+1} \left (2 a^2 x^2+15\right ) \cosh ^{-1}(a x)}{128 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcCosh[a*x]^4,x]

[Out]

(3*a^2*x^2*(15 + a^2*x^2) - 6*a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(15 + 2*a^2*x^2)*ArcCosh[a*x] + 3*(-15 + 24*a^2
*x^2 + 8*a^4*x^4)*ArcCosh[a*x]^2 - 16*a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(3 + 2*a^2*x^2)*ArcCosh[a*x]^3 + 4*(-3
+ 8*a^4*x^4)*ArcCosh[a*x]^4)/(128*a^4)

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Maple [A]  time = 0.045, size = 224, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{4}} \left ({\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{4} \left ( ax-1 \right ) \left ( ax+1 \right ){a}^{2}{x}^{2}}{4}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}{a}^{2}{x}^{2}}{4}}-{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{3}{x}^{3}}{4}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}ax}{8}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}}{32}}+{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2} \left ( ax-1 \right ) \left ( ax+1 \right ){a}^{2}{x}^{2}}{16}}+{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{2}{x}^{2}}{4}}-{\frac{3\,{a}^{3}{x}^{3}{\rm arccosh} \left (ax\right )}{32}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{45\,ax{\rm arccosh} \left (ax\right )}{64}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{45\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{128}}+{\frac{ \left ( 3\,ax-3 \right ) \left ( ax+1 \right ){a}^{2}{x}^{2}}{128}}+{\frac{3\,{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)^4,x)

[Out]

1/a^4*(1/4*arccosh(a*x)^4*(a*x-1)*(a*x+1)*a^2*x^2+1/4*arccosh(a*x)^4*a^2*x^2-1/4*arccosh(a*x)^3*(a*x-1)^(1/2)*
(a*x+1)^(1/2)*a^3*x^3-3/8*arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x-3/32*arccosh(a*x)^4+3/16*arccosh(a*x)
^2*(a*x-1)*(a*x+1)*a^2*x^2+3/4*arccosh(a*x)^2*a^2*x^2-3/32*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^3*x^3-45
/64*arccosh(a*x)*a*x*(a*x-1)^(1/2)*(a*x+1)^(1/2)-45/128*arccosh(a*x)^2+3/128*(a*x-1)*(a*x+1)*a^2*x^2+3/8*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 )^{4} - \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 )^{3}}{a^{3} x^{3} +{\left (a^{2} x^{2} - 1\right )} \sqrt{a x + 1} \sqrt{a x - 1} - a x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccosh(a*x)^4,x, algorithm="maxima")

[Out]

1/4*x^4*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^4 - integrate((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))^3/(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 = 2.4774, size = 402, normalized size = 1.88 \begin{align*} \frac{3 \, a^{4} x^{4} + 4 \,{\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{4} - 16 \,{\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 )^{3} + 45 \, a^{2} x^{2} + 3 \,{\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} - 6 \,{\left (2 \, a^{3} x^{3} + 15 \, a x\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{128 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccosh(a*x)^4,x, algorithm="fricas")

[Out]

1/128*(3*a^4*x^4 + 4*(8*a^4*x^4 - 3)*log(a*x + sqrt(a^2*x^2 - 1))^4 - 16*(2*a^3*x^3 + 3*a*x)*sqrt(a^2*x^2 - 1)
*log(a*x + sqrt(a^2*x^2 - 1))^3 + 45*a^2*x^2 + 3*(8*a^4*x^4 + 24*a^2*x^2 - 15)*log(a*x + sqrt(a^2*x^2 - 1))^2
- 6*(2*a^3*x^3 + 15*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 = 8.66786, size = 197, normalized size = 0.92 \begin{align*} \begin{cases} \frac{x^{4} \operatorname{acosh}^{4}{\left (a x \right )}}{4} + \frac{3 x^{4} \operatorname{acosh}^{2}{\left (a x \right )}}{16} + \frac{3 x^{4}}{128} - \frac{x^{3} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{3}{\left (a x \right )}}{4 a} - \frac{3 x^{3} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{32 a} + \frac{9 x^{2} \operatorname{acosh}^{2}{\left (a x \right )}}{16 a^{2}} + \frac{45 x^{2}}{128 a^{2}} - \frac{3 x \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{3}{\left (a x \right )}}{8 a^{3}} - \frac{45 x \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{64 a^{3}} - \frac{3 \operatorname{acosh}^{4}{\left (a x \right )}}{32 a^{4}} - \frac{45 \operatorname{acosh}^{2}{\left (a x \right )}}{128 a^{4}} & \text{for}\: a \neq 0 \\\frac{\pi ^{4} x^{4}}{64} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*acosh(a*x)**4,x)

[Out]

Piecewise((x**4*acosh(a*x)**4/4 + 3*x**4*acosh(a*x)**2/16 + 3*x**4/128 - x**3*sqrt(a**2*x**2 - 1)*acosh(a*x)**
3/(4*a) - 3*x**3*sqrt(a**2*x**2 - 1)*acosh(a*x)/(32*a) + 9*x**2*acosh(a*x)**2/(16*a**2) + 45*x**2/(128*a**2) -
3*x*sqrt(a**2*x**2 - 1)*acosh(a*x)**3/(8*a**3) - 45*x*sqrt(a**2*x**2 - 1)*acosh(a*x)/(64*a**3) - 3*acosh(a*x)
**4/(32*a**4) - 45*acosh(a*x)**2/(128*a**4), Ne(a, 0)), (pi**4*x**4/64, 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 )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccosh(a*x)^4,x, algorithm="giac")

[Out]

integrate(x^3*arccosh(a*x)^4, x)