3.1.26 \(\int \cosh ^{-1}(a x)^3 \, dx\) [26]

Optimal. Leaf size=68 \[ -\frac {6 \sqrt {-1+a x} \sqrt {1+a x}}{a}+6 x \cosh ^{-1}(a x)-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a}+x \cosh ^{-1}(a x)^3 \]

[Out]

6*x*arccosh(a*x)+x*arccosh(a*x)^3-6*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a-3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)
/a

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Rubi [A]
time = 0.12, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5879, 5915, 75} \begin {gather*} -\frac {6 \sqrt {a x-1} \sqrt {a x+1}}{a}+x \cosh ^{-1}(a x)^3-\frac {3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{a}+6 x \cosh ^{-1}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 75

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

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 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]

Rubi steps

\begin {align*} \int \cosh ^{-1}(a x)^3 \, dx &=x \cosh ^{-1}(a x)^3-(3 a) \int \frac {x \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a}+x \cosh ^{-1}(a x)^3+6 \int \cosh ^{-1}(a x) \, dx\\ &=6 x \cosh ^{-1}(a x)-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a}+x \cosh ^{-1}(a x)^3-(6 a) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {6 \sqrt {-1+a x} \sqrt {1+a x}}{a}+6 x \cosh ^{-1}(a x)-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a}+x \cosh ^{-1}(a x)^3\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 68, normalized size = 1.00 \begin {gather*} -\frac {6 \sqrt {-1+a x} \sqrt {1+a x}}{a}+6 x \cosh ^{-1}(a x)-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a}+x \cosh ^{-1}(a x)^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 1.78, size = 61, normalized size = 0.90

method	result	size
derivativedivides	\(\frac {\mathrm {arccosh}\left (a x \right )^{3} a x -3 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+6 a x \,\mathrm {arccosh}\left (a x \right )-6 \sqrt {a x -1}\, \sqrt {a x +1}}{a}\)	\(61\)
default	\(\frac {\mathrm {arccosh}\left (a x \right )^{3} a x -3 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+6 a x \,\mathrm {arccosh}\left (a x \right )-6 \sqrt {a x -1}\, \sqrt {a x +1}}{a}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a*(arccosh(a*x)^3*a*x-3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+6*a*x*arccosh(a*x)-6*(a*x-1)^(1/2)*(a*x+1
)^(1/2))

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Maxima [A]
time = 0.25, size = 57, normalized size = 0.84 \begin {gather*} x \operatorname {arcosh}\left (a x\right )^{3} - \frac {3 \, \sqrt {a^{2} x^{2} - 1} \operatorname {arcosh}\left (a x\right )^{2}}{a} + \frac {6 \, {\left (a x \operatorname {arcosh}\left (a x\right ) - \sqrt {a^{2} x^{2} - 1}\right )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arccosh(a*x)^3 - 3*sqrt(a^2*x^2 - 1)*arccosh(a*x)^2/a + 6*(a*x*arccosh(a*x) - sqrt(a^2*x^2 - 1))/a

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Fricas [A]
time = 0.41, size = 90, normalized size = 1.32 \begin {gather*} \frac {a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} + 6 \, a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 3 \, \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 6 \, \sqrt {a^{2} x^{2} - 1}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x*log(a*x + sqrt(a^2*x^2 - 1))^3 + 6*a*x*log(a*x + sqrt(a^2*x^2 - 1)) - 3*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(
a^2*x^2 - 1))^2 - 6*sqrt(a^2*x^2 - 1))/a

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Sympy [C] Result contains complex when optimal does not.
time = 0.12, size = 63, normalized size = 0.93 \begin {gather*} \begin {cases} x \operatorname {acosh}^{3}{\left (a x \right )} + 6 x \operatorname {acosh}{\left (a x \right )} - \frac {3 \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{a} - \frac {6 \sqrt {a^{2} x^{2} - 1}}{a} & \text {for}\: a \neq 0 \\- \frac {i \pi ^{3} x}{8} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*acosh(a*x)**3 + 6*x*acosh(a*x) - 3*sqrt(a**2*x**2 - 1)*acosh(a*x)**2/a - 6*sqrt(a**2*x**2 - 1)/a,
Ne(a, 0)), (-I*pi**3*x/8, True))

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Giac [A]
time = 0.40, size = 98, normalized size = 1.44 \begin {gather*} x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 3 \, a {\left (\frac {\sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}}{a^{2}} - \frac {2 \, {\left (x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {\sqrt {a^{2} x^{2} - 1}}{a}\right )}}{a}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {acosh}\left (a\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acosh(a*x)^3, x)

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