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

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, 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 12

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

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]

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 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 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]

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*}

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Mathematica [A]  time = 0.07, size = 71, normalized size = 0.92 \[ -\frac {-8 a^4 x^4 \cosh ^{-1}(a x)+a x \sqrt {a x-1} \sqrt {a x+1} \left (2 a^2 x^2+3\right )+6 \tanh ^{-1}\left (\sqrt {\frac {a x-1}{a x+1}}\right )}{32 a^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/32*(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)]])/a^4

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fricas [A]  time = 0.56, size = 59, normalized size = 0.77 \[ \frac {{\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {a^{2} x^{2} - 1}}{32 \, a^{4}} \]

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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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),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 = 0.02, size = 99, normalized size = 1.29 \[ \frac {x^{4} \mathrm {arccosh}\left (a x \right )}{4}-\frac {x^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{16 a}-\frac {3 x \sqrt {a x -1}\, \sqrt {a x +1}}{32 a^{3}}-\frac {3 \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (a x +\sqrt {a^{2} x^{2}-1}\right )}{32 a^{4} \sqrt {a^{2} x^{2}-1}} \]

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 = 0.69, size = 77, normalized size = 1.00 \[ \frac {1}{4} \, x^{4} \operatorname {arcosh}\left (a x\right ) - \frac {1}{32} \, {\left (\frac {2 \, \sqrt {a^{2} x^{2} - 1} x^{3}}{a^{2}} + \frac {3 \, \sqrt {a^{2} x^{2} - 1} x}{a^{4}} + \frac {3 \, \log \left (2 \, a^{2} x + 2 \, \sqrt {a^{2} x^{2} - 1} a\right )}{a^{5}}\right )} a \]

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)*a)/a^5)*a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*acosh(a*x), x)

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sympy [A]  time = 1.04, size = 68, normalized size = 0.88 \[ \begin {cases} \frac {x^{4} \operatorname {acosh}{\left (a x \right )}}{4} - \frac {x^{3} \sqrt {a^{2} x^{2} - 1}}{16 a} - \frac {3 x \sqrt {a^{2} x^{2} - 1}}{32 a^{3}} - \frac {3 \operatorname {acosh}{\left (a x \right )}}{32 a^{4}} & \text {for}\: a \neq 0 \\\frac {i \pi x^{4}}{8} & \text {otherwise} \end {cases} \]

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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