3.1.34 \(\int x^3 \cosh ^{-1}(a x)^4 \, dx\) [34]

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

[Out]

45/128*x^2/a^2+3/128*x^4-45/128*arccosh(a*x)^2/a^4+9/16*x^2*arccosh(a*x)^2/a^2+3/16*x^4*arccosh(a*x)^2-3/32*ar
ccosh(a*x)^4/a^4+1/4*x^4*arccosh(a*x)^4-45/64*x*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-3/32*x^3*arccosh(
a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a-3/8*x*arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-1/4*x^3*arccosh(a*x)^3
*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

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Rubi [A]
time = 0.85, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5883, 5939, 5893, 30} \begin {gather*} -\frac {3 \cosh ^{-1}(a x)^4}{32 a^4}-\frac {45 \cosh ^{-1}(a x)^2}{128 a^4}-\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 {45 x^2}{128 a^2}+\frac {9 x^2 \cosh ^{-1}(a x)^2}{16 a^2}+\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} \end {gather*}

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 30

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

Rule 5883

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)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1
*e2*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)
^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 +
e2*x)^p/(-1 + c*x)^p], Int[(f*x)^(m - 1)*(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, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IG
tQ[m, 1] && NeQ[m + 2*p + 1, 0]

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*}

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Mathematica [A]
time = 0.08, size = 143, normalized size = 0.67 \begin {gather*} \frac {3 a^2 x^2 \left (15+a^2 x^2\right )-6 a x \sqrt {-1+a x} \sqrt {1+a x} \left (15+2 a^2 x^2\right ) \cosh ^{-1}(a x)+3 \left (-15+24 a^2 x^2+8 a^4 x^4\right ) \cosh ^{-1}(a x)^2-16 a x \sqrt {-1+a x} \sqrt {1+a x} \left (3+2 a^2 x^2\right ) \cosh ^{-1}(a x)^3+4 \left (-3+8 a^4 x^4\right ) \cosh ^{-1}(a x)^4}{128 a^4} \end {gather*}

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 [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{3} \mathrm {arccosh}\left (a x \right )^{4}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arccosh(a*x)^4,x)

[Out]

int(x^3*arccosh(a*x)^4,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.35, size = 176, normalized size = 0.82 \begin {gather*} \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 {gather*}

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 = 0.70, size = 197, normalized size = 0.92 \begin {gather*} \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 {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,{\mathrm {acosh}\left (a\,x\right )}^4 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*acosh(a*x)^4,x)

[Out]

int(x^3*acosh(a*x)^4, x)

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