\(\int x^3 \text {arccosh}(a x)^2 \, dx\) [13]

Optimal result

Integrand size = 10, antiderivative size = 106 \[ \int x^3 \text {arccosh}(a x)^2 \, dx=\frac {3 x^2}{32 a^2}+\frac {x^4}{32}-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{16 a^3}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{8 a}-\frac {3 \text {arccosh}(a x)^2}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^2 \]

[Out]

3/32*x^2/a^2+1/32*x^4-3/32*arccosh(a*x)^2/a^4+1/4*x^4*arccosh(a*x)^2-3/16*x*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)
^(1/2)/a^3-1/8*x^3*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5883, 5939, 5893, 30} \[ \int x^3 \text {arccosh}(a x)^2 \, dx=-\frac {3 \text {arccosh}(a x)^2}{32 a^4}-\frac {3 x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{16 a^3}+\frac {3 x^2}{32 a^2}+\frac {1}{4} x^4 \text {arccosh}(a x)^2-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{8 a}+\frac {x^4}{32} \]

[In]

Int[x^3*ArcCosh[a*x]^2,x]

[Out]

(3*x^2)/(32*a^2) + x^4/32 - (3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(16*a^3) - (x^3*Sqrt[-1 + a*x]*Sqr
t[1 + a*x]*ArcCosh[a*x])/(8*a) - (3*ArcCosh[a*x]^2)/(32*a^4) + (x^4*ArcCosh[a*x]^2)/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*} \text {integral}& = \frac {1}{4} x^4 \text {arccosh}(a x)^2-\frac {1}{2} a \int \frac {x^4 \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{8 a}+\frac {1}{4} x^4 \text {arccosh}(a x)^2+\frac {\int x^3 \, dx}{8}-\frac {3 \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 a} \\ & = \frac {x^4}{32}-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{16 a^3}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{8 a}+\frac {1}{4} x^4 \text {arccosh}(a x)^2-\frac {3 \int \frac {\text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 a^3}+\frac {3 \int x \, dx}{16 a^2} \\ & = \frac {3 x^2}{32 a^2}+\frac {x^4}{32}-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{16 a^3}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{8 a}-\frac {3 \text {arccosh}(a x)^2}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.73 \[ \int x^3 \text {arccosh}(a x)^2 \, dx=\frac {a^2 x^2 \left (3+a^2 x^2\right )-2 a x \sqrt {-1+a x} \sqrt {1+a x} \left (3+2 a^2 x^2\right ) \text {arccosh}(a x)+\left (-3+8 a^4 x^4\right ) \text {arccosh}(a x)^2}{32 a^4} \]

[In]

Integrate[x^3*ArcCosh[a*x]^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {\frac {a^{4} x^{4} \operatorname {arccosh}\left (a x \right )^{2}}{4}-\frac {a^{3} x^{3} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{8}-\frac {3 a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{16}-\frac {3 \operatorname {arccosh}\left (a x \right )^{2}}{32}+\frac {a^{4} x^{4}}{32}+\frac {3 a^{2} x^{2}}{32}}{a^{4}}\)	\(92\)
default	\(\frac {\frac {a^{4} x^{4} \operatorname {arccosh}\left (a x \right )^{2}}{4}-\frac {a^{3} x^{3} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{8}-\frac {3 a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{16}-\frac {3 \operatorname {arccosh}\left (a x \right )^{2}}{32}+\frac {a^{4} x^{4}}{32}+\frac {3 a^{2} x^{2}}{32}}{a^{4}}\)	\(92\)

[In]

int(x^3*arccosh(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/a^4*(1/4*a^4*x^4*arccosh(a*x)^2-1/8*a^3*x^3*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)-3/16*a*x*arccosh(a*x)*(
a*x-1)^(1/2)*(a*x+1)^(1/2)-3/32*arccosh(a*x)^2+1/32*a^4*x^4+3/32*a^2*x^2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int x^3 \text {arccosh}(a x)^2 \, dx=\frac {a^{4} x^{4} + 3 \, a^{2} x^{2} + {\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 2 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{32 \, a^{4}} \]

[In]

integrate(x^3*arccosh(a*x)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int x^3 \text {arccosh}(a x)^2 \, dx=\int x^{3} \operatorname {acosh}^{2}{\left (a x \right )}\, dx \]

[In]

integrate(x**3*acosh(a*x)**2,x)

[Out]

Integral(x**3*acosh(a*x)**2, x)

Maxima [F]

\[ \int x^3 \text {arccosh}(a x)^2 \, dx=\int { x^{3} \operatorname {arcosh}\left (a x\right )^{2} \,d x } \]

[In]

integrate(x^3*arccosh(a*x)^2,x, algorithm="maxima")

[Out]

1/4*x^4*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2 - integrate(1/2*(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))/(a^3*x^3 + (a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*x),
x)

Giac [F(-2)]

Exception generated. \[ \int x^3 \text {arccosh}(a x)^2 \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3*arccosh(a*x)^2,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

Mupad [F(-1)]

Timed out. \[ \int x^3 \text {arccosh}(a x)^2 \, dx=\int x^3\,{\mathrm {acosh}\left (a\,x\right )}^2 \,d x \]

[In]

int(x^3*acosh(a*x)^2,x)

[Out]

int(x^3*acosh(a*x)^2, x)