Integrand size = 8, antiderivative size = 77 \[ \int x^3 \text {arccosh}(a x) \, dx=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x}}{32 a^3}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{16 a}-\frac {3 \text {arccosh}(a x)}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x) \] Output:
-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-3/32*arccosh(a*x)/a^4+1/4*x^4*arccosh(a*x)
Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int x^3 \text {arccosh}(a x) \, dx=-\frac {a x \sqrt {-1+a x} \sqrt {1+a x} \left (3+2 a^2 x^2\right )-8 a^4 x^4 \text {arccosh}(a x)+6 \text {arctanh}\left (\sqrt {\frac {-1+a x}{1+a x}}\right )}{32 a^4} \] Input:
Integrate[x^3*ArcCosh[a*x],x]
Output:
-1/32*(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(3 + 2*a^2*x^2) - 8*a^4*x^4*ArcCos h[a*x] + 6*ArcTanh[Sqrt[(-1 + a*x)/(1 + a*x)]])/a^4
Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6298, 111, 27, 101, 43}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^3 \text {arccosh}(a x) \, dx\) |
\(\Big \downarrow \) 6298 |
\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \int \frac {x^4}{\sqrt {a x-1} \sqrt {a x+1}}dx\) |
\(\Big \downarrow \) 111 |
\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \left (\frac {\int \frac {3 x^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{4 a^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \left (\frac {3 \int \frac {x^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{4 a^2}\right )\) |
\(\Big \downarrow \) 101 |
\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a^2}\right )}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{4 a^2}\right )\) |
\(\Big \downarrow \) 43 |
\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \left (\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{4 a^2}+\frac {3 \left (\frac {\text {arccosh}(a x)}{2 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a^2}\right )}{4 a^2}\right )\) |
Input:
Int[x^3*ArcCosh[a*x],x]
Output:
(x^4*ArcCosh[a*x])/4 - (a*((x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(4*a^2) + (3 *((x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a^2) + ArcCosh[a*x]/(2*a^3)))/(4*a^2 )))/4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[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]
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.12
method | result | size |
orering | \(\frac {\left (14 a^{4} x^{4}+3 a^{2} x^{2}-12\right ) \operatorname {arccosh}\left (a x \right )}{32 a^{4}}-\frac {\left (2 a^{2} x^{2}+3\right ) \left (a x -1\right ) \left (a x +1\right ) \left (3 x^{2} \operatorname {arccosh}\left (a x \right )+\frac {x^{3} a}{\sqrt {a x -1}\, \sqrt {a x +1}}\right )}{32 x^{2} a^{4}}\) | \(86\) |
derivativedivides | \(\frac {\frac {a^{4} x^{4} \operatorname {arccosh}\left (a x \right )}{4}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (2 a^{3} x^{3} \sqrt {a^{2} x^{2}-1}+3 a x \sqrt {a^{2} x^{2}-1}+3 \ln \left (a x +\sqrt {a^{2} x^{2}-1}\right )\right )}{32 \sqrt {a^{2} x^{2}-1}}}{a^{4}}\) | \(98\) |
default | \(\frac {\frac {a^{4} x^{4} \operatorname {arccosh}\left (a x \right )}{4}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (2 a^{3} x^{3} \sqrt {a^{2} x^{2}-1}+3 a x \sqrt {a^{2} x^{2}-1}+3 \ln \left (a x +\sqrt {a^{2} x^{2}-1}\right )\right )}{32 \sqrt {a^{2} x^{2}-1}}}{a^{4}}\) | \(98\) |
parts | \(\frac {x^{4} \operatorname {arccosh}\left (a x \right )}{4}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (2 \,\operatorname {csgn}\left (a \right ) a^{3} x^{3} \sqrt {a^{2} x^{2}-1}+3 x \sqrt {a^{2} x^{2}-1}\, \operatorname {csgn}\left (a \right ) a +3 \ln \left (\left (\sqrt {a^{2} x^{2}-1}\, \operatorname {csgn}\left (a \right )+a x \right ) \operatorname {csgn}\left (a \right )\right )\right ) \operatorname {csgn}\left (a \right )}{32 a^{4} \sqrt {a^{2} x^{2}-1}}\) | \(106\) |
Input:
int(x^3*arccosh(a*x),x,method=_RETURNVERBOSE)
Output:
1/32*(14*a^4*x^4+3*a^2*x^2-12)/a^4*arccosh(a*x)-1/32/x^2*(2*a^2*x^2+3)/a^4 *(a*x-1)*(a*x+1)*(3*x^2*arccosh(a*x)+x^3*a/(a*x-1)^(1/2)/(a*x+1)^(1/2))
Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.77 \[ \int x^3 \text {arccosh}(a x) \, dx=\frac {{\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {a^{2} x^{2} - 1}}{32 \, a^{4}} \] Input:
integrate(x^3*arccosh(a*x),x, algorithm="fricas")
Output:
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)*s qrt(a^2*x^2 - 1))/a^4
\[ \int x^3 \text {arccosh}(a x) \, dx=\int x^{3} \operatorname {acosh}{\left (a x \right )}\, dx \] Input:
integrate(x**3*acosh(a*x),x)
Output:
Integral(x**3*acosh(a*x), x)
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int x^3 \text {arccosh}(a x) \, dx=\frac {1}{4} \, x^{4} \operatorname {arcosh}\left (a x\right ) - \frac {1}{32} \, {\left (\frac {2 \, \sqrt {a^{2} x^{2} - 1} x^{3}}{a^{2}} + \frac {3 \, \sqrt {a^{2} x^{2} - 1} x}{a^{4}} + \frac {3 \, \log \left (2 \, a^{2} x + 2 \, \sqrt {a^{2} x^{2} - 1} a\right )}{a^{5}}\right )} a \] Input:
integrate(x^3*arccosh(a*x),x, algorithm="maxima")
Output:
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
Exception generated. \[ \int x^3 \text {arccosh}(a x) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*arccosh(a*x),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x^3 \text {arccosh}(a x) \, dx=\int x^3\,\mathrm {acosh}\left (a\,x\right ) \,d x \] Input:
int(x^3*acosh(a*x),x)
Output:
int(x^3*acosh(a*x), x)
Time = 0.17 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.87 \[ \int x^3 \text {arccosh}(a x) \, dx=\frac {8 \mathit {acosh} \left (a x \right ) a^{4} x^{4}-2 \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}-3 \sqrt {a^{2} x^{2}-1}\, a x -3 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right )}{32 a^{4}} \] Input:
int(x^3*acosh(a*x),x)
Output:
(8*acosh(a*x)*a**4*x**4 - 2*sqrt(a**2*x**2 - 1)*a**3*x**3 - 3*sqrt(a**2*x* *2 - 1)*a*x - 3*log(sqrt(a**2*x**2 - 1) + a*x))/(32*a**4)