Integrand size = 8, antiderivative size = 120 \[ \int x \text {arccosh}(a x)^4 \, dx=\frac {3 x^2}{4}-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{2 a}-\frac {3 \text {arccosh}(a x)^2}{4 a^2}+\frac {3}{2} x^2 \text {arccosh}(a x)^2-\frac {x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{a}-\frac {\text {arccosh}(a x)^4}{4 a^2}+\frac {1}{2} x^2 \text {arccosh}(a x)^4 \]
3/4*x^2-3/4*arccosh(a*x)^2/a^2+3/2*x^2*arccosh(a*x)^2-1/4*arccosh(a*x)^4/a ^2+1/2*x^2*arccosh(a*x)^4-3/2*x*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a -x*arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a
Time = 0.06 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.87 \[ \int x \text {arccosh}(a x)^4 \, dx=\frac {3 a^2 x^2-6 a x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)+\left (-3+6 a^2 x^2\right ) \text {arccosh}(a x)^2-4 a x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3+\left (-1+2 a^2 x^2\right ) \text {arccosh}(a x)^4}{4 a^2} \]
(3*a^2*x^2 - 6*a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] + (-3 + 6*a^2 *x^2)*ArcCosh[a*x]^2 - 4*a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3 + (-1 + 2*a^2*x^2)*ArcCosh[a*x]^4)/(4*a^2)
Time = 1.40 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6298, 6354, 6298, 6308, 6354, 15, 6308}
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.
\(\displaystyle \int x \text {arccosh}(a x)^4 \, dx\) |
\(\Big \downarrow \) 6298 |
\(\displaystyle \frac {1}{2} x^2 \text {arccosh}(a x)^4-2 a \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {a x-1} \sqrt {a x+1}}dx\) |
\(\Big \downarrow \) 6354 |
\(\displaystyle \frac {1}{2} x^2 \text {arccosh}(a x)^4-2 a \left (\frac {\int \frac {\text {arccosh}(a x)^3}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}-\frac {3 \int x \text {arccosh}(a x)^2dx}{2 a}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 a^2}\right )\) |
\(\Big \downarrow \) 6298 |
\(\displaystyle \frac {1}{2} x^2 \text {arccosh}(a x)^4-2 a \left (\frac {\int \frac {\text {arccosh}(a x)^3}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}-\frac {3 \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 a}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 a^2}\right )\) |
\(\Big \downarrow \) 6308 |
\(\displaystyle \frac {1}{2} x^2 \text {arccosh}(a x)^4-2 a \left (-\frac {3 \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 a}+\frac {\text {arccosh}(a x)^4}{8 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 a^2}\right )\) |
\(\Big \downarrow \) 6354 |
\(\displaystyle \frac {1}{2} x^2 \text {arccosh}(a x)^4-2 a \left (-\frac {3 \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}-\frac {\int xdx}{2 a}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}\right )\right )}{2 a}+\frac {\text {arccosh}(a x)^4}{8 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 a^2}\right )\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {1}{2} x^2 \text {arccosh}(a x)^4-2 a \left (-\frac {3 \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 a}+\frac {\text {arccosh}(a x)^4}{8 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 a^2}\right )\) |
\(\Big \downarrow \) 6308 |
\(\displaystyle \frac {1}{2} x^2 \text {arccosh}(a x)^4-2 a \left (\frac {\text {arccosh}(a x)^4}{8 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{2 a^2}-\frac {3 \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\text {arccosh}(a x)^2}{4 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 a}\right )\) |
(x^2*ArcCosh[a*x]^4)/2 - 2*a*((x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] ^3)/(2*a^2) + ArcCosh[a*x]^4/(8*a^3) - (3*((x^2*ArcCosh[a*x]^2)/2 - a*(-1/ 4*x^2/a + (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(2*a^2) + ArcCosh[ a*x]^2/(4*a^3))))/(2*a))
3.1.36.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(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*ArcCosh[ 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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(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] + (Simp[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] - Simp[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] && IGtQ[m, 1] && N eQ[m + 2*p + 1, 0]
Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{4}}{2}-a x \operatorname {arccosh}\left (a x \right )^{3} \sqrt {a x -1}\, \sqrt {a x +1}-\frac {\operatorname {arccosh}\left (a x \right )^{4}}{4}+\frac {3 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2}}{2}-\frac {3 a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{2}-\frac {3 \operatorname {arccosh}\left (a x \right )^{2}}{4}+\frac {3 a^{2} x^{2}}{4}}{a^{2}}\) | \(104\) |
default | \(\frac {\frac {a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{4}}{2}-a x \operatorname {arccosh}\left (a x \right )^{3} \sqrt {a x -1}\, \sqrt {a x +1}-\frac {\operatorname {arccosh}\left (a x \right )^{4}}{4}+\frac {3 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2}}{2}-\frac {3 a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{2}-\frac {3 \operatorname {arccosh}\left (a x \right )^{2}}{4}+\frac {3 a^{2} x^{2}}{4}}{a^{2}}\) | \(104\) |
1/a^2*(1/2*a^2*x^2*arccosh(a*x)^4-a*x*arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1) ^(1/2)-1/4*arccosh(a*x)^4+3/2*a^2*x^2*arccosh(a*x)^2-3/2*a*x*arccosh(a*x)* (a*x-1)^(1/2)*(a*x+1)^(1/2)-3/4*arccosh(a*x)^2+3/4*a^2*x^2)
Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.15 \[ \int x \text {arccosh}(a x)^4 \, dx=-\frac {4 \, \sqrt {a^{2} x^{2} - 1} a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - {\left (2 \, a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{4} - 3 \, a^{2} x^{2} + 6 \, \sqrt {a^{2} x^{2} - 1} a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}}{4 \, a^{2}} \]
-1/4*(4*sqrt(a^2*x^2 - 1)*a*x*log(a*x + sqrt(a^2*x^2 - 1))^3 - (2*a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1))^4 - 3*a^2*x^2 + 6*sqrt(a^2*x^2 - 1)*a*x* log(a*x + sqrt(a^2*x^2 - 1)) - 3*(2*a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1))^2)/a^2
\[ \int x \text {arccosh}(a x)^4 \, dx=\int x \operatorname {acosh}^{4}{\left (a x \right )}\, dx \]
\[ \int x \text {arccosh}(a x)^4 \, dx=\int { x \operatorname {arcosh}\left (a x\right )^{4} \,d x } \]
1/2*x^2*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^4 - integrate(2*(a^3*x^4 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x^3 - a*x^2)*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 )
Exception generated. \[ \int x \text {arccosh}(a x)^4 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x \text {arccosh}(a x)^4 \, dx=\int x\,{\mathrm {acosh}\left (a\,x\right )}^4 \,d x \]