Integrand size = 14, antiderivative size = 215 \[ \int (d+e x)^2 \text {arccosh}(c x)^2 \, dx=2 d^2 x+\frac {4 e^2 x}{9 c^2}+\frac {1}{2} d e x^2+\frac {2 e^2 x^3}{27}-\frac {2 d^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{c}-\frac {4 e^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{9 c^3}-\frac {d e x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{c}-\frac {2 e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{9 c}-\frac {d^3 \text {arccosh}(c x)^2}{3 e}-\frac {d e \text {arccosh}(c x)^2}{2 c^2}+\frac {(d+e x)^3 \text {arccosh}(c x)^2}{3 e} \]
2*d^2*x+4/9*e^2*x/c^2+1/2*d*e*x^2+2/27*e^2*x^3-1/3*d^3*arccosh(c*x)^2/e-1/ 2*d*e*arccosh(c*x)^2/c^2+1/3*(e*x+d)^3*arccosh(c*x)^2/e-2*d^2*arccosh(c*x) *(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-4/9*e^2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^ (1/2)/c^3-d*e*x*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-2/9*e^2*x^2*arc cosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c
Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.61 \[ \int (d+e x)^2 \text {arccosh}(c x)^2 \, dx=\frac {c x \left (24 e^2+c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )\right )-6 \sqrt {-1+c x} \sqrt {1+c x} \left (4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right ) \text {arccosh}(c x)+9 \left (-3 c d e+2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )\right ) \text {arccosh}(c x)^2}{54 c^3} \]
(c*x*(24*e^2 + c^2*(108*d^2 + 27*d*e*x + 4*e^2*x^2)) - 6*Sqrt[-1 + c*x]*Sq rt[1 + c*x]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2))*ArcCosh[c*x] + 9* (-3*c*d*e + 2*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2))*ArcCosh[c*x]^2)/(54*c^3)
Time = 1.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6378, 6390, 2009}
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.
\(\displaystyle \int \text {arccosh}(c x)^2 (d+e x)^2 \, dx\) |
\(\Big \downarrow \) 6378 |
\(\displaystyle \frac {\text {arccosh}(c x)^2 (d+e x)^3}{3 e}-\frac {2 c \int \frac {(d+e x)^3 \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 e}\) |
\(\Big \downarrow \) 6390 |
\(\displaystyle \frac {\text {arccosh}(c x)^2 (d+e x)^3}{3 e}-\frac {2 c \int \left (\frac {\text {arccosh}(c x) d^3}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 e x \text {arccosh}(c x) d^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 e^2 x^2 \text {arccosh}(c x) d}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {e^3 x^3 \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}\right )dx}{3 e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\text {arccosh}(c x)^2 (d+e x)^3}{3 e}-\frac {2 c \left (\frac {2 e^3 \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{3 c^4}+\frac {3 d e^2 \text {arccosh}(c x)^2}{4 c^3}+\frac {3 d^2 e \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{c^2}+\frac {3 d e^2 x \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{2 c^2}+\frac {e^3 x^2 \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{3 c^2}+\frac {d^3 \text {arccosh}(c x)^2}{2 c}-\frac {2 e^3 x}{3 c^3}-\frac {3 d^2 e x}{c}-\frac {3 d e^2 x^2}{4 c}-\frac {e^3 x^3}{9 c}\right )}{3 e}\) |
((d + e*x)^3*ArcCosh[c*x]^2)/(3*e) - (2*c*((-3*d^2*e*x)/c - (2*e^3*x)/(3*c ^3) - (3*d*e^2*x^2)/(4*c) - (e^3*x^3)/(9*c) + (3*d^2*e*Sqrt[-1 + c*x]*Sqrt [1 + c*x]*ArcCosh[c*x])/c^2 + (2*e^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[ c*x])/(3*c^4) + (3*d*e^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(2*c ^2) + (e^3*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c^2) + (d^3*A rcCosh[c*x]^2)/(2*c) + (3*d*e^2*ArcCosh[c*x]^2)/(4*c^3)))/(3*e)
3.1.9.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*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, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Int[Expand Integrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[ d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1 ] || (EqQ[m, 2] && LtQ[p, -2]))
Time = 0.58 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {54 \operatorname {arccosh}\left (c x \right )^{2} c^{3} d^{2} x +54 \operatorname {arccosh}\left (c x \right )^{2} c^{3} d e \,x^{2}+18 \operatorname {arccosh}\left (c x \right )^{2} e^{2} c^{3} x^{3}-108 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} d^{2}-54 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} d e x -12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2} e^{2}-27 \operatorname {arccosh}\left (c x \right )^{2} c d e -24 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2}+108 c^{3} x \,d^{2}+27 c^{3} x^{2} d e +4 c^{3} x^{3} e^{2}+24 c x \,e^{2}}{54 c^{3}}\) | \(207\) |
default | \(\frac {54 \operatorname {arccosh}\left (c x \right )^{2} c^{3} d^{2} x +54 \operatorname {arccosh}\left (c x \right )^{2} c^{3} d e \,x^{2}+18 \operatorname {arccosh}\left (c x \right )^{2} e^{2} c^{3} x^{3}-108 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} d^{2}-54 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} d e x -12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2} e^{2}-27 \operatorname {arccosh}\left (c x \right )^{2} c d e -24 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2}+108 c^{3} x \,d^{2}+27 c^{3} x^{2} d e +4 c^{3} x^{3} e^{2}+24 c x \,e^{2}}{54 c^{3}}\) | \(207\) |
1/54/c^3*(54*arccosh(c*x)^2*c^3*d^2*x+54*arccosh(c*x)^2*c^3*d*e*x^2+18*arc cosh(c*x)^2*e^2*c^3*x^3-108*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^2*d ^2-54*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^2*d*e*x-12*arccosh(c*x)*( c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2*e^2-27*arccosh(c*x)^2*c*d*e-24*arccosh( c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^2+108*c^3*x*d^2+27*c^3*x^2*d*e+4*c^3*x^ 3*e^2+24*c*x*e^2)
Time = 0.26 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.76 \[ \int (d+e x)^2 \text {arccosh}(c x)^2 \, dx=\frac {4 \, c^{3} e^{2} x^{3} + 27 \, c^{3} d e x^{2} + 9 \, {\left (2 \, c^{3} e^{2} x^{3} + 6 \, c^{3} d e x^{2} + 6 \, c^{3} d^{2} x - 3 \, c d e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (2 \, c^{2} e^{2} x^{2} + 9 \, c^{2} d e x + 18 \, c^{2} d^{2} + 4 \, e^{2}\right )} \sqrt {c^{2} x^{2} - 1} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 12 \, {\left (9 \, c^{3} d^{2} + 2 \, c e^{2}\right )} x}{54 \, c^{3}} \]
1/54*(4*c^3*e^2*x^3 + 27*c^3*d*e*x^2 + 9*(2*c^3*e^2*x^3 + 6*c^3*d*e*x^2 + 6*c^3*d^2*x - 3*c*d*e)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 6*(2*c^2*e^2*x^2 + 9*c^2*d*e*x + 18*c^2*d^2 + 4*e^2)*sqrt(c^2*x^2 - 1)*log(c*x + sqrt(c^2*x^ 2 - 1)) + 12*(9*c^3*d^2 + 2*c*e^2)*x)/c^3
\[ \int (d+e x)^2 \text {arccosh}(c x)^2 \, dx=\int \left (d + e x\right )^{2} \operatorname {acosh}^{2}{\left (c x \right )}\, dx \]
\[ \int (d+e x)^2 \text {arccosh}(c x)^2 \, dx=\int { {\left (e x + d\right )}^{2} \operatorname {arcosh}\left (c x\right )^{2} \,d x } \]
1/3*(e^2*x^3 + 3*d*e*x^2 + 3*d^2*x)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) ^2 - integrate(2/3*(c^3*e^2*x^5 + 3*c^3*d*e*x^4 - 3*c*d*e*x^2 - 3*c*d^2*x + (3*c^3*d^2 - c*e^2)*x^3 + (c^2*e^2*x^4 + 3*c^2*d*e*x^3 + 3*c^2*d^2*x^2)* sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^3*x ^3 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x), x)
Exception generated. \[ \int (d+e x)^2 \text {arccosh}(c x)^2 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int (d+e x)^2 \text {arccosh}(c x)^2 \, dx=\int {\mathrm {acosh}\left (c\,x\right )}^2\,{\left (d+e\,x\right )}^2 \,d x \]