Integrand size = 14, antiderivative size = 181 \[ \int \left (c+d x^2\right )^2 \text {arccosh}(a x) \, dx=\frac {\left (15 a^4 c^2+10 a^2 c d+3 d^2\right ) \left (1-a^2 x^2\right )}{15 a^5 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {2 d \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )^2}{45 a^5 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {d^2 \left (1-a^2 x^2\right )^3}{25 a^5 \sqrt {-1+a x} \sqrt {1+a x}}+c^2 x \text {arccosh}(a x)+\frac {2}{3} c d x^3 \text {arccosh}(a x)+\frac {1}{5} d^2 x^5 \text {arccosh}(a x) \]
c^2*x*arccosh(a*x)+2/3*c*d*x^3*arccosh(a*x)+1/5*d^2*x^5*arccosh(a*x)+1/15* (15*a^4*c^2+10*a^2*c*d+3*d^2)*(-a^2*x^2+1)/a^5/(a*x-1)^(1/2)/(a*x+1)^(1/2) -2/45*d*(5*a^2*c+3*d)*(-a^2*x^2+1)^2/a^5/(a*x-1)^(1/2)/(a*x+1)^(1/2)+1/25* d^2*(-a^2*x^2+1)^3/a^5/(a*x-1)^(1/2)/(a*x+1)^(1/2)
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.57 \[ \int \left (c+d x^2\right )^2 \text {arccosh}(a x) \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (24 d^2+4 a^2 d \left (25 c+3 d x^2\right )+a^4 \left (225 c^2+50 c d x^2+9 d^2 x^4\right )\right )}{225 a^5}+\left (c^2 x+\frac {2}{3} c d x^3+\frac {d^2 x^5}{5}\right ) \text {arccosh}(a x) \]
-1/225*(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(24*d^2 + 4*a^2*d*(25*c + 3*d*x^2) + a^4*(225*c^2 + 50*c*d*x^2 + 9*d^2*x^4)))/a^5 + (c^2*x + (2*c*d*x^3)/3 + (d ^2*x^5)/5)*ArcCosh[a*x]
Time = 0.44 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6323, 27, 1905, 1576, 1140, 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.
\(\displaystyle \int \text {arccosh}(a x) \left (c+d x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 6323 |
\(\displaystyle -a \int \frac {x \left (3 d^2 x^4+10 c d x^2+15 c^2\right )}{15 \sqrt {a x-1} \sqrt {a x+1}}dx+c^2 x \text {arccosh}(a x)+\frac {2}{3} c d x^3 \text {arccosh}(a x)+\frac {1}{5} d^2 x^5 \text {arccosh}(a x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{15} a \int \frac {x \left (3 d^2 x^4+10 c d x^2+15 c^2\right )}{\sqrt {a x-1} \sqrt {a x+1}}dx+c^2 x \text {arccosh}(a x)+\frac {2}{3} c d x^3 \text {arccosh}(a x)+\frac {1}{5} d^2 x^5 \text {arccosh}(a x)\) |
\(\Big \downarrow \) 1905 |
\(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \int \frac {x \left (3 d^2 x^4+10 c d x^2+15 c^2\right )}{\sqrt {a^2 x^2-1}}dx}{15 \sqrt {a x-1} \sqrt {a x+1}}+c^2 x \text {arccosh}(a x)+\frac {2}{3} c d x^3 \text {arccosh}(a x)+\frac {1}{5} d^2 x^5 \text {arccosh}(a x)\) |
\(\Big \downarrow \) 1576 |
\(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \int \frac {3 d^2 x^4+10 c d x^2+15 c^2}{\sqrt {a^2 x^2-1}}dx^2}{30 \sqrt {a x-1} \sqrt {a x+1}}+c^2 x \text {arccosh}(a x)+\frac {2}{3} c d x^3 \text {arccosh}(a x)+\frac {1}{5} d^2 x^5 \text {arccosh}(a x)\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \int \left (\frac {3 \left (a^2 x^2-1\right )^{3/2} d^2}{a^4}+\frac {2 \left (5 c a^2+3 d\right ) \sqrt {a^2 x^2-1} d}{a^4}+\frac {15 c^2 a^4+10 c d a^2+3 d^2}{a^4 \sqrt {a^2 x^2-1}}\right )dx^2}{30 \sqrt {a x-1} \sqrt {a x+1}}+c^2 x \text {arccosh}(a x)+\frac {2}{3} c d x^3 \text {arccosh}(a x)+\frac {1}{5} d^2 x^5 \text {arccosh}(a x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \left (\frac {4 d \left (a^2 x^2-1\right )^{3/2} \left (5 a^2 c+3 d\right )}{3 a^6}+\frac {6 d^2 \left (a^2 x^2-1\right )^{5/2}}{5 a^6}+\frac {2 \sqrt {a^2 x^2-1} \left (15 a^4 c^2+10 a^2 c d+3 d^2\right )}{a^6}\right )}{30 \sqrt {a x-1} \sqrt {a x+1}}+c^2 x \text {arccosh}(a x)+\frac {2}{3} c d x^3 \text {arccosh}(a x)+\frac {1}{5} d^2 x^5 \text {arccosh}(a x)\) |
-1/30*(a*Sqrt[-1 + a^2*x^2]*((2*(15*a^4*c^2 + 10*a^2*c*d + 3*d^2)*Sqrt[-1 + a^2*x^2])/a^6 + (4*d*(5*a^2*c + 3*d)*(-1 + a^2*x^2)^(3/2))/(3*a^6) + (6* d^2*(-1 + a^2*x^2)^(5/2))/(5*a^6)))/(Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + c^2*x *ArcCosh[a*x] + (2*c*d*x^3*ArcCosh[a*x])/3 + (d^2*x^5*ArcCosh[a*x])/5
3.1.43.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) *(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x _Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ q]/(d1*d2 + e1*e2*x^n)^FracPart[q]) Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCosh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] , x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
Time = 0.62 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.59
method | result | size |
parts | \(\frac {d^{2} x^{5} \operatorname {arccosh}\left (a x \right )}{5}+\frac {2 c d \,x^{3} \operatorname {arccosh}\left (a x \right )}{3}+c^{2} x \,\operatorname {arccosh}\left (a x \right )-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (9 a^{4} d^{2} x^{4}+50 a^{4} c d \,x^{2}+225 a^{4} c^{2}+12 a^{2} d^{2} x^{2}+100 a^{2} c d +24 d^{2}\right )}{225 a^{5}}\) | \(106\) |
derivativedivides | \(\frac {\operatorname {arccosh}\left (a x \right ) c^{2} a x +\frac {2 a \,\operatorname {arccosh}\left (a x \right ) c d \,x^{3}}{3}+\frac {a \,\operatorname {arccosh}\left (a x \right ) d^{2} x^{5}}{5}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (9 a^{4} d^{2} x^{4}+50 a^{4} c d \,x^{2}+225 a^{4} c^{2}+12 a^{2} d^{2} x^{2}+100 a^{2} c d +24 d^{2}\right )}{225 a^{4}}}{a}\) | \(113\) |
default | \(\frac {\operatorname {arccosh}\left (a x \right ) c^{2} a x +\frac {2 a \,\operatorname {arccosh}\left (a x \right ) c d \,x^{3}}{3}+\frac {a \,\operatorname {arccosh}\left (a x \right ) d^{2} x^{5}}{5}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (9 a^{4} d^{2} x^{4}+50 a^{4} c d \,x^{2}+225 a^{4} c^{2}+12 a^{2} d^{2} x^{2}+100 a^{2} c d +24 d^{2}\right )}{225 a^{4}}}{a}\) | \(113\) |
1/5*d^2*x^5*arccosh(a*x)+2/3*c*d*x^3*arccosh(a*x)+c^2*x*arccosh(a*x)-1/225 /a^5*(a*x-1)^(1/2)*(a*x+1)^(1/2)*(9*a^4*d^2*x^4+50*a^4*c*d*x^2+225*a^4*c^2 +12*a^2*d^2*x^2+100*a^2*c*d+24*d^2)
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.67 \[ \int \left (c+d x^2\right )^2 \text {arccosh}(a x) \, dx=\frac {15 \, {\left (3 \, a^{5} d^{2} x^{5} + 10 \, a^{5} c d x^{3} + 15 \, a^{5} c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (9 \, a^{4} d^{2} x^{4} + 225 \, a^{4} c^{2} + 100 \, a^{2} c d + 2 \, {\left (25 \, a^{4} c d + 6 \, a^{2} d^{2}\right )} x^{2} + 24 \, d^{2}\right )} \sqrt {a^{2} x^{2} - 1}}{225 \, a^{5}} \]
1/225*(15*(3*a^5*d^2*x^5 + 10*a^5*c*d*x^3 + 15*a^5*c^2*x)*log(a*x + sqrt(a ^2*x^2 - 1)) - (9*a^4*d^2*x^4 + 225*a^4*c^2 + 100*a^2*c*d + 2*(25*a^4*c*d + 6*a^2*d^2)*x^2 + 24*d^2)*sqrt(a^2*x^2 - 1))/a^5
\[ \int \left (c+d x^2\right )^2 \text {arccosh}(a x) \, dx=\int \left (c + d x^{2}\right )^{2} \operatorname {acosh}{\left (a x \right )}\, dx \]
Time = 0.20 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.85 \[ \int \left (c+d x^2\right )^2 \text {arccosh}(a x) \, dx=-\frac {1}{225} \, {\left (\frac {9 \, \sqrt {a^{2} x^{2} - 1} d^{2} x^{4}}{a^{2}} + \frac {50 \, \sqrt {a^{2} x^{2} - 1} c d x^{2}}{a^{2}} + \frac {225 \, \sqrt {a^{2} x^{2} - 1} c^{2}}{a^{2}} + \frac {12 \, \sqrt {a^{2} x^{2} - 1} d^{2} x^{2}}{a^{4}} + \frac {100 \, \sqrt {a^{2} x^{2} - 1} c d}{a^{4}} + \frac {24 \, \sqrt {a^{2} x^{2} - 1} d^{2}}{a^{6}}\right )} a + \frac {1}{15} \, {\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \operatorname {arcosh}\left (a x\right ) \]
-1/225*(9*sqrt(a^2*x^2 - 1)*d^2*x^4/a^2 + 50*sqrt(a^2*x^2 - 1)*c*d*x^2/a^2 + 225*sqrt(a^2*x^2 - 1)*c^2/a^2 + 12*sqrt(a^2*x^2 - 1)*d^2*x^2/a^4 + 100* sqrt(a^2*x^2 - 1)*c*d/a^4 + 24*sqrt(a^2*x^2 - 1)*d^2/a^6)*a + 1/15*(3*d^2* x^5 + 10*c*d*x^3 + 15*c^2*x)*arccosh(a*x)
Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.74 \[ \int \left (c+d x^2\right )^2 \text {arccosh}(a x) \, dx=\frac {1}{15} \, {\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {{\left (15 \, a^{4} c^{2} + 10 \, a^{2} c d + 3 \, d^{2}\right )} \sqrt {a^{2} x^{2} - 1}}{15 \, a^{5}} - \frac {50 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{2} c d + 9 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {5}{2}} d^{2} + 30 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} d^{2}}{225 \, a^{5}} \]
1/15*(3*d^2*x^5 + 10*c*d*x^3 + 15*c^2*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 1/ 15*(15*a^4*c^2 + 10*a^2*c*d + 3*d^2)*sqrt(a^2*x^2 - 1)/a^5 - 1/225*(50*(a^ 2*x^2 - 1)^(3/2)*a^2*c*d + 9*(a^2*x^2 - 1)^(5/2)*d^2 + 30*(a^2*x^2 - 1)^(3 /2)*d^2)/a^5
Timed out. \[ \int \left (c+d x^2\right )^2 \text {arccosh}(a x) \, dx=\int \mathrm {acosh}\left (a\,x\right )\,{\left (d\,x^2+c\right )}^2 \,d x \]