Integrand size = 22, antiderivative size = 143 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=-\frac {8 b d^2 \sqrt {-1+c x} \sqrt {1+c x}}{15 c}+\frac {4 b d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{45 c}-\frac {b d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{25 c}+d^2 x (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x)) \]
4/45*b*d^2*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c-1/25*b*d^2*(c*x-1)^(5/2)*(c*x+1)^ (5/2)/c+d^2*x*(a+b*arccosh(c*x))-2/3*c^2*d^2*x^3*(a+b*arccosh(c*x))+1/5*c^ 4*d^2*x^5*(a+b*arccosh(c*x))-8/15*b*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.69 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \left (b \sqrt {-1+c x} \sqrt {1+c x} \left (-149+38 c^2 x^2-9 c^4 x^4\right )+15 a c x \left (15-10 c^2 x^2+3 c^4 x^4\right )+15 b c x \left (15-10 c^2 x^2+3 c^4 x^4\right ) \text {arccosh}(c x)\right )}{225 c} \]
(d^2*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-149 + 38*c^2*x^2 - 9*c^4*x^4) + 15* a*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 15*b*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^ 4)*ArcCosh[c*x]))/(225*c)
Time = 0.43 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6309, 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.
\(\displaystyle \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx\) |
\(\Big \downarrow \) 6309 |
\(\displaystyle -b c \int \frac {d^2 x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{15 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+d^2 x (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{15} b c d^2 \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+d^2 x (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 1905 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {c^2 x^2-1}}dx}{15 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+d^2 x (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 1576 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \int \frac {3 c^4 x^4-10 c^2 x^2+15}{\sqrt {c^2 x^2-1}}dx^2}{30 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+d^2 x (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \int \left (3 \left (c^2 x^2-1\right )^{3/2}-4 \sqrt {c^2 x^2-1}+\frac {8}{\sqrt {c^2 x^2-1}}\right )dx^2}{30 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+d^2 x (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+d^2 x (a+b \text {arccosh}(c x))-\frac {b c d^2 \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/30*(b*c*d^2*Sqrt[-1 + c^2*x^2]*((16*Sqrt[-1 + c^2*x^2])/c^2 - (8*(-1 + c^2*x^2)^(3/2))/(3*c^2) + (6*(-1 + c^2*x^2)^(5/2))/(5*c^2)))/(Sqrt[-1 + c* x]*Sqrt[1 + c*x]) + d^2*x*(a + b*ArcCosh[c*x]) - (2*c^2*d^2*x^3*(a + b*Arc Cosh[c*x]))/3 + (c^4*d^2*x^5*(a + b*ArcCosh[c*x]))/5
3.1.14.3.1 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] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Time = 0.49 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.69
method | result | size |
parts | \(d^{2} a \left (\frac {1}{5} c^{4} x^{5}-\frac {2}{3} x^{3} c^{2}+x \right )+\frac {d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\) | \(99\) |
derivativedivides | \(\frac {d^{2} a \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\) | \(102\) |
default | \(\frac {d^{2} a \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\) | \(102\) |
d^2*a*(1/5*c^4*x^5-2/3*x^3*c^2+x)+d^2*b/c*(1/5*arccosh(c*x)*c^5*x^5-2/3*c^ 3*x^3*arccosh(c*x)+c*x*arccosh(c*x)-1/225*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(9*c ^4*x^4-38*c^2*x^2+149))
Time = 0.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.93 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {45 \, a c^{5} d^{2} x^{5} - 150 \, a c^{3} d^{2} x^{3} + 225 \, a c d^{2} x + 15 \, {\left (3 \, b c^{5} d^{2} x^{5} - 10 \, b c^{3} d^{2} x^{3} + 15 \, b c d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (9 \, b c^{4} d^{2} x^{4} - 38 \, b c^{2} d^{2} x^{2} + 149 \, b d^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{225 \, c} \]
1/225*(45*a*c^5*d^2*x^5 - 150*a*c^3*d^2*x^3 + 225*a*c*d^2*x + 15*(3*b*c^5* d^2*x^5 - 10*b*c^3*d^2*x^3 + 15*b*c*d^2*x)*log(c*x + sqrt(c^2*x^2 - 1)) - (9*b*c^4*d^2*x^4 - 38*b*c^2*d^2*x^2 + 149*b*d^2)*sqrt(c^2*x^2 - 1))/c
\[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=d^{2} \left (\int a\, dx + \int b \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 2 a c^{2} x^{2}\right )\, dx + \int a c^{4} x^{4}\, dx + \int \left (- 2 b c^{2} x^{2} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{4} x^{4} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
d**2*(Integral(a, x) + Integral(b*acosh(c*x), x) + Integral(-2*a*c**2*x**2 , x) + Integral(a*c**4*x**4, x) + Integral(-2*b*c**2*x**2*acosh(c*x), x) + Integral(b*c**4*x**4*acosh(c*x), x))
Time = 0.23 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.36 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{5} \, a c^{4} d^{2} x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{4} d^{2} - \frac {2}{3} \, a c^{2} d^{2} x^{3} - \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{2} d^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{2}}{c} \]
1/5*a*c^4*d^2*x^5 + 1/75*(15*x^5*arccosh(c*x) - (3*sqrt(c^2*x^2 - 1)*x^4/c ^2 + 4*sqrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c)*b*c^4*d^2 - 2/3*a*c^2*d^2*x^3 - 2/9*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^ 2 + 2*sqrt(c^2*x^2 - 1)/c^4))*b*c^2*d^2 + a*d^2*x + (c*x*arccosh(c*x) - sq rt(c^2*x^2 - 1))*b*d^2/c
Exception generated. \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \]