Integrand size = 25, antiderivative size = 177 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=-\frac {8 b d^2 \sqrt {-1+c x} \sqrt {1+c x}}{105 c^3}+\frac {4 b d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{315 c^3}-\frac {b d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{175 c^3}-\frac {b d^2 (-1+c x)^{7/2} (1+c x)^{7/2}}{49 c^3}+\frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arccosh}(c x)) \]
4/315*b*d^2*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c^3-1/175*b*d^2*(c*x-1)^(5/2)*(c*x +1)^(5/2)/c^3-1/49*b*d^2*(c*x-1)^(7/2)*(c*x+1)^(7/2)/c^3+1/3*d^2*x^3*(a+b* arccosh(c*x))-2/5*c^2*d^2*x^5*(a+b*arccosh(c*x))+1/7*c^4*d^2*x^7*(a+b*arcc osh(c*x))-8/105*b*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3
Time = 0.12 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.66 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \left (105 a c^3 x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right )-b \sqrt {-1+c x} \sqrt {1+c x} \left (818+409 c^2 x^2-612 c^4 x^4+225 c^6 x^6\right )+105 b c^3 x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right ) \text {arccosh}(c x)\right )}{11025 c^3} \]
(d^2*(105*a*c^3*x^3*(35 - 42*c^2*x^2 + 15*c^4*x^4) - b*Sqrt[-1 + c*x]*Sqrt [1 + c*x]*(818 + 409*c^2*x^2 - 612*c^4*x^4 + 225*c^6*x^6) + 105*b*c^3*x^3* (35 - 42*c^2*x^2 + 15*c^4*x^4)*ArcCosh[c*x]))/(11025*c^3)
Time = 0.51 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6336, 27, 1905, 1578, 1195, 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.
| \(\displaystyle \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6336 |
| \(\displaystyle -b c \int \frac {d^2 x^3 \left (15 c^4 x^4-42 c^2 x^2+35\right )}{105 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{105} b c d^2 \int \frac {x^3 \left (15 c^4 x^4-42 c^2 x^2+35\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \int \frac {x^3 \left (15 c^4 x^4-42 c^2 x^2+35\right )}{\sqrt {c^2 x^2-1}}dx}{105 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \int \frac {x^2 \left (15 c^4 x^4-42 c^2 x^2+35\right )}{\sqrt {c^2 x^2-1}}dx^2}{210 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \int \left (\frac {15 \left (c^2 x^2-1\right )^{5/2}}{c^2}+\frac {3 \left (c^2 x^2-1\right )^{3/2}}{c^2}-\frac {4 \sqrt {c^2 x^2-1}}{c^2}+\frac {8}{c^2 \sqrt {c^2 x^2-1}}\right )dx^2}{210 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} c^4 d^2 x^7 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))-\frac {b c d^2 \sqrt {c^2 x^2-1} \left (\frac {30 \left (c^2 x^2-1\right )^{7/2}}{7 c^4}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^4}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^4}+\frac {16 \sqrt {c^2 x^2-1}}{c^4}\right )}{210 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/210*(b*c*d^2*Sqrt[-1 + c^2*x^2]*((16*Sqrt[-1 + c^2*x^2])/c^4 - (8*(-1 + c^2*x^2)^(3/2))/(3*c^4) + (6*(-1 + c^2*x^2)^(5/2))/(5*c^4) + (30*(-1 + c^ 2*x^2)^(7/2))/(7*c^4)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^2*x^3*(a + b*A rcCosh[c*x]))/3 - (2*c^2*d^2*x^5*(a + b*ArcCosh[c*x]))/5 + (c^4*d^2*x^7*(a + b*ArcCosh[c*x]))/7
3.1.12.3.1 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_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(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 egerQ[(m - 1)/2]
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_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(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, f, m}, x] && E qQ[c^2*d + e, 0] && IGtQ[p, 0]
Time = 0.49 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.66
| method | result | size |
| parts | \(d^{2} a \left (\frac {1}{7} c^{4} x^{7}-\frac {2}{5} c^{2} x^{5}+\frac {1}{3} x^{3}\right )+\frac {d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (225 c^{6} x^{6}-612 c^{4} x^{4}+409 c^{2} x^{2}+818\right )}{11025}\right )}{c^{3}}\) | \(116\) |
| derivativedivides | \(\frac {d^{2} a \left (\frac {1}{7} c^{7} x^{7}-\frac {2}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (225 c^{6} x^{6}-612 c^{4} x^{4}+409 c^{2} x^{2}+818\right )}{11025}\right )}{c^{3}}\) | \(120\) |
| default | \(\frac {d^{2} a \left (\frac {1}{7} c^{7} x^{7}-\frac {2}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (225 c^{6} x^{6}-612 c^{4} x^{4}+409 c^{2} x^{2}+818\right )}{11025}\right )}{c^{3}}\) | \(120\) |
d^2*a*(1/7*c^4*x^7-2/5*c^2*x^5+1/3*x^3)+d^2*b/c^3*(1/7*arccosh(c*x)*c^7*x^ 7-2/5*arccosh(c*x)*c^5*x^5+1/3*c^3*x^3*arccosh(c*x)-1/11025*(c*x-1)^(1/2)* (c*x+1)^(1/2)*(225*c^6*x^6-612*c^4*x^4+409*c^2*x^2+818))
Time = 0.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.86 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1575 \, a c^{7} d^{2} x^{7} - 4410 \, a c^{5} d^{2} x^{5} + 3675 \, a c^{3} d^{2} x^{3} + 105 \, {\left (15 \, b c^{7} d^{2} x^{7} - 42 \, b c^{5} d^{2} x^{5} + 35 \, b c^{3} d^{2} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (225 \, b c^{6} d^{2} x^{6} - 612 \, b c^{4} d^{2} x^{4} + 409 \, b c^{2} d^{2} x^{2} + 818 \, b d^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{11025 \, c^{3}} \]
1/11025*(1575*a*c^7*d^2*x^7 - 4410*a*c^5*d^2*x^5 + 3675*a*c^3*d^2*x^3 + 10 5*(15*b*c^7*d^2*x^7 - 42*b*c^5*d^2*x^5 + 35*b*c^3*d^2*x^3)*log(c*x + sqrt( c^2*x^2 - 1)) - (225*b*c^6*d^2*x^6 - 612*b*c^4*d^2*x^4 + 409*b*c^2*d^2*x^2 + 818*b*d^2)*sqrt(c^2*x^2 - 1))/c^3
\[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=d^{2} \left (\int a x^{2}\, dx + \int \left (- 2 a c^{2} x^{4}\right )\, dx + \int a c^{4} x^{6}\, dx + \int b x^{2} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 2 b c^{2} x^{4} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{4} x^{6} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
d**2*(Integral(a*x**2, x) + Integral(-2*a*c**2*x**4, x) + Integral(a*c**4* x**6, x) + Integral(b*x**2*acosh(c*x), x) + Integral(-2*b*c**2*x**4*acosh( c*x), x) + Integral(b*c**4*x**6*acosh(c*x), x))
Time = 0.21 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.47 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{7} \, a c^{4} d^{2} x^{7} - \frac {2}{5} \, a c^{2} d^{2} x^{5} + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{4} d^{2} - \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{2} \]
1/7*a*c^4*d^2*x^7 - 2/5*a*c^2*d^2*x^5 + 1/245*(35*x^7*arccosh(c*x) - (5*sq rt(c^2*x^2 - 1)*x^6/c^2 + 6*sqrt(c^2*x^2 - 1)*x^4/c^4 + 8*sqrt(c^2*x^2 - 1 )*x^2/c^6 + 16*sqrt(c^2*x^2 - 1)/c^8)*c)*b*c^4*d^2 - 2/75*(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^2*d^2 + 1/3*a*d^2*x^3 + 1/9*(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*d^2
Exception generated. \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \]