Integrand size = 22, antiderivative size = 191 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {16 b d^3 \sqrt {-1+c x} \sqrt {1+c x}}{35 c}+\frac {8 b d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}{105 c}-\frac {6 b d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{175 c}+\frac {b d^3 (-1+c x)^{7/2} (1+c x)^{7/2}}{49 c}+d^3 x (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x)) \]
8/105*b*d^3*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c-6/175*b*d^3*(c*x-1)^(5/2)*(c*x+1 )^(5/2)/c+1/49*b*d^3*(c*x-1)^(7/2)*(c*x+1)^(7/2)/c+d^3*x*(a+b*arccosh(c*x) )-c^2*d^3*x^3*(a+b*arccosh(c*x))+3/5*c^4*d^3*x^5*(a+b*arccosh(c*x))-1/7*c^ 6*d^3*x^7*(a+b*arccosh(c*x))-16/35*b*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c
Time = 0.18 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.64 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {d^3 \left (b \sqrt {-1+c x} \sqrt {1+c x} \left (2161-757 c^2 x^2+351 c^4 x^4-75 c^6 x^6\right )+105 a c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+105 b c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right ) \text {arccosh}(c x)\right )}{3675 c} \]
-1/3675*(d^3*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2161 - 757*c^2*x^2 + 351*c^4 *x^4 - 75*c^6*x^6) + 105*a*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + 105*b*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)*ArcCosh[c*x]))/c
Time = 0.65 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6309, 27, 2113, 2331, 2389, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6309 |
| \(\displaystyle -b c \int \frac {d^3 x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{35 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+d^3 x (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{35} b c d^3 \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+d^3 x (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2113 |
| \(\displaystyle -\frac {b c d^3 \sqrt {c^2 x^2-1} \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{\sqrt {c^2 x^2-1}}dx}{35 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+d^3 x (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {b c d^3 \sqrt {c^2 x^2-1} \int \frac {-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35}{\sqrt {c^2 x^2-1}}dx^2}{70 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+d^3 x (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle -\frac {b c d^3 \sqrt {c^2 x^2-1} \int \left (-5 \left (c^2 x^2-1\right )^{5/2}+6 \left (c^2 x^2-1\right )^{3/2}-8 \sqrt {c^2 x^2-1}+\frac {16}{\sqrt {c^2 x^2-1}}\right )dx^2}{70 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+d^3 x (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+d^3 x (a+b \text {arccosh}(c x))-\frac {b c d^3 \sqrt {c^2 x^2-1} \left (-\frac {10 \left (c^2 x^2-1\right )^{7/2}}{7 c^2}+\frac {12 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {16 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {32 \sqrt {c^2 x^2-1}}{c^2}\right )}{70 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/70*(b*c*d^3*Sqrt[-1 + c^2*x^2]*((32*Sqrt[-1 + c^2*x^2])/c^2 - (16*(-1 + c^2*x^2)^(3/2))/(3*c^2) + (12*(-1 + c^2*x^2)^(5/2))/(5*c^2) - (10*(-1 + c ^2*x^2)^(7/2))/(7*c^2)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^3*x*(a + b*Arc Cosh[c*x]) - c^2*d^3*x^3*(a + b*ArcCosh[c*x]) + (3*c^4*d^3*x^5*(a + b*ArcC osh[c*x]))/5 - (c^6*d^3*x^7*(a + b*ArcCosh[c*x]))/7
3.1.23.3.1 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[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_. )*(x_))^(p_.), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[ m]/(a*c + b*d*x^2)^FracPart[m]) Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a *d, 0] && EqQ[m, n] && !IntegerQ[m]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
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 = 0.47 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.68
| method | result | size |
| parts | \(-d^{3} a \left (\frac {1}{7} c^{6} x^{7}-\frac {3}{5} c^{4} x^{5}+x^{3} c^{2}-x \right )-\frac {d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} x^{6}-351 c^{4} x^{4}+757 c^{2} x^{2}-2161\right )}{3675}\right )}{c}\) | \(130\) |
| derivativedivides | \(\frac {-d^{3} a \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} x^{6}-351 c^{4} x^{4}+757 c^{2} x^{2}-2161\right )}{3675}\right )}{c}\) | \(132\) |
| default | \(\frac {-d^{3} a \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} x^{6}-351 c^{4} x^{4}+757 c^{2} x^{2}-2161\right )}{3675}\right )}{c}\) | \(132\) |
-d^3*a*(1/7*c^6*x^7-3/5*c^4*x^5+x^3*c^2-x)-d^3*b/c*(1/7*arccosh(c*x)*c^7*x ^7-3/5*arccosh(c*x)*c^5*x^5+c^3*x^3*arccosh(c*x)-c*x*arccosh(c*x)-1/3675*( c*x-1)^(1/2)*(c*x+1)^(1/2)*(75*c^6*x^6-351*c^4*x^4+757*c^2*x^2-2161))
Time = 0.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.88 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {525 \, a c^{7} d^{3} x^{7} - 2205 \, a c^{5} d^{3} x^{5} + 3675 \, a c^{3} d^{3} x^{3} - 3675 \, a c d^{3} x + 105 \, {\left (5 \, b c^{7} d^{3} x^{7} - 21 \, b c^{5} d^{3} x^{5} + 35 \, b c^{3} d^{3} x^{3} - 35 \, b c d^{3} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (75 \, b c^{6} d^{3} x^{6} - 351 \, b c^{4} d^{3} x^{4} + 757 \, b c^{2} d^{3} x^{2} - 2161 \, b d^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{3675 \, c} \]
-1/3675*(525*a*c^7*d^3*x^7 - 2205*a*c^5*d^3*x^5 + 3675*a*c^3*d^3*x^3 - 367 5*a*c*d^3*x + 105*(5*b*c^7*d^3*x^7 - 21*b*c^5*d^3*x^5 + 35*b*c^3*d^3*x^3 - 35*b*c*d^3*x)*log(c*x + sqrt(c^2*x^2 - 1)) - (75*b*c^6*d^3*x^6 - 351*b*c^ 4*d^3*x^4 + 757*b*c^2*d^3*x^2 - 2161*b*d^3)*sqrt(c^2*x^2 - 1))/c
\[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=- d^{3} \left (\int \left (- a\right )\, dx + \int \left (- b \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int 3 a c^{2} x^{2}\, dx + \int \left (- 3 a c^{4} x^{4}\right )\, dx + \int a c^{6} x^{6}\, dx + \int 3 b c^{2} x^{2} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 3 b c^{4} x^{4} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{6} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
-d**3*(Integral(-a, x) + Integral(-b*acosh(c*x), x) + Integral(3*a*c**2*x* *2, x) + Integral(-3*a*c**4*x**4, x) + Integral(a*c**6*x**6, x) + Integral (3*b*c**2*x**2*acosh(c*x), x) + Integral(-3*b*c**4*x**4*acosh(c*x), x) + I ntegral(b*c**6*x**6*acosh(c*x), x))
Time = 0.24 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.58 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{7} \, a c^{6} d^{3} x^{7} + \frac {3}{5} \, a c^{4} d^{3} x^{5} - \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{25} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{4} d^{3} - a c^{2} d^{3} x^{3} - \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{2} d^{3} + a d^{3} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{3}}{c} \]
-1/7*a*c^6*d^3*x^7 + 3/5*a*c^4*d^3*x^5 - 1/245*(35*x^7*arccosh(c*x) - (5*s qrt(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^6*d^3 + 1/25*(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*sqr t(c^2*x^2 - 1)/c^6)*c)*b*c^4*d^3 - a*c^2*d^3*x^3 - 1/3*(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^3 + a* d^3*x + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*d^3/c
Exception generated. \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \]