Integrand size = 24, antiderivative size = 231 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=\frac {298}{225} b^2 d^2 x-\frac {76}{675} b^2 c^2 d^2 x^3+\frac {2}{125} b^2 c^4 d^2 x^5-\frac {16 b d^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{15 c}+\frac {8 b d^2 (-1+c x)^{3/2} (1+c x)^{3/2} (a+b \text {arccosh}(c x))}{45 c}-\frac {2 b d^2 (-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))}{25 c}+\frac {8}{15} d^2 x (a+b \text {arccosh}(c x))^2+\frac {4}{15} d^2 x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2 \] Output:
298/225*b^2*d^2*x-76/675*b^2*c^2*d^2*x^3+2/125*b^2*c^4*d^2*x^5-16/15*b*d^2 *(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))/c+8/45*b*d^2*(c*x-1)^(3/2) *(c*x+1)^(3/2)*(a+b*arccosh(c*x))/c-2/25*b*d^2*(c*x-1)^(5/2)*(c*x+1)^(5/2) *(a+b*arccosh(c*x))/c+8/15*d^2*x*(a+b*arccosh(c*x))^2+4/15*d^2*x*(-c^2*x^2 +1)*(a+b*arccosh(c*x))^2+1/5*d^2*x*(-c^2*x^2+1)^2*(a+b*arccosh(c*x))^2
Time = 1.25 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.87 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=\frac {d^2 \left (225 a^2 c x \left (15-10 c^2 x^2+3 c^4 x^4\right )-30 a b \sqrt {-1+c x} \sqrt {1+c x} \left (149-38 c^2 x^2+9 c^4 x^4\right )+2 b^2 c x \left (2235-190 c^2 x^2+27 c^4 x^4\right )-30 b \left (-15 a c x \left (15-10 c^2 x^2+3 c^4 x^4\right )+b \sqrt {-1+c x} \sqrt {1+c x} \left (149-38 c^2 x^2+9 c^4 x^4\right )\right ) \text {arccosh}(c x)+225 b^2 c x \left (15-10 c^2 x^2+3 c^4 x^4\right ) \text {arccosh}(c x)^2\right )}{3375 c} \] Input:
Integrate[(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x])^2,x]
Output:
(d^2*(225*a^2*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) - 30*a*b*Sqrt[-1 + c*x]*Sq rt[1 + c*x]*(149 - 38*c^2*x^2 + 9*c^4*x^4) + 2*b^2*c*x*(2235 - 190*c^2*x^2 + 27*c^4*x^4) - 30*b*(-15*a*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(149 - 38*c^2*x^2 + 9*c^4*x^4))*ArcCosh[c*x] + 225*b ^2*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4)*ArcCosh[c*x]^2))/(3375*c)
Time = 2.17 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6312, 27, 6312, 6294, 6330, 24, 25, 39, 210, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6312 |
| \(\displaystyle \frac {4}{5} d \int d \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2dx-\frac {2}{5} b c d^2 \int x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))dx+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{5} d^2 \int \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2dx-\frac {2}{5} b c d^2 \int x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))dx+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6312 |
| \(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} b c \int x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))dx+\frac {2}{3} \int (a+b \text {arccosh}(c x))^2dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2\right )-\frac {2}{5} b c d^2 \int x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))dx+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6294 |
| \(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^2-2 b c \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )+\frac {2}{3} b c \int x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2\right )-\frac {2}{5} b c d^2 \int x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))dx+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^2-2 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )\right )+\frac {2}{3} b c \left (\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 c^2}-\frac {b \int -((1-c x) (c x+1))dx}{3 c}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2\right )-\frac {2}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {b \int (1-c x)^2 (c x+1)^2dx}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} b c \left (\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 c^2}-\frac {b \int -((1-c x) (c x+1))dx}{3 c}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^2-2 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {b \int (1-c x)^2 (c x+1)^2dx}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} b c \left (\frac {b \int (1-c x) (c x+1)dx}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^2-2 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {b \int (1-c x)^2 (c x+1)^2dx}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 39 |
| \(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} b c \left (\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^2-2 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {b \int \left (1-c^2 x^2\right )^2dx}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} b c \left (\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^2-2 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {b \int \left (c^4 x^4-2 c^2 x^2+1\right )dx}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2+\frac {4}{5} d^2 \left (\frac {2}{3} b c \left (\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^2-2 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\) |
Input:
Int[(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x])^2,x]
Output:
(d^2*x*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x])^2)/5 - (2*b*c*d^2*(-1/5*(b*(x - (2*c^2*x^3)/3 + (c^4*x^5)/5))/c + ((-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]))/(5*c^2)))/5 + (4*d^2*((x*(1 - c^2*x^2)*(a + b*ArcCosh[c* x])^2)/3 + (2*b*c*((b*(x - (c^2*x^3)/3))/(3*c) + ((-1 + c*x)^(3/2)*(1 + c* x)^(3/2)*(a + b*ArcCosh[c*x]))/(3*c^2)))/3 + (2*(x*(a + b*ArcCosh[c*x])^2 - 2*b*c*(-((b*x)/c) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/ c^2)))/3))/5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcCosh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt [1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCosh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p )] Int[x*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p _)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 *c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ p] Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
Time = 0.18 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.14
| method | result | size |
| derivativedivides | \(\frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{2} \left (\frac {8 \operatorname {arccosh}\left (c x \right )^{2} c x}{15}+\frac {\operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{5}-\frac {4 \operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{15}-\frac {16 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{125}-\frac {272 c x \left (c x -1\right ) \left (c x +1\right )}{3375}+\frac {8 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{45}\right )+2 d^{2} a b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\) | \(264\) |
| default | \(\frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{2} \left (\frac {8 \operatorname {arccosh}\left (c x \right )^{2} c x}{15}+\frac {\operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{5}-\frac {4 \operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{15}-\frac {16 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{125}-\frac {272 c x \left (c x -1\right ) \left (c x +1\right )}{3375}+\frac {8 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{45}\right )+2 d^{2} a b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\) | \(264\) |
| parts | \(d^{2} a^{2} \left (\frac {1}{5} c^{4} x^{5}-\frac {2}{3} c^{2} x^{3}+x \right )+\frac {d^{2} b^{2} \left (\frac {8 \operatorname {arccosh}\left (c x \right )^{2} c x}{15}+\frac {\operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{5}-\frac {4 \operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{15}-\frac {16 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{125}-\frac {272 c x \left (c x -1\right ) \left (c x +1\right )}{3375}+\frac {8 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{45}\right )}{c}+\frac {2 d^{2} a b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\) | \(264\) |
| orering | \(\frac {x \left (1647 c^{6} x^{6}-8677 c^{4} x^{4}+51845 c^{2} x^{2}-3375\right ) \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{3375 \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )^{2}}-\frac {\left (324 c^{6} x^{6}-2035 c^{4} x^{4}+18450 c^{2} x^{2}-2235\right ) \left (-4 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{2} d x +\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{3375 c^{2} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}+\frac {x \left (27 c^{4} x^{4}-190 c^{2} x^{2}+2235\right ) \left (8 c^{4} d^{2} x^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}-\frac {16 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{3} d x b}{\sqrt {c x -1}\, \sqrt {c x +1}}-4 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{2} d +\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{2} b^{2} c^{2}}{\left (c x -1\right ) \left (c x +1\right )}-\frac {\left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {\left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}\right )}{3375 c^{2} \left (c x -1\right ) \left (c x +1\right )}\) | \(448\) |
Input:
int((-c^2*d*x^2+d)^2*(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/c*(d^2*a^2*(1/5*c^5*x^5-2/3*c^3*x^3+c*x)+d^2*b^2*(8/15*arccosh(c*x)^2*c* x+1/5*arccosh(c*x)^2*c*x*(c*x-1)^2*(c*x+1)^2-4/15*arccosh(c*x)^2*c*x*(c*x- 1)*(c*x+1)-16/15*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4144/3375*c*x-2/ 25*arccosh(c*x)*(c*x-1)^(5/2)*(c*x+1)^(5/2)+2/125*c*x*(c*x-1)^2*(c*x+1)^2- 272/3375*c*x*(c*x-1)*(c*x+1)+8/45*arccosh(c*x)*(c*x-1)^(3/2)*(c*x+1)^(3/2) )+2*d^2*a*b*(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 = 0.28 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.20 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=\frac {27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{5} d^{2} x^{5} - 10 \, {\left (225 \, a^{2} + 38 \, b^{2}\right )} c^{3} d^{2} x^{3} + 15 \, {\left (225 \, a^{2} + 298 \, b^{2}\right )} c d^{2} x + 225 \, {\left (3 \, b^{2} c^{5} d^{2} x^{5} - 10 \, b^{2} c^{3} d^{2} x^{3} + 15 \, b^{2} c d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 30 \, {\left (45 \, a b c^{5} d^{2} x^{5} - 150 \, a b c^{3} d^{2} x^{3} + 225 \, a b c d^{2} x - {\left (9 \, b^{2} c^{4} d^{2} x^{4} - 38 \, b^{2} c^{2} d^{2} x^{2} + 149 \, b^{2} d^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 30 \, {\left (9 \, a b c^{4} d^{2} x^{4} - 38 \, a b c^{2} d^{2} x^{2} + 149 \, a b d^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{3375 \, c} \] Input:
integrate((-c^2*d*x^2+d)^2*(a+b*arccosh(c*x))^2,x, algorithm="fricas")
Output:
1/3375*(27*(25*a^2 + 2*b^2)*c^5*d^2*x^5 - 10*(225*a^2 + 38*b^2)*c^3*d^2*x^ 3 + 15*(225*a^2 + 298*b^2)*c*d^2*x + 225*(3*b^2*c^5*d^2*x^5 - 10*b^2*c^3*d ^2*x^3 + 15*b^2*c*d^2*x)*log(c*x + sqrt(c^2*x^2 - 1))^2 + 30*(45*a*b*c^5*d ^2*x^5 - 150*a*b*c^3*d^2*x^3 + 225*a*b*c*d^2*x - (9*b^2*c^4*d^2*x^4 - 38*b ^2*c^2*d^2*x^2 + 149*b^2*d^2)*sqrt(c^2*x^2 - 1))*log(c*x + sqrt(c^2*x^2 - 1)) - 30*(9*a*b*c^4*d^2*x^4 - 38*a*b*c^2*d^2*x^2 + 149*a*b*d^2)*sqrt(c^2*x ^2 - 1))/c
\[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=d^{2} \left (\int a^{2}\, dx + \int b^{2} \operatorname {acosh}^{2}{\left (c x \right )}\, dx + \int 2 a b \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 2 a^{2} c^{2} x^{2}\right )\, dx + \int a^{2} c^{4} x^{4}\, dx + \int \left (- 2 b^{2} c^{2} x^{2} \operatorname {acosh}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{4} x^{4} \operatorname {acosh}^{2}{\left (c x \right )}\, dx + \int \left (- 4 a b c^{2} x^{2} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int 2 a b c^{4} x^{4} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \] Input:
integrate((-c**2*d*x**2+d)**2*(a+b*acosh(c*x))**2,x)
Output:
d**2*(Integral(a**2, x) + Integral(b**2*acosh(c*x)**2, x) + Integral(2*a*b *acosh(c*x), x) + Integral(-2*a**2*c**2*x**2, x) + Integral(a**2*c**4*x**4 , x) + Integral(-2*b**2*c**2*x**2*acosh(c*x)**2, x) + Integral(b**2*c**4*x **4*acosh(c*x)**2, x) + Integral(-4*a*b*c**2*x**2*acosh(c*x), x) + Integra l(2*a*b*c**4*x**4*acosh(c*x), x))
Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (199) = 398\).
Time = 0.04 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.98 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=\frac {1}{5} \, b^{2} c^{4} d^{2} x^{5} \operatorname {arcosh}\left (c x\right )^{2} + \frac {1}{5} \, a^{2} c^{4} d^{2} x^{5} - \frac {2}{3} \, b^{2} c^{2} d^{2} x^{3} \operatorname {arcosh}\left (c x\right )^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} a b c^{4} d^{2} - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c \operatorname {arcosh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{4} d^{2} - \frac {2}{3} \, a^{2} c^{2} d^{2} x^{3} - \frac {4}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} a b c^{2} d^{2} + \frac {4}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} c^{2} d^{2} + b^{2} d^{2} x \operatorname {arcosh}\left (c x\right )^{2} + 2 \, b^{2} d^{2} {\left (x - \frac {\sqrt {c^{2} x^{2} - 1} \operatorname {arcosh}\left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} a b d^{2}}{c} \] Input:
integrate((-c^2*d*x^2+d)^2*(a+b*arccosh(c*x))^2,x, algorithm="maxima")
Output:
1/5*b^2*c^4*d^2*x^5*arccosh(c*x)^2 + 1/5*a^2*c^4*d^2*x^5 - 2/3*b^2*c^2*d^2 *x^3*arccosh(c*x)^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)*a*b*c^4*d ^2 - 2/1125*(15*(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*arccosh(c*x) - (9*c^4*x^5 + 20*c^2*x^3 + 120 *x)/c^4)*b^2*c^4*d^2 - 2/3*a^2*c^2*d^2*x^3 - 4/9*(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))*a*b*c^2*d^2 + 4/27*( 3*c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4)*arccosh(c*x) - ( c^2*x^3 + 6*x)/c^2)*b^2*c^2*d^2 + b^2*d^2*x*arccosh(c*x)^2 + 2*b^2*d^2*(x - sqrt(c^2*x^2 - 1)*arccosh(c*x)/c) + a^2*d^2*x + 2*(c*x*arccosh(c*x) - sq rt(c^2*x^2 - 1))*a*b*d^2/c
Exception generated. \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)^2*(a+b*arccosh(c*x))^2,x, algorithm="giac")
Output:
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. \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \] Input:
int((a + b*acosh(c*x))^2*(d - c^2*d*x^2)^2,x)
Output:
int((a + b*acosh(c*x))^2*(d - c^2*d*x^2)^2, x)
\[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=\frac {d^{2} \left (90 \mathit {acosh} \left (c x \right ) a b \,c^{5} x^{5}-300 \mathit {acosh} \left (c x \right ) a b \,c^{3} x^{3}+450 \mathit {acosh} \left (c x \right ) a b c x -18 \sqrt {c^{2} x^{2}-1}\, a b \,c^{4} x^{4}+76 \sqrt {c^{2} x^{2}-1}\, a b \,c^{2} x^{2}+152 \sqrt {c^{2} x^{2}-1}\, a b -450 \sqrt {c x +1}\, \sqrt {c x -1}\, a b +225 \left (\int \mathit {acosh} \left (c x \right )^{2}d x \right ) b^{2} c +225 \left (\int \mathit {acosh} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}-450 \left (\int \mathit {acosh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+45 a^{2} c^{5} x^{5}-150 a^{2} c^{3} x^{3}+225 a^{2} c x \right )}{225 c} \] Input:
int((-c^2*d*x^2+d)^2*(a+b*acosh(c*x))^2,x)
Output:
(d**2*(90*acosh(c*x)*a*b*c**5*x**5 - 300*acosh(c*x)*a*b*c**3*x**3 + 450*ac osh(c*x)*a*b*c*x - 18*sqrt(c**2*x**2 - 1)*a*b*c**4*x**4 + 76*sqrt(c**2*x** 2 - 1)*a*b*c**2*x**2 + 152*sqrt(c**2*x**2 - 1)*a*b - 450*sqrt(c*x + 1)*sqr t(c*x - 1)*a*b + 225*int(acosh(c*x)**2,x)*b**2*c + 225*int(acosh(c*x)**2*x **4,x)*b**2*c**5 - 450*int(acosh(c*x)**2*x**2,x)*b**2*c**3 + 45*a**2*c**5* x**5 - 150*a**2*c**3*x**3 + 225*a**2*c*x))/(225*c)