Integrand size = 24, antiderivative size = 314 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\frac {4322 b^2 d^3 x}{3675}-\frac {1514 b^2 c^2 d^3 x^3}{11025}+\frac {234 b^2 c^4 d^3 x^5}{6125}-\frac {2}{343} b^2 c^6 d^3 x^7-\frac {32 b d^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{35 c}+\frac {16 b d^3 (-1+c x)^{3/2} (1+c x)^{3/2} (a+b \text {arccosh}(c x))}{105 c}-\frac {12 b d^3 (-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))}{175 c}+\frac {2 b d^3 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))}{49 c}+\frac {16}{35} d^3 x (a+b \text {arccosh}(c x))^2+\frac {8}{35} d^3 x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2+\frac {6}{35} d^3 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))^2 \] Output:
4322/3675*b^2*d^3*x-1514/11025*b^2*c^2*d^3*x^3+234/6125*b^2*c^4*d^3*x^5-2/ 343*b^2*c^6*d^3*x^7-32/35*b*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c *x))/c+16/105*b*d^3*(c*x-1)^(3/2)*(c*x+1)^(3/2)*(a+b*arccosh(c*x))/c-12/17 5*b*d^3*(c*x-1)^(5/2)*(c*x+1)^(5/2)*(a+b*arccosh(c*x))/c+2/49*b*d^3*(c*x-1 )^(7/2)*(c*x+1)^(7/2)*(a+b*arccosh(c*x))/c+16/35*d^3*x*(a+b*arccosh(c*x))^ 2+8/35*d^3*x*(-c^2*x^2+1)*(a+b*arccosh(c*x))^2+6/35*d^3*x*(-c^2*x^2+1)^2*( a+b*arccosh(c*x))^2+1/7*d^3*x*(-c^2*x^2+1)^3*(a+b*arccosh(c*x))^2
Time = 1.43 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.79 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\frac {d^3 \left (2 b^2 c x \left (226905-26495 c^2 x^2+7371 c^4 x^4-1125 c^6 x^6\right )-11025 a^2 c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+210 a b \sqrt {-1+c x} \sqrt {1+c x} \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )+210 b \left (-105 a c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+b \sqrt {-1+c x} \sqrt {1+c x} \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )\right ) \text {arccosh}(c x)-11025 b^2 c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right ) \text {arccosh}(c x)^2\right )}{385875 c} \] Input:
Integrate[(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x])^2,x]
Output:
(d^3*(2*b^2*c*x*(226905 - 26495*c^2*x^2 + 7371*c^4*x^4 - 1125*c^6*x^6) - 1 1025*a^2*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + 210*a*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) + 2 10*b*(-105*a*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + 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))*ArcC osh[c*x] - 11025*b^2*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)*ArcCo sh[c*x]^2))/(385875*c)
Time = 2.85 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6312, 27, 6312, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6312 |
| \(\displaystyle \frac {6}{7} d \int d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2dx+\frac {2}{7} b c d^3 \int x (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))dx+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{7} d^3 \int \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2dx+\frac {2}{7} b c d^3 \int x (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))dx+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6312 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \int \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2dx-\frac {2}{5} b c \int x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))dx+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\right )+\frac {2}{7} b c d^3 \int x (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))dx+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6312 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} b c \int x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))dx+\frac {2}{3} \int (a+b \text {arccosh}(c x))^2dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2\right )-\frac {2}{5} b c \int x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))dx+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\right )+\frac {2}{7} b c d^3 \int x (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))dx+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6294 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^2-2 b c \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )+\frac {2}{3} b c \int x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2\right )-\frac {2}{5} b c \int x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))dx+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\right )+\frac {2}{7} b c d^3 \int x (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))dx+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^2-2 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )\right )+\frac {2}{3} b c \left (\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 c^2}-\frac {b \int -((1-c x) (c x+1))dx}{3 c}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2\right )-\frac {2}{5} b c \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {b \int (1-c x)^2 (c x+1)^2dx}{5 c}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\right )+\frac {2}{7} b c d^3 \left (\frac {(c x-1)^{7/2} (c x+1)^{7/2} (a+b \text {arccosh}(c x))}{7 c^2}-\frac {b \int -(1-c x)^3 (c x+1)^3dx}{7 c}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} b c \left (\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 c^2}-\frac {b \int -((1-c x) (c x+1))dx}{3 c}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^2-2 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {b \int (1-c x)^2 (c x+1)^2dx}{5 c}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\right )+\frac {2}{7} b c d^3 \left (\frac {(c x-1)^{7/2} (c x+1)^{7/2} (a+b \text {arccosh}(c x))}{7 c^2}-\frac {b \int -(1-c x)^3 (c x+1)^3dx}{7 c}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} b c \left (\frac {b \int (1-c x) (c x+1)dx}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^2-2 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {b \int (1-c x)^2 (c x+1)^2dx}{5 c}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\right )+\frac {2}{7} b c d^3 \left (\frac {b \int (1-c x)^3 (c x+1)^3dx}{7 c}+\frac {(c x-1)^{7/2} (c x+1)^{7/2} (a+b \text {arccosh}(c x))}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 39 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} b c \left (\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^2-2 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {b \int \left (1-c^2 x^2\right )^2dx}{5 c}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\right )+\frac {2}{7} b c d^3 \left (\frac {b \int \left (1-c^2 x^2\right )^3dx}{7 c}+\frac {(c x-1)^{7/2} (c x+1)^{7/2} (a+b \text {arccosh}(c x))}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} b c \left (\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^2-2 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {b \int \left (c^4 x^4-2 c^2 x^2+1\right )dx}{5 c}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2\right )+\frac {2}{7} b c d^3 \left (\frac {b \int \left (-c^6 x^6+3 c^4 x^4-3 c^2 x^2+1\right )dx}{7 c}+\frac {(c x-1)^{7/2} (c x+1)^{7/2} (a+b \text {arccosh}(c x))}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))^2+\frac {6}{7} d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^2+\frac {4}{5} \left (\frac {2}{3} b c \left (\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^2-2 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )+\frac {2}{7} b c d^3 \left (\frac {(c x-1)^{7/2} (c x+1)^{7/2} (a+b \text {arccosh}(c x))}{7 c^2}+\frac {b \left (-\frac {1}{7} c^6 x^7+\frac {3 c^4 x^5}{5}-c^2 x^3+x\right )}{7 c}\right )\) |
Input:
Int[(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x])^2,x]
Output:
(d^3*x*(1 - c^2*x^2)^3*(a + b*ArcCosh[c*x])^2)/7 + (2*b*c*d^3*((b*(x - c^2 *x^3 + (3*c^4*x^5)/5 - (c^6*x^7)/7))/(7*c) + ((-1 + c*x)^(7/2)*(1 + c*x)^( 7/2)*(a + b*ArcCosh[c*x]))/(7*c^2)))/7 + (6*d^3*((x*(1 - c^2*x^2)^2*(a + b *ArcCosh[c*x])^2)/5 - (2*b*c*(-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*((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))/7
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 = 0.38 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(\frac {-d^{3} a^{2} \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b^{2} \left (-\frac {16 \operatorname {arccosh}\left (c x \right )^{2} c x}{35}+\frac {\operatorname {arccosh}\left (c x \right )^{2} \left (c x -1\right )^{3} \left (c x +1\right )^{3} c x}{7}-\frac {6 \operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{35}+\frac {8 \operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{35}+\frac {32 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{35}-\frac {413312 c x}{385875}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {7}{2}} \left (c x +1\right )^{\frac {7}{2}}}{49}+\frac {2 c x \left (c x -1\right )^{3} \left (c x +1\right )^{3}}{343}-\frac {888 c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{42875}+\frac {30256 c x \left (c x -1\right ) \left (c x +1\right )}{385875}+\frac {12 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{175}-\frac {16 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{105}\right )-2 d^{3} a b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} x^{6}-351 c^{4} x^{4}+757 c^{2} x^{2}-2161\right )}{3675}\right )}{c}\) | \(356\) |
| default | \(\frac {-d^{3} a^{2} \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b^{2} \left (-\frac {16 \operatorname {arccosh}\left (c x \right )^{2} c x}{35}+\frac {\operatorname {arccosh}\left (c x \right )^{2} \left (c x -1\right )^{3} \left (c x +1\right )^{3} c x}{7}-\frac {6 \operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{35}+\frac {8 \operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{35}+\frac {32 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{35}-\frac {413312 c x}{385875}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {7}{2}} \left (c x +1\right )^{\frac {7}{2}}}{49}+\frac {2 c x \left (c x -1\right )^{3} \left (c x +1\right )^{3}}{343}-\frac {888 c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{42875}+\frac {30256 c x \left (c x -1\right ) \left (c x +1\right )}{385875}+\frac {12 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{175}-\frac {16 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{105}\right )-2 d^{3} a b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} x^{6}-351 c^{4} x^{4}+757 c^{2} x^{2}-2161\right )}{3675}\right )}{c}\) | \(356\) |
| parts | \(-d^{3} a^{2} \left (\frac {1}{7} c^{6} x^{7}-\frac {3}{5} c^{4} x^{5}+c^{2} x^{3}-x \right )-\frac {d^{3} b^{2} \left (-\frac {16 \operatorname {arccosh}\left (c x \right )^{2} c x}{35}+\frac {\operatorname {arccosh}\left (c x \right )^{2} \left (c x -1\right )^{3} \left (c x +1\right )^{3} c x}{7}-\frac {6 \operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{35}+\frac {8 \operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{35}+\frac {32 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{35}-\frac {413312 c x}{385875}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {7}{2}} \left (c x +1\right )^{\frac {7}{2}}}{49}+\frac {2 c x \left (c x -1\right )^{3} \left (c x +1\right )^{3}}{343}-\frac {888 c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{42875}+\frac {30256 c x \left (c x -1\right ) \left (c x +1\right )}{385875}+\frac {12 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{175}-\frac {16 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{105}\right )}{c}-\frac {2 d^{3} a b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} x^{6}-351 c^{4} x^{4}+757 c^{2} x^{2}-2161\right )}{3675}\right )}{c}\) | \(357\) |
| orering | \(\frac {x \left (47625 c^{8} x^{8}-271212 c^{6} x^{6}+741678 c^{4} x^{4}-3539900 c^{2} x^{2}+128625\right ) \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{128625 \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (c^{2} x^{2}-1\right )^{2}}-\frac {\left (20250 c^{8} x^{8}-125811 c^{6} x^{6}+407785 c^{4} x^{4}-2802345 c^{2} x^{2}+226905\right ) \left (-6 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{2} d x +\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{385875 c^{2} \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (c^{2} x^{2}-1\right )}+\frac {x \left (1125 c^{6} x^{6}-7371 c^{4} x^{4}+26495 c^{2} x^{2}-226905\right ) \left (24 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{4} d^{2} x^{2}-\frac {24 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{3} d x b}{\sqrt {c x -1}\, \sqrt {c x +1}}-6 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{2} d +\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{3} b^{2} c^{2}}{\left (c x -1\right ) \left (c x +1\right )}-\frac {\left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {\left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}\right )}{385875 c^{2} \left (c x -1\right )^{2} \left (c x +1\right )^{2}}\) | \(489\) |
Input:
int((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/c*(-d^3*a^2*(1/7*c^7*x^7-3/5*c^5*x^5+c^3*x^3-c*x)-d^3*b^2*(-16/35*arccos h(c*x)^2*c*x+1/7*arccosh(c*x)^2*(c*x-1)^3*(c*x+1)^3*c*x-6/35*arccosh(c*x)^ 2*c*x*(c*x-1)^2*(c*x+1)^2+8/35*arccosh(c*x)^2*c*x*(c*x-1)*(c*x+1)+32/35*ar ccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)-413312/385875*c*x-2/49*arccosh(c*x) *(c*x-1)^(7/2)*(c*x+1)^(7/2)+2/343*c*x*(c*x-1)^3*(c*x+1)^3-888/42875*c*x*( c*x-1)^2*(c*x+1)^2+30256/385875*c*x*(c*x-1)*(c*x+1)+12/175*arccosh(c*x)*(c *x-1)^(5/2)*(c*x+1)^(5/2)-16/105*arccosh(c*x)*(c*x-1)^(3/2)*(c*x+1)^(3/2)) -2*d^3*a*b*(1/7*arccosh(c*x)*c^7*x^7-3/5*arccosh(c*x)*c^5*x^5+c^3*x^3*arcc osh(c*x)-c*x*arccosh(c*x)-1/3675*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(75*c^6*x^6-3 51*c^4*x^4+757*c^2*x^2-2161)))
Time = 0.25 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.13 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=-\frac {1125 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} d^{3} x^{7} - 189 \, {\left (1225 \, a^{2} + 78 \, b^{2}\right )} c^{5} d^{3} x^{5} + 35 \, {\left (11025 \, a^{2} + 1514 \, b^{2}\right )} c^{3} d^{3} x^{3} - 105 \, {\left (3675 \, a^{2} + 4322 \, b^{2}\right )} c d^{3} x + 11025 \, {\left (5 \, b^{2} c^{7} d^{3} x^{7} - 21 \, b^{2} c^{5} d^{3} x^{5} + 35 \, b^{2} c^{3} d^{3} x^{3} - 35 \, b^{2} c d^{3} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 210 \, {\left (525 \, a b c^{7} d^{3} x^{7} - 2205 \, a b c^{5} d^{3} x^{5} + 3675 \, a b c^{3} d^{3} x^{3} - 3675 \, a b c d^{3} x - {\left (75 \, b^{2} c^{6} d^{3} x^{6} - 351 \, b^{2} c^{4} d^{3} x^{4} + 757 \, b^{2} c^{2} d^{3} x^{2} - 2161 \, b^{2} d^{3}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 210 \, {\left (75 \, a b c^{6} d^{3} x^{6} - 351 \, a b c^{4} d^{3} x^{4} + 757 \, a b c^{2} d^{3} x^{2} - 2161 \, a b d^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{385875 \, c} \] Input:
integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))^2,x, algorithm="fricas")
Output:
-1/385875*(1125*(49*a^2 + 2*b^2)*c^7*d^3*x^7 - 189*(1225*a^2 + 78*b^2)*c^5 *d^3*x^5 + 35*(11025*a^2 + 1514*b^2)*c^3*d^3*x^3 - 105*(3675*a^2 + 4322*b^ 2)*c*d^3*x + 11025*(5*b^2*c^7*d^3*x^7 - 21*b^2*c^5*d^3*x^5 + 35*b^2*c^3*d^ 3*x^3 - 35*b^2*c*d^3*x)*log(c*x + sqrt(c^2*x^2 - 1))^2 + 210*(525*a*b*c^7* d^3*x^7 - 2205*a*b*c^5*d^3*x^5 + 3675*a*b*c^3*d^3*x^3 - 3675*a*b*c*d^3*x - (75*b^2*c^6*d^3*x^6 - 351*b^2*c^4*d^3*x^4 + 757*b^2*c^2*d^3*x^2 - 2161*b^ 2*d^3)*sqrt(c^2*x^2 - 1))*log(c*x + sqrt(c^2*x^2 - 1)) - 210*(75*a*b*c^6*d ^3*x^6 - 351*a*b*c^4*d^3*x^4 + 757*a*b*c^2*d^3*x^2 - 2161*a*b*d^3)*sqrt(c^ 2*x^2 - 1))/c
\[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=- d^{3} \left (\int \left (- a^{2}\right )\, dx + \int \left (- b^{2} \operatorname {acosh}^{2}{\left (c x \right )}\right )\, dx + \int \left (- 2 a b \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int 3 a^{2} c^{2} x^{2}\, dx + \int \left (- 3 a^{2} c^{4} x^{4}\right )\, dx + \int a^{2} c^{6} x^{6}\, dx + \int 3 b^{2} c^{2} x^{2} \operatorname {acosh}^{2}{\left (c x \right )}\, dx + \int \left (- 3 b^{2} c^{4} x^{4} \operatorname {acosh}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{6} x^{6} \operatorname {acosh}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{2} x^{2} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 6 a b c^{4} x^{4} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int 2 a b c^{6} x^{6} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \] Input:
integrate((-c**2*d*x**2+d)**3*(a+b*acosh(c*x))**2,x)
Output:
-d**3*(Integral(-a**2, x) + Integral(-b**2*acosh(c*x)**2, x) + Integral(-2 *a*b*acosh(c*x), x) + Integral(3*a**2*c**2*x**2, x) + Integral(-3*a**2*c** 4*x**4, x) + Integral(a**2*c**6*x**6, x) + Integral(3*b**2*c**2*x**2*acosh (c*x)**2, x) + Integral(-3*b**2*c**4*x**4*acosh(c*x)**2, x) + Integral(b** 2*c**6*x**6*acosh(c*x)**2, x) + Integral(6*a*b*c**2*x**2*acosh(c*x), x) + Integral(-6*a*b*c**4*x**4*acosh(c*x), x) + Integral(2*a*b*c**6*x**6*acosh( c*x), x))
Leaf count of result is larger than twice the leaf count of optimal. 714 vs. \(2 (271) = 542\).
Time = 0.06 (sec) , antiderivative size = 714, normalized size of antiderivative = 2.27 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx =\text {Too large to display} \] Input:
integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))^2,x, algorithm="maxima")
Output:
-1/7*b^2*c^6*d^3*x^7*arccosh(c*x)^2 - 1/7*a^2*c^6*d^3*x^7 + 3/5*b^2*c^4*d^ 3*x^5*arccosh(c*x)^2 + 3/5*a^2*c^4*d^3*x^5 - 2/245*(35*x^7*arccosh(c*x) - (5*sqrt(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)*a*b*c^6*d^3 + 2/25725*(105*( 5*sqrt(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*arccosh(c*x) - (75*c^6*x^7 + 1 26*c^4*x^5 + 280*c^2*x^3 + 1680*x)/c^6)*b^2*c^6*d^3 - b^2*c^2*d^3*x^3*arcc osh(c*x)^2 + 2/25*(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*sqrt(c^2*x^2 - 1)/c^6)*c)*a*b*c^4*d^3 - 2/37 5*(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^3 - a^2*c^2*d^3*x^3 - 2/3*(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))*a*b*c^2*d^3 + 2/9*(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^3 + b^2*d^3*x*arccosh(c*x)^2 + 2*b^2*d^3*(x - sqrt(c^2*x^2 - 1)*arccosh(c*x)/c) + a^2*d^3*x + 2*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1) )*a*b*d^3/c
Exception generated. \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)^3*(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. \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \] Input:
int((a + b*acosh(c*x))^2*(d - c^2*d*x^2)^3,x)
Output:
int((a + b*acosh(c*x))^2*(d - c^2*d*x^2)^3, x)
\[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\frac {d^{3} \left (-1050 \mathit {acosh} \left (c x \right ) a b \,c^{7} x^{7}+4410 \mathit {acosh} \left (c x \right ) a b \,c^{5} x^{5}-7350 \mathit {acosh} \left (c x \right ) a b \,c^{3} x^{3}+7350 \mathit {acosh} \left (c x \right ) a b c x +150 \sqrt {c^{2} x^{2}-1}\, a b \,c^{6} x^{6}-702 \sqrt {c^{2} x^{2}-1}\, a b \,c^{4} x^{4}+1514 \sqrt {c^{2} x^{2}-1}\, a b \,c^{2} x^{2}+3028 \sqrt {c^{2} x^{2}-1}\, a b -7350 \sqrt {c x +1}\, \sqrt {c x -1}\, a b +3675 \left (\int \mathit {acosh} \left (c x \right )^{2}d x \right ) b^{2} c -3675 \left (\int \mathit {acosh} \left (c x \right )^{2} x^{6}d x \right ) b^{2} c^{7}+11025 \left (\int \mathit {acosh} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}-11025 \left (\int \mathit {acosh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}-525 a^{2} c^{7} x^{7}+2205 a^{2} c^{5} x^{5}-3675 a^{2} c^{3} x^{3}+3675 a^{2} c x \right )}{3675 c} \] Input:
int((-c^2*d*x^2+d)^3*(a+b*acosh(c*x))^2,x)
Output:
(d**3*( - 1050*acosh(c*x)*a*b*c**7*x**7 + 4410*acosh(c*x)*a*b*c**5*x**5 - 7350*acosh(c*x)*a*b*c**3*x**3 + 7350*acosh(c*x)*a*b*c*x + 150*sqrt(c**2*x* *2 - 1)*a*b*c**6*x**6 - 702*sqrt(c**2*x**2 - 1)*a*b*c**4*x**4 + 1514*sqrt( c**2*x**2 - 1)*a*b*c**2*x**2 + 3028*sqrt(c**2*x**2 - 1)*a*b - 7350*sqrt(c* x + 1)*sqrt(c*x - 1)*a*b + 3675*int(acosh(c*x)**2,x)*b**2*c - 3675*int(aco sh(c*x)**2*x**6,x)*b**2*c**7 + 11025*int(acosh(c*x)**2*x**4,x)*b**2*c**5 - 11025*int(acosh(c*x)**2*x**2,x)*b**2*c**3 - 525*a**2*c**7*x**7 + 2205*a** 2*c**5*x**5 - 3675*a**2*c**3*x**3 + 3675*a**2*c*x))/(3675*c)