Integrand size = 31, antiderivative size = 479 \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {2 f g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 c^2}-\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {g^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \]
1/2*f^2*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-1/8*g^2*x*(a+b*arccosh(c *x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/4*g^2*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d )^(1/2)-2/3*f*g*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c ^2+2/3*b*f*g*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/4*b*c* f^2*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16*b*g^2*x^2*(- c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/9*b*c*f*g*x^3*(-c^2*d*x ^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/16*b*c*g^2*x^4*(-c^2*d*x^2+d)^(1 /2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/4*f^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d )^(1/2)/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/16*g^2*(a+b*arccosh(c*x))^2*(-c^ 2*d*x^2+d)^(1/2)/b/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.91 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.74 \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {48 a c \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (12 c^2 f^2 x+16 f g \left (-1+c^2 x^2\right )+3 g^2 x \left (-1+2 c^2 x^2\right )\right )-144 a \sqrt {d} \left (4 c^2 f^2+g^2\right ) \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+64 b c f g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)-\cosh (3 \text {arccosh}(c x))\right )-144 b c^2 f^2 \sqrt {d-c^2 d x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))-9 b g^2 \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )}{1152 c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
(48*a*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(12*c^2*f ^2*x + 16*f*g*(-1 + c^2*x^2) + 3*g^2*x*(-1 + 2*c^2*x^2)) - 144*a*Sqrt[d]*( 4*c^2*f^2 + g^2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 64*b*c*f*g*Sqrt[d - c^2*d*x^2]*(9 *c*x + 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] - Cosh[3*A rcCosh[c*x]]) - 144*b*c^2*f^2*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) - 9*b*g^2*Sqrt[d - c ^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4 *ArcCosh[c*x]]))/(1152*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
Time = 1.45 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.59, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6387, 6390, 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.
| \(\displaystyle \int \sqrt {d-c^2 d x^2} (f+g x)^2 (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6387 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \sqrt {c x-1} \sqrt {c x+1} (f+g x)^2 (a+b \text {arccosh}(c x))dx}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6390 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \left (\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) f^2+2 g x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) f+g^2 x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))\right )dx}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {g^2 (a+b \text {arccosh}(c x))^2}{16 b c^3}+\frac {2 f g (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 c^2}-\frac {g^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {1}{2} f^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))-\frac {f^2 (a+b \text {arccosh}(c x))^2}{4 b c}+\frac {1}{4} g^2 x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))-\frac {1}{4} b c f^2 x^2-\frac {2}{9} b c f g x^3+\frac {2 b f g x}{3 c}-\frac {1}{16} b c g^2 x^4+\frac {b g^2 x^2}{16 c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\) |
(Sqrt[d - c^2*d*x^2]*((2*b*f*g*x)/(3*c) - (b*c*f^2*x^2)/4 + (b*g^2*x^2)/(1 6*c) - (2*b*c*f*g*x^3)/9 - (b*c*g^2*x^4)/16 + (f^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/2 - (g^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(8*c^2) + (g^2*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*A rcCosh[c*x]))/4 + (2*f*g*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c *x]))/(3*c^2) - (f^2*(a + b*ArcCosh[c*x])^2)/(4*b*c) - (g^2*(a + b*ArcCosh [c*x])^2)/(16*b*c^3)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.1.54.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d)^IntPart[p]*((d + e*x^2)^Fra cPart[p]/((-1 + c*x)^FracPart[p]*(1 + c*x)^FracPart[p])) Int[(f + g*x)^m* (-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && IntegerQ[m]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Int[Expand Integrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[ d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1 ] || (EqQ[m, 2] && LtQ[p, -2]))
Leaf count of result is larger than twice the leaf count of optimal. \(992\) vs. \(2(407)=814\).
Time = 1.17 (sec) , antiderivative size = 993, normalized size of antiderivative = 2.07
| method | result | size |
| default | \(a \left (f^{2} \left (\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}\right )+g^{2} \left (-\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}}{4 c^{2}}\right )-\frac {2 f g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} \left (4 c^{2} f^{2}+g^{2}\right )}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) g^{2} \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) f g \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{36 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) f^{2} \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x +1\right ) c \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) f g \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{4 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) f g \left (1+\operatorname {arccosh}\left (c x \right )\right )}{4 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) f^{2} \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x +1\right ) c \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) f g \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{36 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) g^{2} \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )\) | \(993\) |
| parts | \(a \left (f^{2} \left (\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}\right )+g^{2} \left (-\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}}{4 c^{2}}\right )-\frac {2 f g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} \left (4 c^{2} f^{2}+g^{2}\right )}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) g^{2} \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) f g \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{36 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) f^{2} \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x +1\right ) c \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) f g \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{4 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) f g \left (1+\operatorname {arccosh}\left (c x \right )\right )}{4 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) f^{2} \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x +1\right ) c \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) f g \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{36 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) g^{2} \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )\) | \(993\) |
a*(f^2*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2 )*x/(-c^2*d*x^2+d)^(1/2)))+g^2*(-1/4*x*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/4/c^2* (1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c ^2*d*x^2+d)^(1/2))))-2/3*f*g*(-c^2*d*x^2+d)^(3/2)/c^2/d)+b*(-1/16*(-d*(c^2 *x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c*x)^2*(4*c^2*f^2+g ^2)+1/256*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*(c*x+1)^(1/2)*(c* x-1)^(1/2)*c^4*x^4+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/ 2)*(c*x+1)^(1/2))*g^2*(-1+4*arccosh(c*x))/(c*x+1)/c^3/(c*x-1)+1/36*(-d*(c^ 2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3 -3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)*f*g*(-1+3*arccosh(c*x))/(c*x+1)/c^2/ (c*x-1)+1/16*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+2*(c*x-1)^(1/2)*(c*x+ 1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*f^2*(-1+2*arccosh(c*x))/(c*x +1)/c/(c*x-1)-1/4*(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+ c^2*x^2-1)*f*g*(-1+arccosh(c*x))/(c*x+1)/c^2/(c*x-1)-1/4*(-d*(c^2*x^2-1))^ (1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*f*g*(1+arccosh(c*x))/(c *x+1)/c^2/(c*x-1)+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1 /2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)*f^2*(1+2*arccosh( c*x))/(c*x+1)/c/(c*x-1)+1/36*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x-1)^(1/2)*(c*x +1)^(1/2)*c^3*x^3+4*c^4*x^4+3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-5*c^2*x^2+1) *f*g*(1+3*arccosh(c*x))/(c*x+1)/c^2/(c*x-1)+1/256*(-d*(c^2*x^2-1))^(1/2...
\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \,d x } \]
integral(sqrt(-c^2*d*x^2 + d)*(a*g^2*x^2 + 2*a*f*g*x + a*f^2 + (b*g^2*x^2 + 2*b*f*g*x + b*f^2)*arccosh(c*x)), x)
\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{2}\, dx \]
\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \,d x } \]
1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a*f^2 + 1/8*a*g^2*(sq rt(-c^2*d*x^2 + d)*x/c^2 - 2*(-c^2*d*x^2 + d)^(3/2)*x/(c^2*d) + sqrt(d)*ar csin(c*x)/c^3) - 2/3*(-c^2*d*x^2 + d)^(3/2)*a*f*g/(c^2*d) + integrate(sqrt (-c^2*d*x^2 + d)*b*g^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 2*sqrt (-c^2*d*x^2 + d)*b*f*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + sqrt(-c^ 2*d*x^2 + d)*b*f^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
Exception generated. \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]