Integrand size = 31, antiderivative size = 713 \[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {b f^2 g x \sqrt {d-c^2 d x^2}}{c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b g^3 x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b f g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f^2 g x^3 \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b g^3 x^3 \sqrt {d-c^2 d x^2}}{45 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c g^3 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {f^2 g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^2}-\frac {2 g^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 c^4}-\frac {g^3 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {f^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 f g^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \]
1/2*f^3*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-3/8*f*g^2*x*(a+b*arccosh (c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+3/4*f*g^2*x^3*(a+b*arccosh(c*x))*(-c^2*d*x ^2+d)^(1/2)-f^2*g*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2) /c^2-2/15*g^3*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4 -1/5*g^3*x^2*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+ b*f^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/15*b*g^3*x* (-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/4*b*c*f^3*x^2*(-c^2 *d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/16*b*f*g^2*x^2*(-c^2*d*x^2+d )^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/3*b*c*f^2*g*x^3*(-c^2*d*x^2+d)^(1/ 2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/45*b*g^3*x^3*(-c^2*d*x^2+d)^(1/2)/c/(c*x- 1)^(1/2)/(c*x+1)^(1/2)-3/16*b*c*f*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/ 2)/(c*x+1)^(1/2)-1/25*b*c*g^3*x^5*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+ 1)^(1/2)-1/4*f^3*(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)-3/16*f*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 = 1.43 (sec) , antiderivative size = 491, normalized size of antiderivative = 0.69 \[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {240 a \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (-16 g^3-c^2 g \left (120 f^2+45 f g x+8 g^2 x^2\right )+6 c^4 x \left (10 f^3+20 f^2 g x+15 f g^2 x^2+4 g^3 x^3\right )\right )-3600 a c \sqrt {d} f \left (4 c^2 f^2+3 g^2\right ) \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+2400 b c^2 f^2 g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)-\cosh (3 \text {arccosh}(c x))\right )-3600 b c^3 f^3 \sqrt {d-c^2 d x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))-675 b c f g^2 \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )+8 b g^3 \sqrt {d-c^2 d x^2} \left (450 c x-450 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-25 \cosh (3 \text {arccosh}(c x))-9 \cosh (5 \text {arccosh}(c x))+75 \text {arccosh}(c x) \sinh (3 \text {arccosh}(c x))+45 \text {arccosh}(c x) \sinh (5 \text {arccosh}(c x))\right )}{28800 c^4 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
(240*a*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(-16*g^3 - c^2*g*(120*f^2 + 45*f*g*x + 8*g^2*x^2) + 6*c^4*x*(10*f^3 + 20*f^2*g*x + 1 5*f*g^2*x^2 + 4*g^3*x^3)) - 3600*a*c*Sqrt[d]*f*(4*c^2*f^2 + 3*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))] + 2400*b*c^2*f^2*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*ArcCosh[c*x]]) - 36 00*b*c^3*f^3*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(A rcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) - 675*b*c*f*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]]) + 8*b*g^3*Sqrt[d - c^2*d*x^2]*(450*c*x - 450*Sqrt[(-1 + c*x)/(1 + c*x )]*(1 + c*x)*ArcCosh[c*x] - 25*Cosh[3*ArcCosh[c*x]] - 9*Cosh[5*ArcCosh[c*x ]] + 75*ArcCosh[c*x]*Sinh[3*ArcCosh[c*x]] + 45*ArcCosh[c*x]*Sinh[5*ArcCosh [c*x]]))/(28800*c^4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
Time = 1.77 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.56, 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.
\(\displaystyle \int \sqrt {d-c^2 d x^2} (f+g x)^3 (a+b \text {arccosh}(c x)) \, dx\) |
\(\Big \downarrow \) 6387 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \sqrt {c x-1} \sqrt {c x+1} (f+g x)^3 (a+b \text {arccosh}(c x))dx}{\sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 6390 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \left (\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) f^3+3 g x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) f^2+3 g^2 x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) f+g^3 x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))\right )dx}{\sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {2 g^3 (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{15 c^4}-\frac {3 f g^2 (a+b \text {arccosh}(c x))^2}{16 b c^3}+\frac {f^2 g (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{c^2}-\frac {3 f g^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {g^3 x^2 (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{5 c^2}+\frac {1}{2} f^3 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))-\frac {f^3 (a+b \text {arccosh}(c x))^2}{4 b c}+\frac {3}{4} f g^2 x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))+\frac {2 b g^3 x}{15 c^3}-\frac {1}{4} b c f^3 x^2-\frac {1}{3} b c f^2 g x^3+\frac {b f^2 g x}{c}-\frac {3}{16} b c f g^2 x^4+\frac {3 b f g^2 x^2}{16 c}-\frac {1}{25} b c g^3 x^5+\frac {b g^3 x^3}{45 c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\) |
(Sqrt[d - c^2*d*x^2]*((b*f^2*g*x)/c + (2*b*g^3*x)/(15*c^3) - (b*c*f^3*x^2) /4 + (3*b*f*g^2*x^2)/(16*c) - (b*c*f^2*g*x^3)/3 + (b*g^3*x^3)/(45*c) - (3* b*c*f*g^2*x^4)/16 - (b*c*g^3*x^5)/25 + (f^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] *(a + b*ArcCosh[c*x]))/2 - (3*f*g^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b* ArcCosh[c*x]))/(8*c^2) + (3*f*g^2*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b* ArcCosh[c*x]))/4 + (f^2*g*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[ c*x]))/c^2 + (2*g^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x])) /(15*c^4) + (g^3*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]) )/(5*c^2) - (f^3*(a + b*ArcCosh[c*x])^2)/(4*b*c) - (3*f*g^2*(a + b*ArcCosh [c*x])^2)/(16*b*c^3)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.1.53.3.1 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. \(1417\) vs. \(2(613)=1226\).
Time = 1.32 (sec) , antiderivative size = 1418, normalized size of antiderivative = 1.99
method | result | size |
default | \(\text {Expression too large to display}\) | \(1418\) |
parts | \(\text {Expression too large to display}\) | \(1418\) |
a*(f^3*(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^3*(-1/5*x^2*(-c^2*d*x^2+d)^(3/2)/c^2/d-2/15/d /c^4*(-c^2*d*x^2+d)^(3/2))+3*f*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))))-f^2*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*f*arccosh(c*x)^2*(4*c^2*f ^2+3*g^2)+1/800*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4+16*(c*x+1)^( 1/2)*(c*x-1)^(1/2)*c^5*x^5+13*c^2*x^2-20*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x ^3+5*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-1)*g^3*(-1+5*arccosh(c*x))/(c*x+1)/c^ 4/(c*x-1)+3/256*(-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))*f*g^2*(-1+4*arccosh(c*x))/(c*x+1)/c^3/(c*x-1)+1/28 8*(-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)*g*(36*arccosh(c*x)*c^2*f^2- 12*c^2*f^2+3*arccosh(c*x)*g^2-g^2)/(c*x+1)/c^4/(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^3*(-1+2*arccosh(c*x))/(c*x+1)/c/(c*x-1)-1/16*(-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)*g*(6*arccosh(c* x)*c^2*f^2-6*c^2*f^2+arccosh(c*x)*g^2-g^2)/(c*x+1)/c^4/(c*x-1)-1/16*(-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)*g*(6*arcc...
\[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \,d x } \]
integral((a*g^3*x^3 + 3*a*f*g^2*x^2 + 3*a*f^2*g*x + a*f^3 + (b*g^3*x^3 + 3 *b*f*g^2*x^2 + 3*b*f^2*g*x + b*f^3)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d), x)
\[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}\, dx \]
\[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \,d x } \]
1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a*f^3 - 1/15*a*g^3*(3 *(-c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(3/2)/(c^4*d)) + 3/8*a*f*g^2*(sqrt(-c^2*d*x^2 + d)*x/c^2 - 2*(-c^2*d*x^2 + d)^(3/2)*x/(c^2* d) + sqrt(d)*arcsin(c*x)/c^3) - (-c^2*d*x^2 + d)^(3/2)*a*f^2*g/(c^2*d) + i ntegrate(sqrt(-c^2*d*x^2 + d)*b*g^3*x^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 3*sqrt(-c^2*d*x^2 + d)*b*f*g^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 3*sqrt(-c^2*d*x^2 + d)*b*f^2*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + sqrt(-c^2*d*x^2 + d)*b*f^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) , x)
Exception generated. \[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]