Integrand size = 24, antiderivative size = 188 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4 \, dx=\frac {35}{8} \left (b^2-c^2\right )^2 x+\frac {35}{8} c \left (b^2-c^2\right )^{3/2} \cosh (x)+\frac {35}{8} b \left (b^2-c^2\right )^{3/2} \sinh (x)+\frac {35}{24} \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac {7}{12} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac {1}{4} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3 \]
35/8*(b^2-c^2)^2*x+35/8*c*(b^2-c^2)^(3/2)*cosh(x)+35/8*b*(b^2-c^2)^(3/2)*s inh(x)+35/24*(b^2-c^2)*(c*cosh(x)+b*sinh(x))*(b*cosh(x)+c*sinh(x)+(b^2-c^2 )^(1/2))+7/12*(c*cosh(x)+b*sinh(x))*(b^2-c^2)^(1/2)*(b*cosh(x)+c*sinh(x)+( b^2-c^2)^(1/2))^2+1/4*(c*cosh(x)+b*sinh(x))*(b*cosh(x)+c*sinh(x)+(b^2-c^2) ^(1/2))^3
Time = 0.33 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.11 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4 \, dx=\frac {35}{8} (b-c)^2 (b+c)^2 x+7 (b-c) c (b+c) \sqrt {b^2-c^2} \cosh (x)+\frac {7}{2} b c \left (b^2-c^2\right ) \cosh (2 x)+\frac {1}{3} c \sqrt {b^2-c^2} \left (3 b^2+c^2\right ) \cosh (3 x)+\frac {1}{8} b c \left (b^2+c^2\right ) \cosh (4 x)+7 b (b-c) (b+c) \sqrt {b^2-c^2} \sinh (x)+\frac {7}{4} \left (b^4-c^4\right ) \sinh (2 x)+\frac {1}{3} b \sqrt {b^2-c^2} \left (b^2+3 c^2\right ) \sinh (3 x)+\frac {1}{32} \left (b^4+6 b^2 c^2+c^4\right ) \sinh (4 x) \]
(35*(b - c)^2*(b + c)^2*x)/8 + 7*(b - c)*c*(b + c)*Sqrt[b^2 - c^2]*Cosh[x] + (7*b*c*(b^2 - c^2)*Cosh[2*x])/2 + (c*Sqrt[b^2 - c^2]*(3*b^2 + c^2)*Cosh [3*x])/3 + (b*c*(b^2 + c^2)*Cosh[4*x])/8 + 7*b*(b - c)*(b + c)*Sqrt[b^2 - c^2]*Sinh[x] + (7*(b^4 - c^4)*Sinh[2*x])/4 + (b*Sqrt[b^2 - c^2]*(b^2 + 3*c ^2)*Sinh[3*x])/3 + ((b^4 + 6*b^2*c^2 + c^4)*Sinh[4*x])/32
Time = 0.58 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {3042, 3592, 3042, 3592, 3042, 3592, 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.
| \(\displaystyle \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (\sqrt {b^2-c^2}+b \cos (i x)-i c \sin (i x)\right )^4dx\) |
| \(\Big \downarrow \) 3592 |
| \(\displaystyle \frac {7}{4} \sqrt {b^2-c^2} \int \left (b \cosh (x)+c \sinh (x)+\sqrt {b^2-c^2}\right )^3dx+\frac {1}{4} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac {7}{4} \sqrt {b^2-c^2} \int \left (b \cos (i x)-i c \sin (i x)+\sqrt {b^2-c^2}\right )^3dx\) |
| \(\Big \downarrow \) 3592 |
| \(\displaystyle \frac {7}{4} \sqrt {b^2-c^2} \left (\frac {5}{3} \sqrt {b^2-c^2} \int \left (b \cosh (x)+c \sinh (x)+\sqrt {b^2-c^2}\right )^2dx+\frac {1}{3} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2\right )+\frac {1}{4} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac {7}{4} \sqrt {b^2-c^2} \left (\frac {1}{3} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac {5}{3} \sqrt {b^2-c^2} \int \left (b \cos (i x)-i c \sin (i x)+\sqrt {b^2-c^2}\right )^2dx\right )\) |
| \(\Big \downarrow \) 3592 |
| \(\displaystyle \frac {7}{4} \sqrt {b^2-c^2} \left (\frac {5}{3} \sqrt {b^2-c^2} \left (\frac {3}{2} \sqrt {b^2-c^2} \int \left (b \cosh (x)+c \sinh (x)+\sqrt {b^2-c^2}\right )dx+\frac {1}{2} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )\right )+\frac {1}{3} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2\right )+\frac {1}{4} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3+\frac {7}{4} \sqrt {b^2-c^2} \left (\frac {1}{3} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac {5}{3} \sqrt {b^2-c^2} \left (\frac {3}{2} \sqrt {b^2-c^2} \left (x \sqrt {b^2-c^2}+b \sinh (x)+c \cosh (x)\right )+\frac {1}{2} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )\right )\right )\) |
((c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^3)/4 + (7*Sqrt[b^2 - c^2]*(((c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2)/3 + (5*Sqrt[b^2 - c^2]*((3*Sqrt[b^2 - c^2]*(Sqrt[b^2 - c^2 ]*x + c*Cosh[x] + b*Sinh[x]))/2 + ((c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2 ] + b*Cosh[x] + c*Sinh[x]))/2))/3))/4
3.8.53.3.1 Defintions of rubi rules used
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (n_), x_Symbol] :> Simp[(-(c*Cos[d + e*x] - b*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)/(e*n)), x] + Simp[a*((2*n - 1)/n) Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[a^2 - b^2 - c^2, 0] && GtQ[n, 0]
Time = 1.94 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.71
| method | result | size |
| parts | \(\left (b^{2}-c^{2}\right )^{2} x +c^{4} \left (\left (\frac {\sinh \left (x \right )^{3}}{4}-\frac {3 \sinh \left (x \right )}{8}\right ) \cosh \left (x \right )+\frac {3 x}{8}\right )+4 b^{3} \left (\frac {c \sinh \left (x \right )^{4}}{4}+\frac {\sqrt {b^{2}-c^{2}}\, \sinh \left (x \right )^{3}}{3}+\frac {c \sinh \left (x \right )^{2}}{2}+\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}\right )+6 b^{2} c^{2} \left (\frac {\cosh \left (x \right )^{3} \sinh \left (x \right )}{4}-\frac {\cosh \left (x \right ) \sinh \left (x \right )}{8}-\frac {x}{8}\right )+4 \sqrt {b^{2}-c^{2}}\, b^{2} c \cosh \left (x \right )^{3}+6 b^{4} \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}+\frac {x}{2}\right )-6 b^{2} c^{2} \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}+\frac {x}{2}\right )+4 b \left (\frac {\sinh \left (x \right )^{4} c^{3}}{4}+\sqrt {\left (b -c \right ) \left (b +c \right )}\, \sinh \left (x \right )^{3} c^{2}+\frac {3 \sinh \left (x \right )^{2} b^{2} c}{2}-\frac {3 \sinh \left (x \right )^{2} c^{3}}{2}+\left (\left (b -c \right ) \left (b +c \right )\right )^{\frac {3}{2}} \sinh \left (x \right )\right )+b^{4} \left (\left (\frac {\cosh \left (x \right )^{3}}{4}+\frac {3 \cosh \left (x \right )}{8}\right ) \sinh \left (x \right )+\frac {3 x}{8}\right )+4 c \left (b^{2}-c^{2}\right )^{\frac {3}{2}} \cosh \left (x \right )+6 c^{2} \left (b^{2}-c^{2}\right ) \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}-\frac {x}{2}\right )+4 \sqrt {b^{2}-c^{2}}\, c^{3} \left (-\frac {2}{3}+\frac {\sinh \left (x \right )^{2}}{3}\right ) \cosh \left (x \right )\) | \(321\) |
| default | \(b^{4} x +c^{4} x -2 b^{2} c^{2} x +4 \sqrt {b^{2}-c^{2}}\, b^{3} \left (\frac {2}{3}+\frac {\cosh \left (x \right )^{2}}{3}\right ) \sinh \left (x \right )-6 b^{2} c^{2} \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}+\frac {x}{2}\right )+4 \sqrt {b^{2}-c^{2}}\, b^{3} \sinh \left (x \right )+4 \sqrt {b^{2}-c^{2}}\, c^{3} \left (-\frac {2}{3}+\frac {\sinh \left (x \right )^{2}}{3}\right ) \cosh \left (x \right )+6 b^{2} c^{2} \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}-\frac {x}{2}\right )-4 \sqrt {b^{2}-c^{2}}\, c^{3} \cosh \left (x \right )+b^{4} \left (\left (\frac {\cosh \left (x \right )^{3}}{4}+\frac {3 \cosh \left (x \right )}{8}\right ) \sinh \left (x \right )+\frac {3 x}{8}\right )+6 b^{4} \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}+\frac {x}{2}\right )+c^{4} \left (\left (\frac {\sinh \left (x \right )^{3}}{4}-\frac {3 \sinh \left (x \right )}{8}\right ) \cosh \left (x \right )+\frac {3 x}{8}\right )-6 c^{4} \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}-\frac {x}{2}\right )+4 \sqrt {b^{2}-c^{2}}\, b^{2} c \cosh \left (x \right )^{3}+4 \sqrt {b^{2}-c^{2}}\, b \,c^{2} \sinh \left (x \right )^{3}+b^{3} c \cosh \left (x \right )^{4}+6 b^{2} c^{2} \left (\frac {\cosh \left (x \right )^{3} \sinh \left (x \right )}{4}-\frac {\cosh \left (x \right ) \sinh \left (x \right )}{8}-\frac {x}{8}\right )+b \,c^{3} \sinh \left (x \right )^{4}+6 b^{3} c \cosh \left (x \right )^{2}-6 b \,c^{3} \cosh \left (x \right )^{2}-4 \sqrt {b^{2}-c^{2}}\, b \,c^{2} \sinh \left (x \right )+4 \sqrt {b^{2}-c^{2}}\, b^{2} c \cosh \left (x \right )\) | \(363\) |
| risch | \(-\frac {35 b^{2} c^{2} x}{4}+\frac {{\mathrm e}^{4 x} b^{4}}{64}+\frac {7 \,{\mathrm e}^{2 x} b^{4}}{8}+\frac {35 b^{4} x}{8}+\frac {35 c^{4} x}{8}+\frac {{\mathrm e}^{3 x} \sqrt {b^{2}-c^{2}}\, b^{2} c}{2}+\frac {{\mathrm e}^{3 x} \sqrt {b^{2}-c^{2}}\, b \,c^{2}}{2}+\frac {7 \,{\mathrm e}^{x} \sqrt {b^{2}-c^{2}}\, b^{2} c}{2}-\frac {7 \,{\mathrm e}^{x} \sqrt {b^{2}-c^{2}}\, b \,c^{2}}{2}+\frac {7 \,{\mathrm e}^{-x} \sqrt {b^{2}-c^{2}}\, b^{2} c}{2}+\frac {7 \,{\mathrm e}^{-x} \sqrt {b^{2}-c^{2}}\, b \,c^{2}}{2}+\frac {{\mathrm e}^{-3 x} \sqrt {b^{2}-c^{2}}\, b^{2} c}{2}-\frac {{\mathrm e}^{-3 x} \sqrt {b^{2}-c^{2}}\, b \,c^{2}}{2}-\frac {7 \,{\mathrm e}^{-2 x} b^{4}}{8}+\frac {{\mathrm e}^{4 x} c^{4}}{64}-\frac {7 \,{\mathrm e}^{2 x} c^{4}}{8}+\frac {7 \,{\mathrm e}^{-2 x} c^{4}}{8}-\frac {{\mathrm e}^{-4 x} b^{4}}{64}-\frac {{\mathrm e}^{-4 x} c^{4}}{64}+\frac {{\mathrm e}^{4 x} b^{3} c}{16}+\frac {3 \,{\mathrm e}^{4 x} b^{2} c^{2}}{32}+\frac {{\mathrm e}^{4 x} b \,c^{3}}{16}+\frac {7 \,{\mathrm e}^{2 x} b^{3} c}{4}-\frac {7 \,{\mathrm e}^{2 x} b \,c^{3}}{4}+\frac {7 \,{\mathrm e}^{-2 x} b^{3} c}{4}-\frac {7 \,{\mathrm e}^{-2 x} b \,c^{3}}{4}+\frac {{\mathrm e}^{-4 x} b^{3} c}{16}-\frac {3 \,{\mathrm e}^{-4 x} b^{2} c^{2}}{32}+\frac {{\mathrm e}^{-4 x} b \,c^{3}}{16}+\frac {{\mathrm e}^{3 x} \sqrt {b^{2}-c^{2}}\, b^{3}}{6}+\frac {{\mathrm e}^{3 x} \sqrt {b^{2}-c^{2}}\, c^{3}}{6}+\frac {7 \,{\mathrm e}^{x} \sqrt {b^{2}-c^{2}}\, b^{3}}{2}-\frac {7 \,{\mathrm e}^{x} \sqrt {b^{2}-c^{2}}\, c^{3}}{2}-\frac {7 \,{\mathrm e}^{-x} \sqrt {b^{2}-c^{2}}\, b^{3}}{2}-\frac {7 \,{\mathrm e}^{-x} \sqrt {b^{2}-c^{2}}\, c^{3}}{2}-\frac {{\mathrm e}^{-3 x} \sqrt {b^{2}-c^{2}}\, b^{3}}{6}+\frac {{\mathrm e}^{-3 x} \sqrt {b^{2}-c^{2}}\, c^{3}}{6}\) | \(519\) |
(b^2-c^2)^2*x+c^4*((1/4*sinh(x)^3-3/8*sinh(x))*cosh(x)+3/8*x)+4*b^3*(1/4*c *sinh(x)^4+1/3*(b^2-c^2)^(1/2)*sinh(x)^3+1/2*c*sinh(x)^2+sinh(x)*(b^2-c^2) ^(1/2))+6*b^2*c^2*(1/4*cosh(x)^3*sinh(x)-1/8*cosh(x)*sinh(x)-1/8*x)+4*(b^2 -c^2)^(1/2)*b^2*c*cosh(x)^3+6*b^4*(1/2*cosh(x)*sinh(x)+1/2*x)-6*b^2*c^2*(1 /2*cosh(x)*sinh(x)+1/2*x)+4*b*(1/4*sinh(x)^4*c^3+((b-c)*(b+c))^(1/2)*sinh( x)^3*c^2+3/2*sinh(x)^2*b^2*c-3/2*sinh(x)^2*c^3+((b-c)*(b+c))^(3/2)*sinh(x) )+b^4*((1/4*cosh(x)^3+3/8*cosh(x))*sinh(x)+3/8*x)+4*c*(b^2-c^2)^(3/2)*cosh (x)+6*c^2*(b^2-c^2)*(1/2*cosh(x)*sinh(x)-1/2*x)+4*(b^2-c^2)^(1/2)*c^3*(-2/ 3+1/3*sinh(x)^2)*cosh(x)
Leaf count of result is larger than twice the leaf count of optimal. 1293 vs. \(2 (164) = 328\).
Time = 0.27 (sec) , antiderivative size = 1293, normalized size of antiderivative = 6.88 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4 \, dx=\text {Too large to display} \]
1/192*(3*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^8 + 24*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)*sinh(x)^7 + 3*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*sinh(x)^8 + 168*(b^4 + 2*b^3*c - 2*b*c^3 - c ^4)*cosh(x)^6 + 84*(2*b^4 + 4*b^3*c - 4*b*c^3 - 2*c^4 + (b^4 + 4*b^3*c + 6 *b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^2)*sinh(x)^6 + 840*(b^4 - 2*b^2*c^2 + c^ 4)*x*cosh(x)^4 + 168*((b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^ 3 + 6*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x))*sinh(x)^5 + 210*((b^4 + 4*b ^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^4 + 12*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^2 + 4*(b^4 - 2*b^2*c^2 + c^4)*x)*sinh(x)^4 - 3*b^4 + 12*b^3 *c - 18*b^2*c^2 + 12*b*c^3 - 3*c^4 + 168*((b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b *c^3 + c^4)*cosh(x)^5 + 20*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^3 + 20* (b^4 - 2*b^2*c^2 + c^4)*x*cosh(x))*sinh(x)^3 - 168*(b^4 - 2*b^3*c + 2*b*c^ 3 - c^4)*cosh(x)^2 + 84*((b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh( x)^6 + 30*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^4 - 2*b^4 + 4*b^3*c - 4* b*c^3 + 2*c^4 + 60*(b^4 - 2*b^2*c^2 + c^4)*x*cosh(x)^2)*sinh(x)^2 + 24*((b ^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^7 + 42*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^5 + 140*(b^4 - 2*b^2*c^2 + c^4)*x*cosh(x)^3 - 14*(b ^4 - 2*b^3*c + 2*b*c^3 - c^4)*cosh(x))*sinh(x) + 32*((b^3 + 3*b^2*c + 3*b* c^2 + c^3)*cosh(x)^7 + 7*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)*sinh(x)^6 + (b^3 + 3*b^2*c + 3*b*c^2 + c^3)*sinh(x)^7 + 21*(b^3 + b^2*c - b*c^2 ...
Leaf count of result is larger than twice the leaf count of optimal. 626 vs. \(2 (178) = 356\).
Time = 0.31 (sec) , antiderivative size = 626, normalized size of antiderivative = 3.33 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4 \, dx=\frac {3 b^{4} x \sinh ^{4}{\left (x \right )}}{8} - \frac {3 b^{4} x \sinh ^{2}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{4} - 3 b^{4} x \sinh ^{2}{\left (x \right )} + \frac {3 b^{4} x \cosh ^{4}{\left (x \right )}}{8} + 3 b^{4} x \cosh ^{2}{\left (x \right )} + b^{4} x - \frac {3 b^{4} \sinh ^{3}{\left (x \right )} \cosh {\left (x \right )}}{8} + \frac {5 b^{4} \sinh {\left (x \right )} \cosh ^{3}{\left (x \right )}}{8} + 3 b^{4} \sinh {\left (x \right )} \cosh {\left (x \right )} + b^{3} c \cosh ^{4}{\left (x \right )} + 6 b^{3} c \cosh ^{2}{\left (x \right )} - \frac {8 b^{3} \sqrt {b^{2} - c^{2}} \sinh ^{3}{\left (x \right )}}{3} + 4 b^{3} \sqrt {b^{2} - c^{2}} \sinh {\left (x \right )} \cosh ^{2}{\left (x \right )} + 4 b^{3} \sqrt {b^{2} - c^{2}} \sinh {\left (x \right )} - \frac {3 b^{2} c^{2} x \sinh ^{4}{\left (x \right )}}{4} + \frac {3 b^{2} c^{2} x \sinh ^{2}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{2} + 6 b^{2} c^{2} x \sinh ^{2}{\left (x \right )} - \frac {3 b^{2} c^{2} x \cosh ^{4}{\left (x \right )}}{4} - 6 b^{2} c^{2} x \cosh ^{2}{\left (x \right )} - 2 b^{2} c^{2} x + \frac {3 b^{2} c^{2} \sinh ^{3}{\left (x \right )} \cosh {\left (x \right )}}{4} + \frac {3 b^{2} c^{2} \sinh {\left (x \right )} \cosh ^{3}{\left (x \right )}}{4} + 4 b^{2} c \sqrt {b^{2} - c^{2}} \cosh ^{3}{\left (x \right )} + 4 b^{2} c \sqrt {b^{2} - c^{2}} \cosh {\left (x \right )} + b c^{3} \sinh ^{4}{\left (x \right )} - 6 b c^{3} \cosh ^{2}{\left (x \right )} + 4 b c^{2} \sqrt {b^{2} - c^{2}} \sinh ^{3}{\left (x \right )} - 4 b c^{2} \sqrt {b^{2} - c^{2}} \sinh {\left (x \right )} + \frac {3 c^{4} x \sinh ^{4}{\left (x \right )}}{8} - \frac {3 c^{4} x \sinh ^{2}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{4} - 3 c^{4} x \sinh ^{2}{\left (x \right )} + \frac {3 c^{4} x \cosh ^{4}{\left (x \right )}}{8} + 3 c^{4} x \cosh ^{2}{\left (x \right )} + c^{4} x + \frac {5 c^{4} \sinh ^{3}{\left (x \right )} \cosh {\left (x \right )}}{8} - \frac {3 c^{4} \sinh {\left (x \right )} \cosh ^{3}{\left (x \right )}}{8} - 3 c^{4} \sinh {\left (x \right )} \cosh {\left (x \right )} + 4 c^{3} \sqrt {b^{2} - c^{2}} \sinh ^{2}{\left (x \right )} \cosh {\left (x \right )} - \frac {8 c^{3} \sqrt {b^{2} - c^{2}} \cosh ^{3}{\left (x \right )}}{3} - 4 c^{3} \sqrt {b^{2} - c^{2}} \cosh {\left (x \right )} \]
3*b**4*x*sinh(x)**4/8 - 3*b**4*x*sinh(x)**2*cosh(x)**2/4 - 3*b**4*x*sinh(x )**2 + 3*b**4*x*cosh(x)**4/8 + 3*b**4*x*cosh(x)**2 + b**4*x - 3*b**4*sinh( x)**3*cosh(x)/8 + 5*b**4*sinh(x)*cosh(x)**3/8 + 3*b**4*sinh(x)*cosh(x) + b **3*c*cosh(x)**4 + 6*b**3*c*cosh(x)**2 - 8*b**3*sqrt(b**2 - c**2)*sinh(x)* *3/3 + 4*b**3*sqrt(b**2 - c**2)*sinh(x)*cosh(x)**2 + 4*b**3*sqrt(b**2 - c* *2)*sinh(x) - 3*b**2*c**2*x*sinh(x)**4/4 + 3*b**2*c**2*x*sinh(x)**2*cosh(x )**2/2 + 6*b**2*c**2*x*sinh(x)**2 - 3*b**2*c**2*x*cosh(x)**4/4 - 6*b**2*c* *2*x*cosh(x)**2 - 2*b**2*c**2*x + 3*b**2*c**2*sinh(x)**3*cosh(x)/4 + 3*b** 2*c**2*sinh(x)*cosh(x)**3/4 + 4*b**2*c*sqrt(b**2 - c**2)*cosh(x)**3 + 4*b* *2*c*sqrt(b**2 - c**2)*cosh(x) + b*c**3*sinh(x)**4 - 6*b*c**3*cosh(x)**2 + 4*b*c**2*sqrt(b**2 - c**2)*sinh(x)**3 - 4*b*c**2*sqrt(b**2 - c**2)*sinh(x ) + 3*c**4*x*sinh(x)**4/8 - 3*c**4*x*sinh(x)**2*cosh(x)**2/4 - 3*c**4*x*si nh(x)**2 + 3*c**4*x*cosh(x)**4/8 + 3*c**4*x*cosh(x)**2 + c**4*x + 5*c**4*s inh(x)**3*cosh(x)/8 - 3*c**4*sinh(x)*cosh(x)**3/8 - 3*c**4*sinh(x)*cosh(x) + 4*c**3*sqrt(b**2 - c**2)*sinh(x)**2*cosh(x) - 8*c**3*sqrt(b**2 - c**2)* cosh(x)**3/3 - 4*c**3*sqrt(b**2 - c**2)*cosh(x)
Time = 0.20 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.47 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4 \, dx=b^{3} c \cosh \left (x\right )^{4} + b c^{3} \sinh \left (x\right )^{4} + \frac {1}{64} \, b^{4} {\left (24 \, x + e^{\left (4 \, x\right )} + 8 \, e^{\left (2 \, x\right )} - 8 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )}\right )} + \frac {1}{64} \, c^{4} {\left (24 \, x + e^{\left (4 \, x\right )} - 8 \, e^{\left (2 \, x\right )} + 8 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )}\right )} - \frac {3}{32} \, b^{2} c^{2} {\left (8 \, x - e^{\left (4 \, x\right )} + e^{\left (-4 \, x\right )}\right )} + {\left (b^{2} - c^{2}\right )}^{2} x + 4 \, {\left (b^{2} - c^{2}\right )}^{\frac {3}{2}} {\left (c \cosh \left (x\right ) + b \sinh \left (x\right )\right )} + \frac {3}{4} \, {\left (8 \, b c \cosh \left (x\right )^{2} + b^{2} {\left (4 \, x + e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right )} - c^{2} {\left (4 \, x - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}\right )} {\left (b^{2} - c^{2}\right )} + \frac {1}{6} \, {\left (24 \, b^{2} c \cosh \left (x\right )^{3} + 24 \, b c^{2} \sinh \left (x\right )^{3} + c^{3} {\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} + e^{\left (-3 \, x\right )} - 9 \, e^{x}\right )} + b^{3} {\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} - e^{\left (-3 \, x\right )} + 9 \, e^{x}\right )}\right )} \sqrt {b^{2} - c^{2}} \]
b^3*c*cosh(x)^4 + b*c^3*sinh(x)^4 + 1/64*b^4*(24*x + e^(4*x) + 8*e^(2*x) - 8*e^(-2*x) - e^(-4*x)) + 1/64*c^4*(24*x + e^(4*x) - 8*e^(2*x) + 8*e^(-2*x ) - e^(-4*x)) - 3/32*b^2*c^2*(8*x - e^(4*x) + e^(-4*x)) + (b^2 - c^2)^2*x + 4*(b^2 - c^2)^(3/2)*(c*cosh(x) + b*sinh(x)) + 3/4*(8*b*c*cosh(x)^2 + b^2 *(4*x + e^(2*x) - e^(-2*x)) - c^2*(4*x - e^(2*x) + e^(-2*x)))*(b^2 - c^2) + 1/6*(24*b^2*c*cosh(x)^3 + 24*b*c^2*sinh(x)^3 + c^3*(e^(3*x) - 9*e^(-x) + e^(-3*x) - 9*e^x) + b^3*(e^(3*x) - 9*e^(-x) - e^(-3*x) + 9*e^x))*sqrt(b^2 - c^2)
Leaf count of result is larger than twice the leaf count of optimal. 390 vs. \(2 (164) = 328\).
Time = 0.28 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.07 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4 \, dx=\frac {7}{2} \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \sqrt {b^{2} - c^{2}} e^{x} + \frac {35}{8} \, {\left (b^{4} - 2 \, b^{2} c^{2} + c^{4}\right )} x + \frac {1}{64} \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} e^{\left (4 \, x\right )} + \frac {1}{6} \, {\left (\sqrt {b^{2} - c^{2}} b^{3} + 3 \, \sqrt {b^{2} - c^{2}} b^{2} c + 3 \, \sqrt {b^{2} - c^{2}} b c^{2} + \sqrt {b^{2} - c^{2}} c^{3}\right )} e^{\left (3 \, x\right )} + \frac {7}{8} \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} e^{\left (2 \, x\right )} - \frac {1}{192} \, {\left (3 \, b^{4} - 12 \, b^{3} c + 18 \, b^{2} c^{2} - 12 \, b c^{3} + 3 \, c^{4} + 672 \, {\left (\sqrt {b^{2} - c^{2}} b^{3} - \sqrt {b^{2} - c^{2}} b^{2} c - \sqrt {b^{2} - c^{2}} b c^{2} + \sqrt {b^{2} - c^{2}} c^{3}\right )} e^{\left (3 \, x\right )} + 168 \, {\left (b^{4} - 2 \, b^{3} c + 2 \, b c^{3} - c^{4}\right )} e^{\left (2 \, x\right )} + 32 \, {\left (\sqrt {b^{2} - c^{2}} b^{3} - 3 \, \sqrt {b^{2} - c^{2}} b^{2} c + 3 \, \sqrt {b^{2} - c^{2}} b c^{2} - \sqrt {b^{2} - c^{2}} c^{3}\right )} e^{x}\right )} e^{\left (-4 \, x\right )} \]
7/2*(b^3 + b^2*c - b*c^2 - c^3)*sqrt(b^2 - c^2)*e^x + 35/8*(b^4 - 2*b^2*c^ 2 + c^4)*x + 1/64*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*e^(4*x) + 1/ 6*(sqrt(b^2 - c^2)*b^3 + 3*sqrt(b^2 - c^2)*b^2*c + 3*sqrt(b^2 - c^2)*b*c^2 + sqrt(b^2 - c^2)*c^3)*e^(3*x) + 7/8*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*e^(2 *x) - 1/192*(3*b^4 - 12*b^3*c + 18*b^2*c^2 - 12*b*c^3 + 3*c^4 + 672*(sqrt( b^2 - c^2)*b^3 - sqrt(b^2 - c^2)*b^2*c - sqrt(b^2 - c^2)*b*c^2 + sqrt(b^2 - c^2)*c^3)*e^(3*x) + 168*(b^4 - 2*b^3*c + 2*b*c^3 - c^4)*e^(2*x) + 32*(sq rt(b^2 - c^2)*b^3 - 3*sqrt(b^2 - c^2)*b^2*c + 3*sqrt(b^2 - c^2)*b*c^2 - sq rt(b^2 - c^2)*c^3)*e^x)*e^(-4*x)
Time = 0.45 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.92 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4 \, dx=x\,{\left (b^2-c^2\right )}^2-{\mathrm {cosh}\left (x\right )}^2\,\left (6\,b\,c^3-6\,b^3\,c\right )-{\mathrm {cosh}\left (x\right )}^4\,\left (b\,c^3-b^3\,c\right )+\mathrm {cosh}\left (x\right )\,{\mathrm {sinh}\left (x\right )}^3\,\left (-\frac {3\,b^4}{8}+\frac {3\,b^2\,c^2}{4}+\frac {5\,c^4}{8}\right )+{\mathrm {cosh}\left (x\right )}^3\,\mathrm {sinh}\left (x\right )\,\left (\frac {5\,b^4}{8}+\frac {3\,b^2\,c^2}{4}-\frac {3\,c^4}{8}\right )+4\,c\,\mathrm {cosh}\left (x\right )\,{\left (b^2-c^2\right )}^{3/2}+4\,b\,\mathrm {sinh}\left (x\right )\,{\left (b^2-c^2\right )}^{3/2}+3\,x\,{\mathrm {cosh}\left (x\right )}^2\,{\left (b^2-c^2\right )}^2+\frac {3\,x\,{\mathrm {cosh}\left (x\right )}^4\,{\left (b^2-c^2\right )}^2}{8}-3\,x\,{\mathrm {sinh}\left (x\right )}^2\,{\left (b^2-c^2\right )}^2+\frac {3\,x\,{\mathrm {sinh}\left (x\right )}^4\,{\left (b^2-c^2\right )}^2}{8}+\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\,\left (3\,b^4-3\,c^4\right )+2\,b\,c^3\,{\mathrm {cosh}\left (x\right )}^2\,{\mathrm {sinh}\left (x\right )}^2+\frac {4\,c\,{\mathrm {cosh}\left (x\right )}^3\,\sqrt {b^2-c^2}\,\left (3\,b^2-2\,c^2\right )}{3}-\frac {4\,b\,{\mathrm {sinh}\left (x\right )}^3\,\sqrt {b^2-c^2}\,\left (2\,b^2-3\,c^2\right )}{3}+4\,b^3\,{\mathrm {cosh}\left (x\right )}^2\,\mathrm {sinh}\left (x\right )\,\sqrt {b^2-c^2}+4\,c^3\,\mathrm {cosh}\left (x\right )\,{\mathrm {sinh}\left (x\right )}^2\,\sqrt {b^2-c^2}-\frac {3\,x\,{\mathrm {cosh}\left (x\right )}^2\,{\mathrm {sinh}\left (x\right )}^2\,{\left (b^2-c^2\right )}^2}{4} \]
x*(b^2 - c^2)^2 - cosh(x)^2*(6*b*c^3 - 6*b^3*c) - cosh(x)^4*(b*c^3 - b^3*c ) + cosh(x)*sinh(x)^3*((5*c^4)/8 - (3*b^4)/8 + (3*b^2*c^2)/4) + cosh(x)^3* sinh(x)*((5*b^4)/8 - (3*c^4)/8 + (3*b^2*c^2)/4) + 4*c*cosh(x)*(b^2 - c^2)^ (3/2) + 4*b*sinh(x)*(b^2 - c^2)^(3/2) + 3*x*cosh(x)^2*(b^2 - c^2)^2 + (3*x *cosh(x)^4*(b^2 - c^2)^2)/8 - 3*x*sinh(x)^2*(b^2 - c^2)^2 + (3*x*sinh(x)^4 *(b^2 - c^2)^2)/8 + cosh(x)*sinh(x)*(3*b^4 - 3*c^4) + 2*b*c^3*cosh(x)^2*si nh(x)^2 + (4*c*cosh(x)^3*(b^2 - c^2)^(1/2)*(3*b^2 - 2*c^2))/3 - (4*b*sinh( x)^3*(b^2 - c^2)^(1/2)*(2*b^2 - 3*c^2))/3 + 4*b^3*cosh(x)^2*sinh(x)*(b^2 - c^2)^(1/2) + 4*c^3*cosh(x)*sinh(x)^2*(b^2 - c^2)^(1/2) - (3*x*cosh(x)^2*s inh(x)^2*(b^2 - c^2)^2)/4