Integrand size = 24, antiderivative size = 90 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2 \, dx=\frac {3}{2} \left (b^2-c^2\right ) x+\frac {3}{2} c \sqrt {b^2-c^2} \cosh (x)+\frac {3}{2} b \sqrt {b^2-c^2} \sinh (x)+\frac {1}{2} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \]
3/2*(b^2-c^2)*x+3/2*c*cosh(x)*(b^2-c^2)^(1/2)+3/2*b*sinh(x)*(b^2-c^2)^(1/2 )+1/2*(c*cosh(x)+b*sinh(x))*(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))
Time = 0.10 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.80 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2 \, dx=\frac {1}{4} \left (6 (b-c) (b+c) x+8 c \sqrt {b^2-c^2} \cosh (x)+2 b c \cosh (2 x)+8 b \sqrt {b^2-c^2} \sinh (x)+\left (b^2+c^2\right ) \sinh (2 x)\right ) \]
(6*(b - c)*(b + c)*x + 8*c*Sqrt[b^2 - c^2]*Cosh[x] + 2*b*c*Cosh[2*x] + 8*b *Sqrt[b^2 - c^2]*Sinh[x] + (b^2 + c^2)*Sinh[2*x])/4
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {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.
\(\displaystyle \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (\sqrt {b^2-c^2}+b \cos (i x)-i c \sin (i x)\right )^2dx\) |
\(\Big \downarrow \) 3592 |
\(\displaystyle \frac {3}{2} \sqrt {b^2-c^2} \int \left (b \cosh (x)+c \sinh (x)+\sqrt {b^2-c^2}\right )dx+\frac {1}{2} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{2} \sqrt {b^2-c^2} \left (x \sqrt {b^2-c^2}+b \sinh (x)+c \cosh (x)\right )+\frac {1}{2} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )\) |
(3*Sqrt[b^2 - c^2]*(Sqrt[b^2 - c^2]*x + c*Cosh[x] + b*Sinh[x]))/2 + ((c*Co sh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]))/2
3.8.55.3.1 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 = 0.44 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.89
method | result | size |
default | \(c^{2} \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}-\frac {x}{2}\right )+c b \cosh \left (x \right )^{2}+b^{2} \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}+\frac {x}{2}\right )+2 c \cosh \left (x \right ) \sqrt {b^{2}-c^{2}}+2 b \sinh \left (x \right ) \sqrt {b^{2}-c^{2}}+b^{2} x -c^{2} x\) | \(80\) |
parts | \(b^{2} x +2 b \left (\frac {c \sinh \left (x \right )^{2}}{2}+\sinh \left (x \right ) \sqrt {\left (b -c \right ) \left (b +c \right )}\right )+b^{2} \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}+\frac {x}{2}\right )+c^{2} \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}-\frac {x}{2}\right )-c^{2} x +2 c \cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\) | \(82\) |
risch | \(\frac {3 b^{2} x}{2}-\frac {3 c^{2} x}{2}+\frac {b^{2} {\mathrm e}^{2 x}}{8}+\frac {{\mathrm e}^{2 x} c b}{4}+\frac {{\mathrm e}^{2 x} c^{2}}{8}+\sqrt {b^{2}-c^{2}}\, {\mathrm e}^{x} b +\sqrt {b^{2}-c^{2}}\, {\mathrm e}^{x} c -\sqrt {b^{2}-c^{2}}\, {\mathrm e}^{-x} b +\sqrt {b^{2}-c^{2}}\, {\mathrm e}^{-x} c -\frac {{\mathrm e}^{-2 x} b^{2}}{8}+\frac {{\mathrm e}^{-2 x} c b}{4}-\frac {{\mathrm e}^{-2 x} c^{2}}{8}\) | \(131\) |
c^2*(1/2*cosh(x)*sinh(x)-1/2*x)+c*b*cosh(x)^2+b^2*(1/2*cosh(x)*sinh(x)+1/2 *x)+2*c*cosh(x)*(b^2-c^2)^(1/2)+2*b*sinh(x)*(b^2-c^2)^(1/2)+b^2*x-c^2*x
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (76) = 152\).
Time = 0.26 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.64 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2 \, dx=\frac {{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{4} + 12 \, {\left (b^{2} - c^{2}\right )} x \cosh \left (x\right )^{2} + 6 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (b^{2} - c^{2}\right )} x\right )} \sinh \left (x\right )^{2} - b^{2} + 2 \, b c - c^{2} + 4 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{3} + 6 \, {\left (b^{2} - c^{2}\right )} x \cosh \left (x\right )\right )} \sinh \left (x\right ) + 8 \, {\left ({\left (b + c\right )} \cosh \left (x\right )^{3} + 3 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (b + c\right )} \sinh \left (x\right )^{3} - {\left (b - c\right )} \cosh \left (x\right ) + {\left (3 \, {\left (b + c\right )} \cosh \left (x\right )^{2} - b + c\right )} \sinh \left (x\right )\right )} \sqrt {b^{2} - c^{2}}}{8 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \]
1/8*((b^2 + 2*b*c + c^2)*cosh(x)^4 + 4*(b^2 + 2*b*c + c^2)*cosh(x)*sinh(x) ^3 + (b^2 + 2*b*c + c^2)*sinh(x)^4 + 12*(b^2 - c^2)*x*cosh(x)^2 + 6*((b^2 + 2*b*c + c^2)*cosh(x)^2 + 2*(b^2 - c^2)*x)*sinh(x)^2 - b^2 + 2*b*c - c^2 + 4*((b^2 + 2*b*c + c^2)*cosh(x)^3 + 6*(b^2 - c^2)*x*cosh(x))*sinh(x) + 8* ((b + c)*cosh(x)^3 + 3*(b + c)*cosh(x)*sinh(x)^2 + (b + c)*sinh(x)^3 - (b - c)*cosh(x) + (3*(b + c)*cosh(x)^2 - b + c)*sinh(x))*sqrt(b^2 - c^2))/(co sh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)
Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.36 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2 \, dx=- \frac {b^{2} x \sinh ^{2}{\left (x \right )}}{2} + \frac {b^{2} x \cosh ^{2}{\left (x \right )}}{2} + b^{2} x + \frac {b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )}}{2} + b c \cosh ^{2}{\left (x \right )} + 2 b \sqrt {b^{2} - c^{2}} \sinh {\left (x \right )} + \frac {c^{2} x \sinh ^{2}{\left (x \right )}}{2} - \frac {c^{2} x \cosh ^{2}{\left (x \right )}}{2} - c^{2} x + \frac {c^{2} \sinh {\left (x \right )} \cosh {\left (x \right )}}{2} + 2 c \sqrt {b^{2} - c^{2}} \cosh {\left (x \right )} \]
-b**2*x*sinh(x)**2/2 + b**2*x*cosh(x)**2/2 + b**2*x + b**2*sinh(x)*cosh(x) /2 + b*c*cosh(x)**2 + 2*b*sqrt(b**2 - c**2)*sinh(x) + c**2*x*sinh(x)**2/2 - c**2*x*cosh(x)**2/2 - c**2*x + c**2*sinh(x)*cosh(x)/2 + 2*c*sqrt(b**2 - c**2)*cosh(x)
Time = 0.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.88 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2 \, dx=b c \cosh \left (x\right )^{2} + \frac {1}{8} \, b^{2} {\left (4 \, x + e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right )} - \frac {1}{8} \, c^{2} {\left (4 \, x - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + b^{2} x - c^{2} x + 2 \, \sqrt {b^{2} - c^{2}} {\left (c \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \]
b*c*cosh(x)^2 + 1/8*b^2*(4*x + e^(2*x) - e^(-2*x)) - 1/8*c^2*(4*x - e^(2*x ) + e^(-2*x)) + b^2*x - c^2*x + 2*sqrt(b^2 - c^2)*(c*cosh(x) + b*sinh(x))
Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.07 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2 \, dx=\sqrt {b^{2} - c^{2}} {\left (b + c\right )} e^{x} + \frac {3}{2} \, {\left (b^{2} - c^{2}\right )} x + \frac {1}{8} \, {\left (b^{2} + 2 \, b c + c^{2}\right )} e^{\left (2 \, x\right )} - \frac {1}{8} \, {\left (b^{2} - 2 \, b c + c^{2} + 8 \, {\left (\sqrt {b^{2} - c^{2}} b - \sqrt {b^{2} - c^{2}} c\right )} e^{x}\right )} e^{\left (-2 \, x\right )} \]
sqrt(b^2 - c^2)*(b + c)*e^x + 3/2*(b^2 - c^2)*x + 1/8*(b^2 + 2*b*c + c^2)* e^(2*x) - 1/8*(b^2 - 2*b*c + c^2 + 8*(sqrt(b^2 - c^2)*b - sqrt(b^2 - c^2)* c)*e^x)*e^(-2*x)
Time = 2.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.78 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2 \, dx=\frac {3\,b^2\,x}{2}-\frac {3\,c^2\,x}{2}+2\,c\,\mathrm {cosh}\left (x\right )\,\sqrt {b^2-c^2}+2\,b\,\mathrm {sinh}\left (x\right )\,\sqrt {b^2-c^2}+b\,c\,{\mathrm {cosh}\left (x\right )}^2+\frac {b^2\,\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )}{2}+\frac {c^2\,\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )}{2} \]