Integrand size = 14, antiderivative size = 89 \[ \int (a+a \cosh (c+d x))^{5/2} \, dx=\frac {64 a^3 \sinh (c+d x)}{15 d \sqrt {a+a \cosh (c+d x)}}+\frac {16 a^2 \sqrt {a+a \cosh (c+d x)} \sinh (c+d x)}{15 d}+\frac {2 a (a+a \cosh (c+d x))^{3/2} \sinh (c+d x)}{5 d} \]
2/5*a*(a+a*cosh(d*x+c))^(3/2)*sinh(d*x+c)/d+64/15*a^3*sinh(d*x+c)/d/(a+a*c osh(d*x+c))^(1/2)+16/15*a^2*sinh(d*x+c)*(a+a*cosh(d*x+c))^(1/2)/d
Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.80 \[ \int (a+a \cosh (c+d x))^{5/2} \, dx=\frac {a^2 \sqrt {a (1+\cosh (c+d x))} \text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (150 \sinh \left (\frac {1}{2} (c+d x)\right )+25 \sinh \left (\frac {3}{2} (c+d x)\right )+3 \sinh \left (\frac {5}{2} (c+d x)\right )\right )}{30 d} \]
(a^2*Sqrt[a*(1 + Cosh[c + d*x])]*Sech[(c + d*x)/2]*(150*Sinh[(c + d*x)/2] + 25*Sinh[(3*(c + d*x))/2] + 3*Sinh[(5*(c + d*x))/2]))/(30*d)
Time = 0.36 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3126, 3042, 3126, 3042, 3125}
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.
\(\displaystyle \int (a \cosh (c+d x)+a)^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a+a \sin \left (i c+i d x+\frac {\pi }{2}\right )\right )^{5/2}dx\) |
\(\Big \downarrow \) 3126 |
\(\displaystyle \frac {8}{5} a \int (\cosh (c+d x) a+a)^{3/2}dx+\frac {2 a \sinh (c+d x) (a \cosh (c+d x)+a)^{3/2}}{5 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \sinh (c+d x) (a \cosh (c+d x)+a)^{3/2}}{5 d}+\frac {8}{5} a \int \left (\sin \left (i c+i d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}dx\) |
\(\Big \downarrow \) 3126 |
\(\displaystyle \frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {\cosh (c+d x) a+a}dx+\frac {2 a \sinh (c+d x) \sqrt {a \cosh (c+d x)+a}}{3 d}\right )+\frac {2 a \sinh (c+d x) (a \cosh (c+d x)+a)^{3/2}}{5 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \sinh (c+d x) (a \cosh (c+d x)+a)^{3/2}}{5 d}+\frac {8}{5} a \left (\frac {2 a \sinh (c+d x) \sqrt {a \cosh (c+d x)+a}}{3 d}+\frac {4}{3} a \int \sqrt {\sin \left (i c+i d x+\frac {\pi }{2}\right ) a+a}dx\right )\) |
\(\Big \downarrow \) 3125 |
\(\displaystyle \frac {8}{5} a \left (\frac {8 a^2 \sinh (c+d x)}{3 d \sqrt {a \cosh (c+d x)+a}}+\frac {2 a \sinh (c+d x) \sqrt {a \cosh (c+d x)+a}}{3 d}\right )+\frac {2 a \sinh (c+d x) (a \cosh (c+d x)+a)^{3/2}}{5 d}\) |
(2*a*(a + a*Cosh[c + d*x])^(3/2)*Sinh[c + d*x])/(5*d) + (8*a*((8*a^2*Sinh[ c + d*x])/(3*d*Sqrt[a + a*Cosh[c + d*x]]) + (2*a*Sqrt[a + a*Cosh[c + d*x]] *Sinh[c + d*x])/(3*d)))/5
3.1.42.3.1 Defintions of rubi rules used
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos [c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[a*((2*n - 1)/n) Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ a^2 - b^2, 0] && IGtQ[n - 1/2, 0]
Time = 0.14 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {8 a^{3} \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (3 \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+4 \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+8\right ) \sqrt {2}}{15 \sqrt {a \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, d}\) | \(73\) |
8/15*a^3*cosh(1/2*d*x+1/2*c)*sinh(1/2*d*x+1/2*c)*(3*cosh(1/2*d*x+1/2*c)^4+ 4*cosh(1/2*d*x+1/2*c)^2+8)*2^(1/2)/(a*cosh(1/2*d*x+1/2*c)^2)^(1/2)/d
Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (77) = 154\).
Time = 0.26 (sec) , antiderivative size = 327, normalized size of antiderivative = 3.67 \[ \int (a+a \cosh (c+d x))^{5/2} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (3 \, a^{2} \cosh \left (d x + c\right )^{5} + 3 \, a^{2} \sinh \left (d x + c\right )^{5} + 25 \, a^{2} \cosh \left (d x + c\right )^{4} + 150 \, a^{2} \cosh \left (d x + c\right )^{3} + 5 \, {\left (3 \, a^{2} \cosh \left (d x + c\right ) + 5 \, a^{2}\right )} \sinh \left (d x + c\right )^{4} - 150 \, a^{2} \cosh \left (d x + c\right )^{2} + 10 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} + 10 \, a^{2} \cosh \left (d x + c\right ) + 15 \, a^{2}\right )} \sinh \left (d x + c\right )^{3} - 25 \, a^{2} \cosh \left (d x + c\right ) + 30 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} + 5 \, a^{2} \cosh \left (d x + c\right )^{2} + 15 \, a^{2} \cosh \left (d x + c\right ) - 5 \, a^{2}\right )} \sinh \left (d x + c\right )^{2} - 3 \, a^{2} + 5 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{4} + 20 \, a^{2} \cosh \left (d x + c\right )^{3} + 90 \, a^{2} \cosh \left (d x + c\right )^{2} - 60 \, a^{2} \cosh \left (d x + c\right ) - 5 \, a^{2}\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{30 \, {\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2}\right )}} \]
1/30*sqrt(1/2)*(3*a^2*cosh(d*x + c)^5 + 3*a^2*sinh(d*x + c)^5 + 25*a^2*cos h(d*x + c)^4 + 150*a^2*cosh(d*x + c)^3 + 5*(3*a^2*cosh(d*x + c) + 5*a^2)*s inh(d*x + c)^4 - 150*a^2*cosh(d*x + c)^2 + 10*(3*a^2*cosh(d*x + c)^2 + 10* a^2*cosh(d*x + c) + 15*a^2)*sinh(d*x + c)^3 - 25*a^2*cosh(d*x + c) + 30*(a ^2*cosh(d*x + c)^3 + 5*a^2*cosh(d*x + c)^2 + 15*a^2*cosh(d*x + c) - 5*a^2) *sinh(d*x + c)^2 - 3*a^2 + 5*(3*a^2*cosh(d*x + c)^4 + 20*a^2*cosh(d*x + c) ^3 + 90*a^2*cosh(d*x + c)^2 - 60*a^2*cosh(d*x + c) - 5*a^2)*sinh(d*x + c)) *sqrt(a/(cosh(d*x + c) + sinh(d*x + c)))/(d*cosh(d*x + c)^2 + 2*d*cosh(d*x + c)*sinh(d*x + c) + d*sinh(d*x + c)^2)
\[ \int (a+a \cosh (c+d x))^{5/2} \, dx=\int \left (a \cosh {\left (c + d x \right )} + a\right )^{\frac {5}{2}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.36 \[ \int (a+a \cosh (c+d x))^{5/2} \, dx=\frac {\sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}}{20 \, d} + \frac {5 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}}{12 \, d} + \frac {5 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} - \frac {5 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{2 \, d} - \frac {5 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {3}{2} \, d x - \frac {3}{2} \, c\right )}}{12 \, d} - \frac {\sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {5}{2} \, d x - \frac {5}{2} \, c\right )}}{20 \, d} \]
1/20*sqrt(2)*a^(5/2)*e^(5/2*d*x + 5/2*c)/d + 5/12*sqrt(2)*a^(5/2)*e^(3/2*d *x + 3/2*c)/d + 5/2*sqrt(2)*a^(5/2)*e^(1/2*d*x + 1/2*c)/d - 5/2*sqrt(2)*a^ (5/2)*e^(-1/2*d*x - 1/2*c)/d - 5/12*sqrt(2)*a^(5/2)*e^(-3/2*d*x - 3/2*c)/d - 1/20*sqrt(2)*a^(5/2)*e^(-5/2*d*x - 5/2*c)/d
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.18 \[ \int (a+a \cosh (c+d x))^{5/2} \, dx=-\frac {\sqrt {2} {\left ({\left (150 \, a^{\frac {5}{2}} e^{\left (2 \, d x + \frac {5}{2} \, c\right )} + 25 \, a^{\frac {5}{2}} e^{\left (d x + \frac {3}{2} \, c\right )} + 3 \, a^{\frac {5}{2}} e^{\left (\frac {1}{2} \, c\right )}\right )} e^{\left (-\frac {5}{2} \, d x - 3 \, c\right )} - {\left (3 \, a^{\frac {5}{2}} e^{\left (\frac {5}{2} \, d x + \frac {35}{2} \, c\right )} + 25 \, a^{\frac {5}{2}} e^{\left (\frac {3}{2} \, d x + \frac {33}{2} \, c\right )} + 150 \, a^{\frac {5}{2}} e^{\left (\frac {1}{2} \, d x + \frac {31}{2} \, c\right )}\right )} e^{\left (-15 \, c\right )}\right )}}{60 \, d} \]
-1/60*sqrt(2)*((150*a^(5/2)*e^(2*d*x + 5/2*c) + 25*a^(5/2)*e^(d*x + 3/2*c) + 3*a^(5/2)*e^(1/2*c))*e^(-5/2*d*x - 3*c) - (3*a^(5/2)*e^(5/2*d*x + 35/2* c) + 25*a^(5/2)*e^(3/2*d*x + 33/2*c) + 150*a^(5/2)*e^(1/2*d*x + 31/2*c))*e ^(-15*c))/d
Timed out. \[ \int (a+a \cosh (c+d x))^{5/2} \, dx=\int {\left (a+a\,\mathrm {cosh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]