Integrand size = 8, antiderivative size = 48 \[ \int (a \cosh (x))^{5/2} \, dx=-\frac {6 i a^2 \sqrt {a \cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )}{5 \sqrt {\cosh (x)}}+\frac {2}{5} a (a \cosh (x))^{3/2} \sinh (x) \] Output:
-6/5*I*a^2*(a*cosh(x))^(1/2)*EllipticE(I*sinh(1/2*x),2^(1/2))/cosh(x)^(1/2 )+2/5*a*(a*cosh(x))^(3/2)*sinh(x)
Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.85 \[ \int (a \cosh (x))^{5/2} \, dx=\frac {2 (a \cosh (x))^{5/2} \left (-3 i E\left (\left .\frac {i x}{2}\right |2\right )+\cosh ^{\frac {3}{2}}(x) \sinh (x)\right )}{5 \cosh ^{\frac {5}{2}}(x)} \] Input:
Integrate[(a*Cosh[x])^(5/2),x]
Output:
(2*(a*Cosh[x])^(5/2)*((-3*I)*EllipticE[(I/2)*x, 2] + Cosh[x]^(3/2)*Sinh[x] ))/(5*Cosh[x]^(5/2))
Time = 0.33 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3115, 3042, 3121, 3042, 3119}
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.
\(\displaystyle \int (a \cosh (x))^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sin \left (\frac {\pi }{2}+i x\right )\right )^{5/2}dx\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {3}{5} a^2 \int \sqrt {a \cosh (x)}dx+\frac {2}{5} a \sinh (x) (a \cosh (x))^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} a \sinh (x) (a \cosh (x))^{3/2}+\frac {3}{5} a^2 \int \sqrt {a \sin \left (i x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {3 a^2 \sqrt {a \cosh (x)} \int \sqrt {\cosh (x)}dx}{5 \sqrt {\cosh (x)}}+\frac {2}{5} a \sinh (x) (a \cosh (x))^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} a \sinh (x) (a \cosh (x))^{3/2}+\frac {3 a^2 \sqrt {a \cosh (x)} \int \sqrt {\sin \left (i x+\frac {\pi }{2}\right )}dx}{5 \sqrt {\cosh (x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2}{5} a \sinh (x) (a \cosh (x))^{3/2}-\frac {6 i a^2 E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{5 \sqrt {\cosh (x)}}\) |
Input:
Int[(a*Cosh[x])^(5/2),x]
Output:
(((-6*I)/5)*a^2*Sqrt[a*Cosh[x]]*EllipticE[(I/2)*x, 2])/Sqrt[Cosh[x]] + (2* a*(a*Cosh[x])^(3/2)*Sinh[x])/5
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Leaf count of result is larger than twice the leaf count of optimal. \(183\) vs. \(2(39)=78\).
Time = 3.99 (sec) , antiderivative size = 184, normalized size of antiderivative = 3.83
method | result | size |
default | \(\frac {\sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right ) a \sinh \left (\frac {x}{2}\right )^{2}}\, a^{3} \left (16 \cosh \left (\frac {x}{2}\right ) \sinh \left (\frac {x}{2}\right )^{6}+16 \sinh \left (\frac {x}{2}\right )^{4} \cosh \left (\frac {x}{2}\right )+3 \sqrt {2}\, \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2}-1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-6 \sqrt {2}\, \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2}-1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticE}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )+4 \sinh \left (\frac {x}{2}\right )^{2} \cosh \left (\frac {x}{2}\right )\right )}{5 \sqrt {a \left (2 \sinh \left (\frac {x}{2}\right )^{4}+\sinh \left (\frac {x}{2}\right )^{2}\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right ) a}}\) | \(184\) |
Input:
int((cosh(x)*a)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/5*((2*cosh(1/2*x)^2-1)*a*sinh(1/2*x)^2)^(1/2)*a^3*(16*cosh(1/2*x)*sinh(1 /2*x)^6+16*sinh(1/2*x)^4*cosh(1/2*x)+3*2^(1/2)*(-2*sinh(1/2*x)^2-1)^(1/2)* (-sinh(1/2*x)^2)^(1/2)*EllipticF(2^(1/2)*cosh(1/2*x),1/2*2^(1/2))-6*2^(1/2 )*(-2*sinh(1/2*x)^2-1)^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticE(2^(1/2)*cosh (1/2*x),1/2*2^(1/2))+4*sinh(1/2*x)^2*cosh(1/2*x))/(a*(2*sinh(1/2*x)^4+sinh (1/2*x)^2))^(1/2)/sinh(1/2*x)/((2*cosh(1/2*x)^2-1)*a)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (37) = 74\).
Time = 0.10 (sec) , antiderivative size = 155, normalized size of antiderivative = 3.23 \[ \int (a \cosh (x))^{5/2} \, dx=-\frac {24 \, \sqrt {\frac {1}{2}} {\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - {\left (a^{2} \cosh \left (x\right )^{4} + 4 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{2} \sinh \left (x\right )^{4} - 12 \, a^{2} \cosh \left (x\right )^{2} + 6 \, {\left (a^{2} \cosh \left (x\right )^{2} - 2 \, a^{2}\right )} \sinh \left (x\right )^{2} - a^{2} + 4 \, {\left (a^{2} \cosh \left (x\right )^{3} - 6 \, a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a \cosh \left (x\right )}}{10 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \] Input:
integrate((a*cosh(x))^(5/2),x, algorithm="fricas")
Output:
-1/10*(24*sqrt(1/2)*(a^2*cosh(x)^2 + 2*a^2*cosh(x)*sinh(x) + a^2*sinh(x)^2 )*sqrt(a)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(x) + sinh (x))) - (a^2*cosh(x)^4 + 4*a^2*cosh(x)*sinh(x)^3 + a^2*sinh(x)^4 - 12*a^2* cosh(x)^2 + 6*(a^2*cosh(x)^2 - 2*a^2)*sinh(x)^2 - a^2 + 4*(a^2*cosh(x)^3 - 6*a^2*cosh(x))*sinh(x))*sqrt(a*cosh(x)))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)
Timed out. \[ \int (a \cosh (x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a*cosh(x))**(5/2),x)
Output:
Timed out
\[ \int (a \cosh (x))^{5/2} \, dx=\int { \left (a \cosh \left (x\right )\right )^{\frac {5}{2}} \,d x } \] Input:
integrate((a*cosh(x))^(5/2),x, algorithm="maxima")
Output:
integrate((a*cosh(x))^(5/2), x)
\[ \int (a \cosh (x))^{5/2} \, dx=\int { \left (a \cosh \left (x\right )\right )^{\frac {5}{2}} \,d x } \] Input:
integrate((a*cosh(x))^(5/2),x, algorithm="giac")
Output:
integrate((a*cosh(x))^(5/2), x)
Timed out. \[ \int (a \cosh (x))^{5/2} \, dx=\int {\left (a\,\mathrm {cosh}\left (x\right )\right )}^{5/2} \,d x \] Input:
int((a*cosh(x))^(5/2),x)
Output:
int((a*cosh(x))^(5/2), x)
\[ \int (a \cosh (x))^{5/2} \, dx=\sqrt {a}\, \left (\int \sqrt {\cosh \left (x \right )}\, \cosh \left (x \right )^{2}d x \right ) a^{2} \] Input:
int((a*cosh(x))^(5/2),x)
Output:
sqrt(a)*int(sqrt(cosh(x))*cosh(x)**2,x)*a**2