Integrand size = 8, antiderivative size = 46 \[ \int \frac {1}{(a \cosh (x))^{3/2}} \, dx=\frac {2 i \sqrt {a \cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )}{a^2 \sqrt {\cosh (x)}}+\frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}} \] Output:
2*I*(a*cosh(x))^(1/2)*EllipticE(I*sinh(1/2*x),2^(1/2))/a^2/cosh(x)^(1/2)+2 *sinh(x)/a/(a*cosh(x))^(1/2)
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(a \cosh (x))^{3/2}} \, dx=\frac {2 \cosh (x) \left (i \sqrt {\cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )+\sinh (x)\right )}{(a \cosh (x))^{3/2}} \] Input:
Integrate[(a*Cosh[x])^(-3/2),x]
Output:
(2*Cosh[x]*(I*Sqrt[Cosh[x]]*EllipticE[(I/2)*x, 2] + Sinh[x]))/(a*Cosh[x])^ (3/2)
Time = 0.30 (sec) , antiderivative size = 46, 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, 3116, 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.
\(\displaystyle \int \frac {1}{(a \cosh (x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a \sin \left (\frac {\pi }{2}+i x\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle \frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}-\frac {\int \sqrt {a \cosh (x)}dx}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}-\frac {\int \sqrt {a \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}-\frac {\sqrt {a \cosh (x)} \int \sqrt {\cosh (x)}dx}{a^2 \sqrt {\cosh (x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}-\frac {\sqrt {a \cosh (x)} \int \sqrt {\sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2 \sqrt {\cosh (x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}+\frac {2 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{a^2 \sqrt {\cosh (x)}}\) |
Input:
Int[(a*Cosh[x])^(-3/2),x]
Output:
((2*I)*Sqrt[a*Cosh[x]]*EllipticE[(I/2)*x, 2])/(a^2*Sqrt[Cosh[x]]) + (2*Sin h[x])/(a*Sqrt[a*Cosh[x]])
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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. \(158\) vs. \(2(41)=82\).
Time = 0.99 (sec) , antiderivative size = 159, normalized size of antiderivative = 3.46
method | result | size |
default | \(\frac {\sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} a +\sinh \left (\frac {x}{2}\right )^{2} a}\, \left (-\sqrt {2}\, \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2}-1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )+2 \sqrt {2}\, \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2}-1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticE}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )+4 \sinh \left (\frac {x}{2}\right )^{2} \cosh \left (\frac {x}{2}\right )\right )}{a \sqrt {a \left (2 \sinh \left (\frac {x}{2}\right )^{4}+\sinh \left (\frac {x}{2}\right )^{2}\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right ) a}}\) | \(159\) |
Input:
int(1/(cosh(x)*a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/a*(2*sinh(1/2*x)^4*a+sinh(1/2*x)^2*a)^(1/2)*(-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)) +2*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. 90 vs. \(2 (39) = 78\).
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.96 \[ \int \frac {1}{(a \cosh (x))^{3/2}} \, dx=\frac {4 \, {\left (\sqrt {\frac {1}{2}} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) + \sqrt {a \cosh \left (x\right )} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}\right )}}{a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} + a^{2}} \] Input:
integrate(1/(a*cosh(x))^(3/2),x, algorithm="fricas")
Output:
4*(sqrt(1/2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(a)*weier strassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(x) + sinh(x))) + sqrt(a* cosh(x))*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2))/(a^2*cosh(x)^2 + 2*a ^2*cosh(x)*sinh(x) + a^2*sinh(x)^2 + a^2)
\[ \int \frac {1}{(a \cosh (x))^{3/2}} \, dx=\int \frac {1}{\left (a \cosh {\left (x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(a*cosh(x))**(3/2),x)
Output:
Integral((a*cosh(x))**(-3/2), x)
\[ \int \frac {1}{(a \cosh (x))^{3/2}} \, dx=\int { \frac {1}{\left (a \cosh \left (x\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a*cosh(x))^(3/2),x, algorithm="maxima")
Output:
integrate((a*cosh(x))^(-3/2), x)
\[ \int \frac {1}{(a \cosh (x))^{3/2}} \, dx=\int { \frac {1}{\left (a \cosh \left (x\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a*cosh(x))^(3/2),x, algorithm="giac")
Output:
integrate((a*cosh(x))^(-3/2), x)
Timed out. \[ \int \frac {1}{(a \cosh (x))^{3/2}} \, dx=\int \frac {1}{{\left (a\,\mathrm {cosh}\left (x\right )\right )}^{3/2}} \,d x \] Input:
int(1/(a*cosh(x))^(3/2),x)
Output:
int(1/(a*cosh(x))^(3/2), x)
\[ \int \frac {1}{(a \cosh (x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\cosh \left (x \right )}}{\cosh \left (x \right )^{2}}d x \right )}{a^{2}} \] Input:
int(1/(a*cosh(x))^(3/2),x)
Output:
(sqrt(a)*int(sqrt(cosh(x))/cosh(x)**2,x))/a**2