Integrand size = 10, antiderivative size = 46 \[ \int \frac {1}{\sqrt {a \cosh ^3(x)}} \, dx=\frac {2 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )}{\sqrt {a \cosh ^3(x)}}+\frac {2 \cosh (x) \sinh (x)}{\sqrt {a \cosh ^3(x)}} \] Output:
2*I*cosh(x)^(3/2)*EllipticE(I*sinh(1/2*x),2^(1/2))/(a*cosh(x)^3)^(1/2)+2*c osh(x)*sinh(x)/(a*cosh(x)^3)^(1/2)
Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {a \cosh ^3(x)}} \, dx=\frac {2 \cosh (x) \left (i \sqrt {\cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )+\sinh (x)\right )}{\sqrt {a \cosh ^3(x)}} \] Input:
Integrate[1/Sqrt[a*Cosh[x]^3],x]
Output:
(2*Cosh[x]*(I*Sqrt[Cosh[x]]*EllipticE[(I/2)*x, 2] + Sinh[x]))/Sqrt[a*Cosh[ x]^3]
Time = 0.34 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3686, 3042, 3116, 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.
\(\displaystyle \int \frac {1}{\sqrt {a \cosh ^3(x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {a \sin \left (\frac {\pi }{2}+i x\right )^3}}dx\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \int \frac {1}{\cosh ^{\frac {3}{2}}(x)}dx}{\sqrt {a \cosh ^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \int \frac {1}{\sin \left (i x+\frac {\pi }{2}\right )^{3/2}}dx}{\sqrt {a \cosh ^3(x)}}\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{\sqrt {\cosh (x)}}-\int \sqrt {\cosh (x)}dx\right )}{\sqrt {a \cosh ^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{\sqrt {\cosh (x)}}-\int \sqrt {\sin \left (i x+\frac {\pi }{2}\right )}dx\right )}{\sqrt {a \cosh ^3(x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{\sqrt {\cosh (x)}}+2 i E\left (\left .\frac {i x}{2}\right |2\right )\right )}{\sqrt {a \cosh ^3(x)}}\) |
Input:
Int[1/Sqrt[a*Cosh[x]^3],x]
Output:
(Cosh[x]^(3/2)*((2*I)*EllipticE[(I/2)*x, 2] + (2*Sinh[x])/Sqrt[Cosh[x]]))/ Sqrt[a*Cosh[x]^3]
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[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Si n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
\[\int \frac {1}{\sqrt {a \cosh \left (x \right )^{3}}}d x\]
Input:
int(1/(a*cosh(x)^3)^(1/2),x)
Output:
int(1/(a*cosh(x)^3)^(1/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (39) = 78\).
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.78 \[ \int \frac {1}{\sqrt {a \cosh ^3(x)}} \, dx=\frac {4 \, {\left (\sqrt {\frac {1}{2}} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) + \sqrt {a \cosh \left (x\right )} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}\right )}}{a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} + a} \] Input:
integrate(1/(a*cosh(x)^3)^(1/2),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*cosh(x)^2 + 2*a*c osh(x)*sinh(x) + a*sinh(x)^2 + a)
\[ \int \frac {1}{\sqrt {a \cosh ^3(x)}} \, dx=\int \frac {1}{\sqrt {a \cosh ^{3}{\left (x \right )}}}\, dx \] Input:
integrate(1/(a*cosh(x)**3)**(1/2),x)
Output:
Integral(1/sqrt(a*cosh(x)**3), x)
\[ \int \frac {1}{\sqrt {a \cosh ^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \cosh \left (x\right )^{3}}} \,d x } \] Input:
integrate(1/(a*cosh(x)^3)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(a*cosh(x)^3), x)
\[ \int \frac {1}{\sqrt {a \cosh ^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \cosh \left (x\right )^{3}}} \,d x } \] Input:
integrate(1/(a*cosh(x)^3)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(a*cosh(x)^3), x)
Timed out. \[ \int \frac {1}{\sqrt {a \cosh ^3(x)}} \, dx=\int \frac {1}{\sqrt {a\,{\mathrm {cosh}\left (x\right )}^3}} \,d x \] Input:
int(1/(a*cosh(x)^3)^(1/2),x)
Output:
int(1/(a*cosh(x)^3)^(1/2), x)
\[ \int \frac {1}{\sqrt {a \cosh ^3(x)}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\cosh \left (x \right )}}{\cosh \left (x \right )^{2}}d x \right )}{a} \] Input:
int(1/(a*cosh(x)^3)^(1/2),x)
Output:
(sqrt(a)*int(sqrt(cosh(x))/cosh(x)**2,x))/a