Integrand size = 10, antiderivative size = 46 \[ \int \sqrt {a \text {sech}^3(x)} \, dx=2 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)}+2 \cosh (x) \sqrt {a \text {sech}^3(x)} \sinh (x) \] Output:
2*I*cosh(x)^(3/2)*EllipticE(I*sinh(1/2*x),2^(1/2))*(a*sech(x)^3)^(1/2)+2*c osh(x)*(a*sech(x)^3)^(1/2)*sinh(x)
Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \sqrt {a \text {sech}^3(x)} \, dx=2 \cosh (x) \sqrt {a \text {sech}^3(x)} \left (i \sqrt {\cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )+\sinh (x)\right ) \] Input:
Integrate[Sqrt[a*Sech[x]^3],x]
Output:
2*Cosh[x]*Sqrt[a*Sech[x]^3]*(I*Sqrt[Cosh[x]]*EllipticE[(I/2)*x, 2] + Sinh[ x])
Time = 0.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4611, 3042, 4255, 3042, 4258, 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.
\(\displaystyle \int \sqrt {a \text {sech}^3(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {a \sec (i x)^3}dx\) |
\(\Big \downarrow \) 4611 |
\(\displaystyle \frac {\sqrt {a \text {sech}^3(x)} \int \text {sech}^{\frac {3}{2}}(x)dx}{\text {sech}^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a \text {sech}^3(x)} \int \csc \left (i x+\frac {\pi }{2}\right )^{3/2}dx}{\text {sech}^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {\sqrt {a \text {sech}^3(x)} \left (2 \sinh (x) \sqrt {\text {sech}(x)}-\int \frac {1}{\sqrt {\text {sech}(x)}}dx\right )}{\text {sech}^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a \text {sech}^3(x)} \left (2 \sinh (x) \sqrt {\text {sech}(x)}-\int \frac {1}{\sqrt {\csc \left (i x+\frac {\pi }{2}\right )}}dx\right )}{\text {sech}^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {\sqrt {a \text {sech}^3(x)} \left (2 \sinh (x) \sqrt {\text {sech}(x)}-\sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \int \sqrt {\cosh (x)}dx\right )}{\text {sech}^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a \text {sech}^3(x)} \left (2 \sinh (x) \sqrt {\text {sech}(x)}-\sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \int \sqrt {\sin \left (i x+\frac {\pi }{2}\right )}dx\right )}{\text {sech}^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\sqrt {a \text {sech}^3(x)} \left (2 \sinh (x) \sqrt {\text {sech}(x)}+2 i \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} E\left (\left .\frac {i x}{2}\right |2\right )\right )}{\text {sech}^{\frac {3}{2}}(x)}\) |
Input:
Int[Sqrt[a*Sech[x]^3],x]
Output:
(Sqrt[a*Sech[x]^3]*((2*I)*Sqrt[Cosh[x]]*EllipticE[(I/2)*x, 2]*Sqrt[Sech[x] ] + 2*Sqrt[Sech[x]]*Sinh[x]))/Sech[x]^(3/2)
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[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Simp[b^ IntPart[p]*((b*(c*Sec[e + f*x])^n)^FracPart[p]/(c*Sec[e + f*x])^(n*FracPart [p])) Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p}, x] && !IntegerQ[p]
\[\int \sqrt {a \operatorname {sech}\left (x \right )^{3}}d x\]
Input:
int((a*sech(x)^3)^(1/2),x)
Output:
int((a*sech(x)^3)^(1/2),x)
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30 \[ \int \sqrt {a \text {sech}^3(x)} \, dx=2 \, \sqrt {2} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 2 \, \sqrt {2} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) \] Input:
integrate((a*sech(x)^3)^(1/2),x, algorithm="fricas")
Output:
2*sqrt(2)*sqrt((a*cosh(x) + a*sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + si nh(x)^2 + 1))*(cosh(x) + sinh(x)) + 2*sqrt(2)*sqrt(a)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(x) + sinh(x)))
\[ \int \sqrt {a \text {sech}^3(x)} \, dx=\int \sqrt {a \operatorname {sech}^{3}{\left (x \right )}}\, dx \] Input:
integrate((a*sech(x)**3)**(1/2),x)
Output:
Integral(sqrt(a*sech(x)**3), x)
\[ \int \sqrt {a \text {sech}^3(x)} \, dx=\int { \sqrt {a \operatorname {sech}\left (x\right )^{3}} \,d x } \] Input:
integrate((a*sech(x)^3)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(a*sech(x)^3), x)
\[ \int \sqrt {a \text {sech}^3(x)} \, dx=\int { \sqrt {a \operatorname {sech}\left (x\right )^{3}} \,d x } \] Input:
integrate((a*sech(x)^3)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(a*sech(x)^3), x)
Timed out. \[ \int \sqrt {a \text {sech}^3(x)} \, dx=\int \sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^3}} \,d x \] Input:
int((a/cosh(x)^3)^(1/2),x)
Output:
int((a/cosh(x)^3)^(1/2), x)
\[ \int \sqrt {a \text {sech}^3(x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\mathrm {sech}\left (x \right )}\, \mathrm {sech}\left (x \right )d x \right ) \] Input:
int((a*sech(x)^3)^(1/2),x)
Output:
sqrt(a)*int(sqrt(sech(x))*sech(x),x)