Integrand size = 10, antiderivative size = 62 \[ \int \text {sech}^{\frac {3}{2}}(a+b x) \, dx=\frac {2 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{b}+\frac {2 \sqrt {\text {sech}(a+b x)} \sinh (a+b x)}{b} \] Output:
2*I*cosh(b*x+a)^(1/2)*EllipticE(I*sinh(1/2*a+1/2*b*x),2^(1/2))*sech(b*x+a) ^(1/2)/b+2*sech(b*x+a)^(1/2)*sinh(b*x+a)/b
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.79 \[ \int \text {sech}^{\frac {3}{2}}(a+b x) \, dx=\frac {2 \sqrt {\text {sech}(a+b x)} \left (i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )+\sinh (a+b x)\right )}{b} \] Input:
Integrate[Sech[a + b*x]^(3/2),x]
Output:
(2*Sqrt[Sech[a + b*x]]*(I*Sqrt[Cosh[a + b*x]]*EllipticE[(I/2)*(a + b*x), 2 ] + Sinh[a + b*x]))/b
Time = 0.33 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {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.
\(\displaystyle \int \text {sech}^{\frac {3}{2}}(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc \left (i a+i b x+\frac {\pi }{2}\right )^{3/2}dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {2 \sinh (a+b x) \sqrt {\text {sech}(a+b x)}}{b}-\int \frac {1}{\sqrt {\text {sech}(a+b x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \sinh (a+b x) \sqrt {\text {sech}(a+b x)}}{b}-\int \frac {1}{\sqrt {\csc \left (i a+i b x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {2 \sinh (a+b x) \sqrt {\text {sech}(a+b x)}}{b}-\sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \int \sqrt {\cosh (a+b x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \sinh (a+b x) \sqrt {\text {sech}(a+b x)}}{b}-\sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \int \sqrt {\sin \left (i a+i b x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 \sinh (a+b x) \sqrt {\text {sech}(a+b x)}}{b}+\frac {2 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b}\) |
Input:
Int[Sech[a + b*x]^(3/2),x]
Output:
((2*I)*Sqrt[Cosh[a + b*x]]*EllipticE[(I/2)*(a + b*x), 2]*Sqrt[Sech[a + b*x ]])/b + (2*Sqrt[Sech[a + b*x]]*Sinh[a + b*x])/b
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]
Time = 0.39 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.66
method | result | size |
default | \(\frac {4 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )+2 \sqrt {-2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \operatorname {EllipticE}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )}{\sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, b}\) | \(103\) |
Input:
int(sech(b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*(2*sinh(1/2*b*x+1/2*a)^2*cosh(1/2*b*x+1/2*a)+(-2*sinh(1/2*b*x+1/2*a)^2-1 )^(1/2)*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*EllipticE(cosh(1/2*b*x+1/2*a),2^(1/ 2)))/sinh(1/2*b*x+1/2*a)/(2*cosh(1/2*b*x+1/2*a)^2-1)^(1/2)/b
Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.55 \[ \int \text {sech}^{\frac {3}{2}}(a+b x) \, dx=\frac {2 \, {\left (\sqrt {2} \sqrt {\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right )\right )}}{b} \] Input:
integrate(sech(b*x+a)^(3/2),x, algorithm="fricas")
Output:
2*(sqrt(2)*sqrt((cosh(b*x + a) + sinh(b*x + a))/(cosh(b*x + a)^2 + 2*cosh( b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 + 1))*(cosh(b*x + a) + sinh(b*x + a)) + sqrt(2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(b*x + a) + sinh(b*x + a))))/b
\[ \int \text {sech}^{\frac {3}{2}}(a+b x) \, dx=\int \operatorname {sech}^{\frac {3}{2}}{\left (a + b x \right )}\, dx \] Input:
integrate(sech(b*x+a)**(3/2),x)
Output:
Integral(sech(a + b*x)**(3/2), x)
\[ \int \text {sech}^{\frac {3}{2}}(a+b x) \, dx=\int { \operatorname {sech}\left (b x + a\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(sech(b*x+a)^(3/2),x, algorithm="maxima")
Output:
integrate(sech(b*x + a)^(3/2), x)
\[ \int \text {sech}^{\frac {3}{2}}(a+b x) \, dx=\int { \operatorname {sech}\left (b x + a\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(sech(b*x+a)^(3/2),x, algorithm="giac")
Output:
integrate(sech(b*x + a)^(3/2), x)
Timed out. \[ \int \text {sech}^{\frac {3}{2}}(a+b x) \, dx=\int {\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{3/2} \,d x \] Input:
int((1/cosh(a + b*x))^(3/2),x)
Output:
int((1/cosh(a + b*x))^(3/2), x)
\[ \int \text {sech}^{\frac {3}{2}}(a+b x) \, dx=\int \sqrt {\mathrm {sech}\left (b x +a \right )}\, \mathrm {sech}\left (b x +a \right )d x \] Input:
int(sech(b*x+a)^(3/2),x)
Output:
int(sqrt(sech(a + b*x))*sech(a + b*x),x)