Integrand size = 10, antiderivative size = 66 \[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=-\frac {2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{3 b}+\frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}} \] Output:
-2/3*I*cosh(b*x+a)^(1/2)*InverseJacobiAM(1/2*I*(b*x+a),2^(1/2))*sech(b*x+a )^(1/2)/b+2/3*sinh(b*x+a)/b/sech(b*x+a)^(1/2)
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\frac {\sqrt {\text {sech}(a+b x)} \left (-2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )+\sinh (2 (a+b x))\right )}{3 b} \] Input:
Integrate[Sech[a + b*x]^(-3/2),x]
Output:
(Sqrt[Sech[a + b*x]]*((-2*I)*Sqrt[Cosh[a + b*x]]*EllipticF[(I/2)*(a + b*x) , 2] + Sinh[2*(a + b*x)]))/(3*b)
Time = 0.33 (sec) , antiderivative size = 66, 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, 4256, 3042, 4258, 3042, 3120}
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.
\(\displaystyle \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\csc \left (i a+i b x+\frac {\pi }{2}\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {1}{3} \int \sqrt {\text {sech}(a+b x)}dx+\frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}+\frac {1}{3} \int \sqrt {\csc \left (i a+i b x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {1}{3} \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \int \frac {1}{\sqrt {\cosh (a+b x)}}dx+\frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}+\frac {1}{3} \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \int \frac {1}{\sqrt {\sin \left (i a+i b x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}-\frac {2 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{3 b}\) |
Input:
Int[Sech[a + b*x]^(-3/2),x]
Output:
(((-2*I)/3)*Sqrt[Cosh[a + b*x]]*EllipticF[(I/2)*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b + (2*Sinh[a + b*x])/(3*b*Sqrt[Sech[a + b*x]])
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(54)=108\).
Time = 1.35 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.64
method | result | size |
default | \(\frac {2 \sqrt {\left (2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \left (4 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}-6 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}+\sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 \sqrt {2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, b}\) | \(174\) |
Input:
int(1/sech(b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/3*((2*cosh(1/2*b*x+1/2*a)^2-1)*sinh(1/2*b*x+1/2*a)^2)^(1/2)*(4*cosh(1/2* b*x+1/2*a)^5-6*cosh(1/2*b*x+1/2*a)^3+(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*co sh(1/2*b*x+1/2*a)^2+1)^(1/2)*EllipticF(cosh(1/2*b*x+1/2*a),2^(1/2))+2*cosh (1/2*b*x+1/2*a))/(2*sinh(1/2*b*x+1/2*a)^4+sinh(1/2*b*x+1/2*a)^2)^(1/2)/sin h(1/2*b*x+1/2*a)/(2*cosh(1/2*b*x+1/2*a)^2-1)^(1/2)/b
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (53) = 106\).
Time = 0.10 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.38 \[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\frac {\sqrt {2} {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} - 1\right )} \sqrt {\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}} + 4 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sqrt {2} \sinh \left (b x + a\right )^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{6 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \] Input:
integrate(1/sech(b*x+a)^(3/2),x, algorithm="fricas")
Output:
1/6*(sqrt(2)*(cosh(b*x + a)^4 + 4*cosh(b*x + a)^3*sinh(b*x + a) + 6*cosh(b *x + a)^2*sinh(b*x + a)^2 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a )^4 - 1)*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)) + 4*(sqrt(2)*cosh(b*x + a)^2 + 2*sqrt(2)*cosh(b*x + a)*sinh(b*x + a) + sqrt(2)*sinh(b*x + a)^2)*weierst rassPInverse(-4, 0, cosh(b*x + a) + sinh(b*x + a)))/(b*cosh(b*x + a)^2 + 2 *b*cosh(b*x + a)*sinh(b*x + a) + b*sinh(b*x + a)^2)
\[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {1}{\operatorname {sech}^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \] Input:
integrate(1/sech(b*x+a)**(3/2),x)
Output:
Integral(sech(a + b*x)**(-3/2), x)
\[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {1}{\operatorname {sech}\left (b x + a\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/sech(b*x+a)^(3/2),x, algorithm="maxima")
Output:
integrate(sech(b*x + a)^(-3/2), x)
\[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {1}{\operatorname {sech}\left (b x + a\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/sech(b*x+a)^(3/2),x, algorithm="giac")
Output:
integrate(sech(b*x + a)^(-3/2), x)
Timed out. \[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {1}{{\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{3/2}} \,d x \] Input:
int(1/(1/cosh(a + b*x))^(3/2),x)
Output:
int(1/(1/cosh(a + b*x))^(3/2), x)
\[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (b x +a \right )}}{\mathrm {sech}\left (b x +a \right )^{2}}d x \] Input:
int(1/sech(b*x+a)^(3/2),x)
Output:
int(sqrt(sech(a + b*x))/sech(a + b*x)**2,x)