3.14 \(\int \frac {1}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx\)

Optimal. Leaf size=66 \[ \frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)}-\frac {6 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b} \]

[Out]

2/5*sinh(b*x+a)/b/sech(b*x+a)^(3/2)-6/5*I*(cosh(1/2*a+1/2*b*x)^2)^(1/2)/cosh(1/2*a+1/2*b*x)*EllipticE(I*sinh(1
/2*a+1/2*b*x),2^(1/2))*cosh(b*x+a)^(1/2)*sech(b*x+a)^(1/2)/b

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Rubi [A]  time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3769, 3771, 2639} \[ \frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)}-\frac {6 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*x]^(-5/2),x]

[Out]

(((-6*I)/5)*Sqrt[Cosh[a + b*x]]*EllipticE[(I/2)*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b + (2*Sinh[a + b*x])/(5*b*
Sech[a + b*x]^(3/2))

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 3769

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
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(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]

Rubi steps

\begin {align*} \int \frac {1}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx &=\frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)}+\frac {3}{5} \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx\\ &=\frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)}+\frac {1}{5} \left (3 \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \sqrt {\cosh (a+b x)} \, dx\\ &=-\frac {6 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)}}{5 b}+\frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 59, normalized size = 0.89 \[ \frac {\sqrt {\text {sech}(a+b x)} \left (\sinh (a+b x)+\sinh (3 (a+b x))-12 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )\right )}{10 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + b*x]^(-5/2),x]

[Out]

(Sqrt[Sech[a + b*x]]*((-12*I)*Sqrt[Cosh[a + b*x]]*EllipticE[(I/2)*(a + b*x), 2] + Sinh[a + b*x] + Sinh[3*(a +
b*x)]))/(10*b)

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fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\operatorname {sech}\left (b x + a\right )^{\frac {5}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sech(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

integral(sech(b*x + a)^(-5/2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {sech}\left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sech(b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate(sech(b*x + a)^(-5/2), x)

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maple [B]  time = 0.52, size = 188, normalized size = 2.85 \[ \frac {2 \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (8 \left (\cosh ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-16 \left (\cosh ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+10 \left (\cosh ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-3 \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticE \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 \sqrt {2 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sech(b*x+a)^(5/2),x)

[Out]

2/5*((2*cosh(1/2*b*x+1/2*a)^2-1)*sinh(1/2*b*x+1/2*a)^2)^(1/2)*(8*cosh(1/2*b*x+1/2*a)^7-16*cosh(1/2*b*x+1/2*a)^
5+10*cosh(1/2*b*x+1/2*a)^3-3*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cosh(1/2*b*x+1/2*a)^2+1)^(1/2)*EllipticE(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)/sinh(1/2*
b*x+1/2*a)/(2*cosh(1/2*b*x+1/2*a)^2-1)^(1/2)/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {sech}\left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sech(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

integrate(sech(b*x + a)^(-5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/cosh(a + b*x))^(5/2),x)

[Out]

int(1/(1/cosh(a + b*x))^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {sech}^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sech(b*x+a)**(5/2),x)

[Out]

Integral(sech(a + b*x)**(-5/2), x)

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