3.10 \(\int \text{sech}^{\frac{3}{2}}(a+b x) \, dx\)

Optimal. Leaf size=62 \[ \frac{2 \sinh (a+b x) \sqrt{\text{sech}(a+b x)}}{b}+\frac{2 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b} \]

[Out]

((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

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Rubi [A]  time = 0.0300481, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3768, 3771, 2639} \[ \frac{2 \sinh (a+b x) \sqrt{\text{sech}(a+b x)}}{b}+\frac{2 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*x]^(3/2),x]

[Out]

((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

Rule 3768

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] + Dist[(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]

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.0428349, size = 49, normalized size = 0.79 \[ \frac{2 \sqrt{\text{sech}(a+b x)} \left (\sinh (a+b x)+i \sqrt{\cosh (a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + b*x]^(3/2),x]

[Out]

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

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Maple [A]  time = 0.287, size = 103, normalized size = 1.7 \begin{align*} 2\,{\frac{{\it EllipticE} \left ( \cosh \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \sqrt{- \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}+2\,\cosh \left ( 1/2\,bx+a/2 \right ) \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\sinh \left ( 1/2\,bx+a/2 \right ) \sqrt{2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(b*x+a)^(3/2),x)

[Out]

2*(EllipticE(cosh(1/2*b*x+1/2*a),2^(1/2))*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*sinh(1/2*b*x+1/2*a)^2-1)^(1/2)+2*
cosh(1/2*b*x+1/2*a)*sinh(1/2*b*x+1/2*a)^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., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}\left (b x + a\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(sech(b*x + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\operatorname{sech}\left (b x + a\right )^{\frac{3}{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

integral(sech(b*x + a)^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}^{\frac{3}{2}}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)**(3/2),x)

[Out]

Integral(sech(a + b*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}\left (b x + a\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^(3/2),x, algorithm="giac")

[Out]

integrate(sech(b*x + a)^(3/2), x)