3.94 \(\int \frac{1}{\sqrt{a+b \text{sech}(c+d x)}} \, dx\)

Optimal. Leaf size=106 \[ \frac{2 \sqrt{a+b} \coth (c+d x) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (\text{sech}(c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a d} \]

[Out]

(2*Sqrt[a + b]*Coth[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a -
b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*d)

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Rubi [A]  time = 0.0292714, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3784} \[ \frac{2 \sqrt{a+b} \coth (c+d x) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (\text{sech}(c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a d} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*Sech[c + d*x]],x]

[Out]

(2*Sqrt[a + b]*Coth[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a -
b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*d)

Rule 3784

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(2*Rt[a + b, 2]*Sqrt[(b*(1 - Csc[c + d*x])
)/(a + b)]*Sqrt[-((b*(1 + Csc[c + d*x]))/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[c + d*x]]/Rt[a
+ b, 2]], (a + b)/(a - b)])/(a*d*Cot[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+b \text{sech}(c+d x)}} \, dx &=\frac{2 \sqrt{a+b} \coth (c+d x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (1+\text{sech}(c+d x))}{a-b}}}{a d}\\ \end{align*}

Mathematica [A]  time = 2.34509, size = 168, normalized size = 1.58 \[ \frac{2 b \tanh \left (\frac{1}{2} (c+d x)\right ) \sqrt{a \cosh (c+d x)+b} \sqrt{\frac{b (\text{sech}(c+d x)+1)}{b-a}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a} \sqrt{b+a \cosh (c+d x)}}{\sqrt{a+b} \sqrt{a \cosh (c+d x)}}\right )|\frac{a+b}{a-b}\right )}{\sqrt{a} d \sqrt{a+b} \sqrt{a \cosh (c+d x)} \sqrt{-\frac{b (\text{sech}(c+d x)-1)}{a+b}} \sqrt{a+b \text{sech}(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b*Sech[c + d*x]],x]

[Out]

(2*b*Sqrt[b + a*Cosh[c + d*x]]*EllipticPi[(a + b)/a, ArcSin[(Sqrt[a]*Sqrt[b + a*Cosh[c + d*x]])/(Sqrt[a + b]*S
qrt[a*Cosh[c + d*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 + Sech[c + d*x]))/(-a + b)]*Tanh[(c + d*x)/2])/(Sqrt[a]*Sq
rt[a + b]*d*Sqrt[a*Cosh[c + d*x]]*Sqrt[-((b*(-1 + Sech[c + d*x]))/(a + b))]*Sqrt[a + b*Sech[c + d*x]])

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Maple [F]  time = 0.235, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{a+b{\rm sech} \left (dx+c\right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \operatorname{sech}\left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*sech(d*x + c) + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{b \operatorname{sech}\left (d x + c\right ) + a}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(b*sech(d*x + c) + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b \operatorname{sech}{\left (c + d x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a + b*sech(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \operatorname{sech}\left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*sech(d*x + c) + a), x)