\(\int \sqrt {a+b \text {sech}(c+d x)} \, dx\) [131]

Optimal result

Integrand size = 14, antiderivative size = 125 \[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\frac {2 \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a}{a+b},\arcsin \left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right ),\frac {a-b}{a+b}\right ) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (1+\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x))}{\sqrt {a+b} d} \]

[Out]

2*coth(d*x+c)*EllipticPi((a+b)^(1/2)/(a+b*sech(d*x+c))^(1/2),a/(a+b),((a-b)/(a+b))^(1/2))*(a+b*sech(d*x+c))*(-
b*(1-sech(d*x+c))/(a+b*sech(d*x+c)))^(1/2)*(b*(1+sech(d*x+c))/(a+b*sech(d*x+c)))^(1/2)/d/(a+b)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3865} \[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\frac {2 \coth (c+d x) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (\text {sech}(c+d x)+1)}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x)) \operatorname {EllipticPi}\left (\frac {a}{a+b},\arcsin \left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right ),\frac {a-b}{a+b}\right )}{d \sqrt {a+b}} \]

[In]

Int[Sqrt[a + b*Sech[c + d*x]],x]

[Out]

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

Rule 3865

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a}{a+b},\arcsin \left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right ),\frac {a-b}{a+b}\right ) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (1+\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x))}{\sqrt {a+b} d} \\ \end{align*}

Mathematica [F]

\[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\int \sqrt {a+b \text {sech}(c+d x)} \, dx \]

[In]

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

[Out]

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

Maple [F]

\[\int \sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\int { \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\int \sqrt {a + b \operatorname {sech}{\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\int { \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\int { \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\int \sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]

[In]

int((a + b/cosh(c + d*x))^(1/2),x)

[Out]

int((a + b/cosh(c + d*x))^(1/2), x)