\(\int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx\) [140]

Optimal result

Integrand size = 14, antiderivative size = 106 \[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 106, 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 = {3869} \[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 \sqrt {a+b} \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d} \]

[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 3869

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

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

Time = 0.57 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 b \sqrt {b+a \cosh (c+d x)} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a+b} \sqrt {a \cosh (c+d x)}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1+\text {sech}(c+d x))}{-a+b}} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a} \sqrt {a+b} d \sqrt {a \cosh (c+d x)} \sqrt {-\frac {b (-1+\text {sech}(c+d x))}{a+b}} \sqrt {a+b \text {sech}(c+d x)}} \]

[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]])

Maple [F]

\[\int \frac {1}{\sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {1}{\sqrt {a + b \operatorname {sech}{\left (c + d x \right )}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {1}{\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}} \,d x \]

[In]

int(1/(a + b/cosh(c + d*x))^(1/2),x)

[Out]

int(1/(a + b/cosh(c + d*x))^(1/2), x)