Integrand size = 14, antiderivative size = 125 \[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\frac {2 \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a}{a+b},\arcsin \left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right ),\frac {a-b}{a+b}\right ) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (1+\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x))}{\sqrt {a+b} d} \] Output:
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))*(-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)*(a+b*sech(d*x+c))/(a+b)^(1/2)/d
\[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\int \sqrt {a+b \text {sech}(c+d x)} \, dx \] Input:
Integrate[Sqrt[a + b*Sech[c + d*x]],x]
Output:
Integrate[Sqrt[a + b*Sech[c + d*x]], x]
Time = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 4267}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {a+b \text {sech}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4267 |
\(\displaystyle \frac {2 \coth (c+d x) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (\text {sech}(c+d x)+1)}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x)) \operatorname {EllipticPi}\left (\frac {a}{a+b},\arcsin \left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right ),\frac {a-b}{a+b}\right )}{d \sqrt {a+b}}\) |
Input:
Int[Sqrt[a + b*Sech[c + d*x]],x]
Output:
(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*Sech[ c + d*x]))/(Sqrt[a + b]*d)
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]))]*E llipticPi[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]
\[\int \sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}d x\]
Input:
int((a+b*sech(d*x+c))^(1/2),x)
Output:
int((a+b*sech(d*x+c))^(1/2),x)
\[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\int { \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \,d x } \] Input:
integrate((a+b*sech(d*x+c))^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(b*sech(d*x + c) + a), x)
\[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\int \sqrt {a + b \operatorname {sech}{\left (c + d x \right )}}\, dx \] Input:
integrate((a+b*sech(d*x+c))**(1/2),x)
Output:
Integral(sqrt(a + b*sech(c + d*x)), x)
\[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\int { \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \,d x } \] Input:
integrate((a+b*sech(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*sech(d*x + c) + a), x)
\[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\int { \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \,d x } \] Input:
integrate((a+b*sech(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*sech(d*x + c) + a), x)
Timed out. \[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\int \sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}} \,d x \] Input:
int((a + b/cosh(c + d*x))^(1/2),x)
Output:
int((a + b/cosh(c + d*x))^(1/2), x)
\[ \int \sqrt {a+b \text {sech}(c+d x)} \, dx=\int \sqrt {\mathrm {sech}\left (d x +c \right ) b +a}d x \] Input:
int((a+b*sech(d*x+c))^(1/2),x)
Output:
int(sqrt(sech(c + d*x)*b + a),x)