3.11 \(\int \frac{1}{\sqrt{\sinh (a+b x)}} \, dx\)

Optimal. Leaf size=54 \[ -\frac{2 i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right ),2\right )}{b \sqrt{\sinh (a+b x)}} \]

[Out]

((-2*I)*EllipticF[(I*a - Pi/2 + I*b*x)/2, 2]*Sqrt[I*Sinh[a + b*x]])/(b*Sqrt[Sinh[a + b*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.019737, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2642, 2641} \[ -\frac{2 i \sqrt{i \sinh (a+b x)} F\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{b \sqrt{\sinh (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Sinh[a + b*x]],x]

[Out]

((-2*I)*EllipticF[(I*a - Pi/2 + I*b*x)/2, 2]*Sqrt[I*Sinh[a + b*x]])/(b*Sqrt[Sinh[a + b*x]])

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{\sinh (a+b x)}} \, dx &=\frac{\sqrt{i \sinh (a+b x)} \int \frac{1}{\sqrt{i \sinh (a+b x)}} \, dx}{\sqrt{\sinh (a+b x)}}\\ &=-\frac{2 i F\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right ) \sqrt{i \sinh (a+b x)}}{b \sqrt{\sinh (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.114061, size = 48, normalized size = 0.89 \[ -\frac{2 \sqrt{\sinh (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 i a-2 i b x+\pi ),2\right )}{b \sqrt{i \sinh (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Sinh[a + b*x]],x]

[Out]

(-2*EllipticF[((-2*I)*a + Pi - (2*I)*b*x)/4, 2]*Sqrt[Sinh[a + b*x]])/(b*Sqrt[I*Sinh[a + b*x]])

________________________________________________________________________________________

Maple [A]  time = 0.036, size = 87, normalized size = 1.6 \begin{align*}{\frac{i\sqrt{2}}{b\cosh \left ( bx+a \right ) }\sqrt{-i \left ( \sinh \left ( bx+a \right ) +i \right ) }\sqrt{-i \left ( -\sinh \left ( bx+a \right ) +i \right ) }\sqrt{i\sinh \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{-i \left ( \sinh \left ( bx+a \right ) +i \right ) },{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{\sinh \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sinh(b*x+a)^(1/2),x)

[Out]

I*(-I*(sinh(b*x+a)+I))^(1/2)*2^(1/2)*(-I*(-sinh(b*x+a)+I))^(1/2)*(I*sinh(b*x+a))^(1/2)*EllipticF((-I*(sinh(b*x
+a)+I))^(1/2),1/2*2^(1/2))/cosh(b*x+a)/sinh(b*x+a)^(1/2)/b

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(sinh(b*x + a)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(sinh(b*x + a)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sinh(b*x+a)**(1/2),x)

[Out]

Integral(1/sqrt(sinh(a + b*x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(sinh(b*x + a)), x)