3.27 \(\int \frac{1}{\sqrt{i \sinh (c+d x)}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{2 i \text{EllipticF}\left (\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right ),2\right )}{d} \]

[Out]

((-2*I)*EllipticF[(I*c - Pi/2 + I*d*x)/2, 2])/d

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Rubi [A]  time = 0.0086927, antiderivative size = 30, 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 = {2641} \[ -\frac{2 i F\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[I*Sinh[c + d*x]],x]

[Out]

((-2*I)*EllipticF[(I*c - Pi/2 + I*d*x)/2, 2])/d

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{i \sinh (c+d x)}} \, dx &=-\frac{2 i F\left (\left .\frac{1}{2} \left (i c-\frac{\pi }{2}+i d x\right )\right |2\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.0226246, size = 28, normalized size = 0.93 \[ \frac{2 i \text{EllipticF}\left (\frac{1}{2} \left (\frac{\pi }{2}-i (c+d x)\right ),2\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[I*Sinh[c + d*x]],x]

[Out]

((2*I)*EllipticF[(Pi/2 - I*(c + d*x))/2, 2])/d

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Maple [A]  time = 0.038, size = 68, normalized size = 2.3 \begin{align*}{\frac{i\sqrt{2}}{d\cosh \left ( dx+c \right ) }\sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) }\sqrt{-i \left ( i-\sinh \left ( dx+c \right ) \right ) }{\it EllipticF} \left ( \sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) },{\frac{\sqrt{2}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(I*sinh(d*x+c))^(1/2),x)

[Out]

I*(-I*(sinh(d*x+c)+I))^(1/2)*2^(1/2)*(-I*(I-sinh(d*x+c)))^(1/2)*EllipticF((-I*(sinh(d*x+c)+I))^(1/2),1/2*2^(1/
2))/cosh(d*x+c)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(I*sinh(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(I*sinh(d*x + c)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{2 i \, \sqrt{\frac{1}{2}} \sqrt{i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{d e^{\left (2 \, d x + 2 \, c\right )} - d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(I*sinh(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(-2*I*sqrt(1/2)*sqrt(I*e^(2*d*x + 2*c) - I)*e^(-1/2*d*x - 1/2*c)/(d*e^(2*d*x + 2*c) - d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(I*sinh(d*x+c))**(1/2),x)

[Out]

Integral(1/sqrt(I*sinh(c + d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(I*sinh(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(I*sinh(d*x + c)), x)