3.26 \(\int \sqrt{i \sinh (c+d x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0091243, 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 = {2639} \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[I*Sinh[c + d*x]],x]

[Out]

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

Rule 2639

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

Rubi steps

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

Mathematica [A]  time = 0.0204305, size = 28, normalized size = 0.93 \[ \frac{2 i E\left (\left .\frac{1}{2} \left (\frac{\pi }{2}-i (c+d x)\right )\right |2\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[I*Sinh[c + d*x]],x]

[Out]

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

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Maple [A]  time = 0.045, size = 91, normalized size = 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 ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( dx+c \right ) },{\frac{\sqrt{2}}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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)*(2*EllipticE((1-I*sinh(d*x+c))^(1/2),1/2*2^(1/
2))-EllipticF((1-I*sinh(d*x+c))^(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 \sqrt{i \, \sinh \left (d x + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(I*sinh(d*x + c)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, \sqrt{\frac{1}{2}} \sqrt{i \, e^{\left (2 \, d x + 2 \, c\right )} - i}{\left (e^{\left (d x + c\right )} + 2\right )} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )} +{\left (d e^{\left (d x + c\right )} - 2 \, d\right )}{\rm integral}\left (\frac{2 \, \sqrt{\frac{1}{2}}{\left (2 \, e^{\left (2 \, d x + 2 \, c\right )} + 3 \, e^{\left (d x + c\right )} - 2\right )} \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 (4 \, d x + 4 \, c\right )} - 4 \, d e^{\left (3 \, d x + 3 \, c\right )} + 3 \, d e^{\left (2 \, d x + 2 \, c\right )} + 4 \, d e^{\left (d x + c\right )} - 4 \, d}, x\right )}{d e^{\left (d x + c\right )} - 2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*sqrt(1/2)*sqrt(I*e^(2*d*x + 2*c) - I)*(e^(d*x + c) + 2)*e^(-1/2*d*x - 1/2*c) + (d*e^(d*x + c) - 2*d)*integr
al(2*sqrt(1/2)*(2*e^(2*d*x + 2*c) + 3*e^(d*x + c) - 2)*sqrt(I*e^(2*d*x + 2*c) - I)*e^(-1/2*d*x - 1/2*c)/(d*e^(
4*d*x + 4*c) - 4*d*e^(3*d*x + 3*c) + 3*d*e^(2*d*x + 2*c) + 4*d*e^(d*x + c) - 4*d), x))/(d*e^(d*x + c) - 2*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(I*sinh(c + d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(I*sinh(d*x + c)), x)