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*(sin(1/2*I*c+1/4*Pi+1/2*I*d*x)^2)^(1/2)/sin(1/2*I*c+1/4*Pi+1/2*I*d*x)*EllipticE(cos(1/2*I*c+1/4*Pi+1/2*I*d
*x),2^(1/2))/d

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, 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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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ \frac {2 \, \sqrt {\frac {1}{2}} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} {\left (e^{\left (d x + c\right )} + 2\right )} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} + {\left (d e^{\left (d x + c\right )} - 2 \, d\right )} {\rm integral}\left (\frac {2 \, \sqrt {\frac {1}{2}} {\left (2 \, e^{\left (2 \, d x + 2 \, c\right )} + 3 \, e^{\left (d x + c\right )} - 2\right )} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{d e^{\left (4 \, d x + 4 \, c\right )} - 4 \, d e^{\left (3 \, d x + 3 \, c\right )} + 3 \, d e^{\left (2 \, d x + 2 \, c\right )} + 4 \, d e^{\left (d x + c\right )} - 4 \, d}, x\right )}{d e^{\left (d x + c\right )} - 2 \, d} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {i \, \sinh \left (d x + c\right )}\,{d x} \]

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)

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maple [A]  time = 0.07, size = 91, normalized size = 3.03 \[ \frac {i \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (d x +c \right )+i\right )}\, \left (2 \EllipticE \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cosh \left (d x +c \right ) d} \]

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*(-sinh(d*x+c)+I))^(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {i \, \sinh \left (d x + c\right )}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sinh(c + d*x)*1i)^(1/2),x)

[Out]

int((sinh(c + d*x)*1i)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {i \sinh {\left (c + d x \right )}}\, dx \]

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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