3.1.26 \(\int \sqrt {i \sinh (c+d x)} \, dx\) [26]

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

[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 = {2719} \begin {gather*} -\frac {2 i E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{d} \end {gather*}

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 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], 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 \begin {gather*} \frac {2 i E\left (\left .\frac {1}{2} \left (\frac {\pi }{2}-i (c+d x)\right )\right |2\right )}{d} \end {gather*}

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 = 0.80, size = 91, normalized size = 3.03

method	result	size
default	\(\frac {i \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (i-\sinh \left (d x +c \right )\right )}\, \left (2 \EllipticE \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cosh \left (d x +c \right ) d}\)	\(91\)
risch	\(\frac {\sqrt {2}\, \sqrt {i \left ({\mathrm e}^{2 d x +2 c}-1\right ) {\mathrm e}^{-d x -c}}}{d}-\frac {\left (-\frac {2 i \left (-i+i {\mathrm e}^{2 d x +2 c}\right )}{\sqrt {{\mathrm e}^{d x +c} \left (-i+i {\mathrm e}^{2 d x +2 c}\right )}}-\frac {\sqrt {{\mathrm e}^{d x +c}+1}\, \sqrt {-2 \,{\mathrm e}^{d x +c}+2}\, \sqrt {-{\mathrm e}^{d x +c}}\, \left (-2 \EllipticE \left (\sqrt {{\mathrm e}^{d x +c}+1}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {{\mathrm e}^{d x +c}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {i {\mathrm e}^{3 d x +3 c}-i {\mathrm e}^{d x +c}}}\right ) \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{2 d x +2 c}-1\right ) {\mathrm e}^{-d x -c}}\, \sqrt {i \left ({\mathrm e}^{2 d x +2 c}-1\right ) {\mathrm e}^{d x +c}}}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )}\)	\(229\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.08, size = 53, normalized size = 1.77 \begin {gather*} -\frac {2 \, {\left (\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 )} + \sqrt {2} \sqrt {i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, e^{\left (d x + c\right )}\right )\right )\right )}}{d} \end {gather*}

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^(-1/2*d*x - 1/2*c) + sqrt(2)*sqrt(I)*weierstrassZeta(4, 0, weierst
rassPInverse(4, 0, e^(d*x + c))))/d

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

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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