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

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

[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)*EllipticF(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 = {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*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 0.93 \[ \frac {2 i F\left (\left .\frac {1}{2} \left (\frac {\pi }{2}-i (c+d x)\right )\right |2\right )}{d} \]

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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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {2 i \, \sqrt {\frac {1}{2}} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{d e^{\left (2 \, d x + 2 \, c\right )} - d}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {i \, \sinh \left (d x + c\right )}}\,{d x} \]

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)

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maple [A]  time = 0.06, size = 68, normalized size = 2.27 \[ \frac {i \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (d x +c \right )+i\right )}\, \EllipticF \left (\sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}, \frac {\sqrt {2}}{2}\right )}{\cosh \left (d x +c \right ) d} \]

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*(-sinh(d*x+c)+I))^(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {i \, \sinh \left (d x + c\right )}}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)*1i)^(1/2),x)

[Out]

int(1/(sinh(c + d*x)*1i)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {i \sinh {\left (c + d x \right )}}}\, dx \]

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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