3.28 \(\int \frac {1}{(i \sinh (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=58 \[ \frac {2 i E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{d}+\frac {2 i \cosh (c+d x)}{d \sqrt {i \sinh (c+d x)}} \]

[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+2*I*cosh(d*x+c)/d/(I*sinh(d*x+c))^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2636, 2639} \[ \frac {2 i E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{d}+\frac {2 i \cosh (c+d x)}{d \sqrt {i \sinh (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(I*Sinh[c + d*x])^(-3/2),x]

[Out]

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

Rule 2636

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

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 \frac {1}{(i \sinh (c+d x))^{3/2}} \, dx &=\frac {2 i \cosh (c+d x)}{d \sqrt {i \sinh (c+d x)}}-\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}+\frac {2 i \cosh (c+d x)}{d \sqrt {i \sinh (c+d x)}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 50, normalized size = 0.86 \[ \frac {2 \left (\sqrt {i \sinh (c+d x)} \coth (c+d x)-i E\left (\left .\frac {1}{4} (-2 i c-2 i d x+\pi )\right |2\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(I*Sinh[c + d*x])^(-3/2),x]

[Out]

(2*((-I)*EllipticE[((-2*I)*c + Pi - (2*I)*d*x)/4, 2] + Coth[c + d*x]*Sqrt[I*Sinh[c + d*x]]))/d

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fricas [F]  time = 1.27, size = 0, normalized size = 0.00 \[ \frac {4 \, \sqrt {\frac {1}{2}} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )} + {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} {\rm integral}\left (-\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{d e^{\left (2 \, d x + 2 \, c\right )} - d}, x\right )}{d e^{\left (2 \, d x + 2 \, c\right )} - d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(I*sinh(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

(4*sqrt(1/2)*sqrt(I*e^(2*d*x + 2*c) - I)*e^(3/2*d*x + 3/2*c) + (d*e^(2*d*x + 2*c) - d)*integral(-2*sqrt(1/2)*s
qrt(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))/(d*e^(2*d*x + 2*c) - d)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (i \, \sinh \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(I*sinh(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((I*sinh(d*x + c))^(-3/2), x)

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maple [A]  time = 0.08, size = 159, normalized size = 2.74 \[ -\frac {i \left (2 \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticE \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{\cosh \left (d x +c \right ) \sqrt {i \sinh \left (d x +c \right )}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(I*sinh(d*x+c))^(3/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (i \, \sinh \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(I*sinh(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((I*sinh(d*x + c))^(-3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)*1i)^(3/2),x)

[Out]

int(1/(sinh(c + d*x)*1i)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (i \sinh {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(I*sinh(d*x+c))**(3/2),x)

[Out]

Integral((I*sinh(c + d*x))**(-3/2), x)

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