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

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Rubi [A]  time = 0.0186412, antiderivative size = 58, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0938688, 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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Maple [A]  time = 0.052, size = 159, normalized size = 2.7 \begin{align*}{\frac{-i}{d\cosh \left ( dx+c \right ) } \left ( 2\,\sqrt{1-i\sinh \left ( dx+c \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( dx+c \right ) }\sqrt{i\sinh \left ( dx+c \right ) }{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) -\sqrt{1-i\sinh \left ( dx+c \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( dx+c \right ) }\sqrt{i\sinh \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( dx+c \right ) },{\frac{\sqrt{2}}{2}} \right ) -2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{i\sinh \left ( dx+c \right ) }}}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (i \, \sinh \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (i \sinh{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (i \, \sinh \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}

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)