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

Optimal. Leaf size=62 \[ \frac{2 i \sqrt{i \sinh (c+d x)} \cosh (c+d x)}{3 d}-\frac{2 i \text{EllipticF}\left (\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right ),2\right )}{3 d} \]

[Out]

(((-2*I)/3)*EllipticF[(I*c - Pi/2 + I*d*x)/2, 2])/d + (((2*I)/3)*Cosh[c + d*x]*Sqrt[I*Sinh[c + d*x]])/d

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Rubi [A]  time = 0.0205847, antiderivative size = 62, 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 = {2635, 2641} \[ \frac{2 i \sqrt{i \sinh (c+d x)} \cosh (c+d x)}{3 d}-\frac{2 i F\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(I*Sinh[c + d*x])^(3/2),x]

[Out]

(((-2*I)/3)*EllipticF[(I*c - Pi/2 + I*d*x)/2, 2])/d + (((2*I)/3)*Cosh[c + d*x]*Sqrt[I*Sinh[c + d*x]])/d

Rule 2635

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

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

Mathematica [C]  time = 0.134912, size = 94, normalized size = 1.52 \[ -\frac{2 i \sqrt{i \sinh (c+d x)} \left (\text{csch}(c+d x) \sqrt{-\sinh (2 c+2 d x)-\cosh (2 c+2 d x)+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\cosh (2 (c+d x))+\sinh (2 (c+d x))\right )-\cosh (c+d x)\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(I*Sinh[c + d*x])^(3/2),x]

[Out]

(((-2*I)/3)*Sqrt[I*Sinh[c + d*x]]*(-Cosh[c + d*x] + Csch[c + d*x]*Hypergeometric2F1[1/4, 1/2, 5/4, Cosh[2*(c +
d*x)] + Sinh[2*(c + d*x)]]*Sqrt[1 - Cosh[2*c + 2*d*x] - Sinh[2*c + 2*d*x]]))/d

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Maple [A]  time = 0.048, size = 104, normalized size = 1.7 \begin{align*}{\frac{{\frac{i}{3}}}{d\cosh \left ( dx+c \right ) } \left ( \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\,i\sinh \left ( dx+c \right ) \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((I*sinh(d*x+c))^(3/2),x)

[Out]

1/3*I*((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*I*sinh(d*x+c)*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 \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((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{{\left (\sqrt{\frac{1}{2}}{\left (i \, e^{\left (2 \, d x + 2 \, c\right )} + i\right )} \sqrt{i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )} + 3 \, d e^{\left (d x + c\right )}{\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 )}}{3 \,{\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )}}, x\right )\right )} e^{\left (-d x - c\right )}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(sqrt(1/2)*(I*e^(2*d*x + 2*c) + I)*sqrt(I*e^(2*d*x + 2*c) - I)*e^(-1/2*d*x - 1/2*c) + 3*d*e^(d*x + c)*inte
gral(-2/3*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))*e^(-d*x -
c)/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((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 \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((I*sinh(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((I*sinh(d*x + c))^(3/2), x)