3.23 \(\int (i \sinh (c+d x))^{7/2} \, dx\)

Optimal. Leaf size=91 \[ -\frac{10 i \text{EllipticF}\left (\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right ),2\right )}{21 d}+\frac{2 i (i \sinh (c+d x))^{5/2} \cosh (c+d x)}{7 d}+\frac{10 i \sqrt{i \sinh (c+d x)} \cosh (c+d x)}{21 d} \]

[Out]

(((-10*I)/21)*EllipticF[(I*c - Pi/2 + I*d*x)/2, 2])/d + (((10*I)/21)*Cosh[c + d*x]*Sqrt[I*Sinh[c + d*x]])/d +
(((2*I)/7)*Cosh[c + d*x]*(I*Sinh[c + d*x])^(5/2))/d

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Rubi [A]  time = 0.0343493, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2635, 2641} \[ -\frac{10 i F\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{21 d}+\frac{2 i (i \sinh (c+d x))^{5/2} \cosh (c+d x)}{7 d}+\frac{10 i \sqrt{i \sinh (c+d x)} \cosh (c+d x)}{21 d} \]

Antiderivative was successfully verified.

[In]

Int[(I*Sinh[c + d*x])^(7/2),x]

[Out]

(((-10*I)/21)*EllipticF[(I*c - Pi/2 + I*d*x)/2, 2])/d + (((10*I)/21)*Cosh[c + d*x]*Sqrt[I*Sinh[c + d*x]])/d +
(((2*I)/7)*Cosh[c + d*x]*(I*Sinh[c + d*x])^(5/2))/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))^{7/2} \, dx &=\frac{2 i \cosh (c+d x) (i \sinh (c+d x))^{5/2}}{7 d}+\frac{5}{7} \int (i \sinh (c+d x))^{3/2} \, dx\\ &=\frac{10 i \cosh (c+d x) \sqrt{i \sinh (c+d x)}}{21 d}+\frac{2 i \cosh (c+d x) (i \sinh (c+d x))^{5/2}}{7 d}+\frac{5}{21} \int \frac{1}{\sqrt{i \sinh (c+d x)}} \, dx\\ &=-\frac{10 i F\left (\left .\frac{1}{2} \left (i c-\frac{\pi }{2}+i d x\right )\right |2\right )}{21 d}+\frac{10 i \cosh (c+d x) \sqrt{i \sinh (c+d x)}}{21 d}+\frac{2 i \cosh (c+d x) (i \sinh (c+d x))^{5/2}}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.164432, size = 65, normalized size = 0.71 \[ \frac{i \left (20 \text{EllipticF}\left (\frac{1}{4} (-2 i c-2 i d x+\pi ),2\right )+\sqrt{i \sinh (c+d x)} (23 \cosh (c+d x)-3 \cosh (3 (c+d x)))\right )}{42 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(I*Sinh[c + d*x])^(7/2),x]

[Out]

((I/42)*(20*EllipticF[((-2*I)*c + Pi - (2*I)*d*x)/4, 2] + (23*Cosh[c + d*x] - 3*Cosh[3*(c + d*x)])*Sqrt[I*Sinh
[c + d*x]]))/d

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Maple [A]  time = 0.048, size = 122, normalized size = 1.3 \begin{align*}{\frac{-{\frac{i}{21}}}{d\cosh \left ( dx+c \right ) } \left ( 6\,i\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{4}-5\,\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 ) },1/2\,\sqrt{2} \right ) -16\,i \left ( \cosh \left ( dx+c \right ) \right ) ^{2}\sinh \left ( dx+c \right ) \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))^(7/2),x)

[Out]

-1/21*I*(6*I*sinh(d*x+c)*cosh(d*x+c)^4-5*(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))-16*I*cosh(d*x+c)^2*sinh(d*x+c))/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{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((I*sinh(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

integrate((I*sinh(d*x + c))^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (\sqrt{\frac{1}{2}}{\left (-3 i \, e^{\left (6 \, d x + 6 \, c\right )} + 23 i \, e^{\left (4 \, d x + 4 \, c\right )} + 23 i \, e^{\left (2 \, d x + 2 \, c\right )} - 3 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 )} + 84 \, d e^{\left (3 \, d x + 3 \, c\right )}{\rm integral}\left (-\frac{10 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 )}}{21 \,{\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )}}, x\right )\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{84 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((I*sinh(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

1/84*(sqrt(1/2)*(-3*I*e^(6*d*x + 6*c) + 23*I*e^(4*d*x + 4*c) + 23*I*e^(2*d*x + 2*c) - 3*I)*sqrt(I*e^(2*d*x + 2
*c) - I)*e^(-1/2*d*x - 1/2*c) + 84*d*e^(3*d*x + 3*c)*integral(-10/21*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^(-3*d*x - 3*c)/d

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((I*sinh(d*x+c))**(7/2),x)

[Out]

Timed out

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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{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((I*sinh(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((I*sinh(d*x + c))^(7/2), x)