∫ (i sinh(c + dx))7∕2 dx [23]

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

10/21*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

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Rubi [A]
time = 0.03, antiderivative size = 91, normalized size of antiderivative = 1.00, 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 = {2715, 2720} \begin {gather*} -\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} \end {gather*}

(((-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 +

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] && Integ

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ

\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*}

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Mathematica [A]
time = 0.12, size = 65, normalized size = 0.71 \begin {gather*} \frac {i \left (20 F\left (\left .\frac {1}{4} (-2 i c+\pi -2 i d x)\right |2\right )+(23 \cosh (c+d x)-3 \cosh (3 (c+d x))) \sqrt {i \sinh (c+d x)}\right )}{42 d} \end {gather*}

((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

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method	result	size

default	\(-\frac {i \left (6 i \left (\cosh ^{4}\left (d x +c \right )\right ) \sinh \left (d x +c \right )-5 \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-16 i \left (\cosh ^{2}\left (d x +c \right )\right ) \sinh \left (d x +c \right )\right )}{21 \cosh \left (d x +c \right ) \sqrt {i \sinh \left (d x +c \right )}\, d}\)	\(122\)

-1/21*I*(6*I*cosh(d*x+c)^4*sinh(d*x+c)-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*

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.09, size = 104, normalized size = 1.14 \begin {gather*} \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 )} - 40 i \, \sqrt {2} \sqrt {i} e^{\left (3 \, d x + 3 \, c\right )} {\rm weierstrassPInverse}\left (4, 0, e^{\left (d x + c\right )}\right )\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{84 \, d} \end {gather*}

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) - 40*I*sqrt(2)*sqrt(I)*e^(3*d*x + 3*c)*weierstrassPInverse(4, 0, e^(d*x + c)))*e

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2} \,d x \end {gather*}

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3.1.23 \(\int (i \sinh (c+d x))^{7/2} \, dx\) [23]