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

Optimal result

Integrand size = 14, antiderivative size = 91 \[ \int (i \sinh (c+d x))^{7/2} \, dx=-\frac {10 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right ),2\right )}{21 d}+\frac {10 i \cosh (c+d x) \sqrt {i \sinh (c+d x)}}{21 d}+\frac {2 i \cosh (c+d x) (i \sinh (c+d x))^{5/2}}{7 d} \]

[Out]

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
*I*d*x),2^(1/2))/d+2/7*I*cosh(d*x+c)*(I*sinh(d*x+c))^(5/2)/d+10/21*I*cosh(d*x+c)*(I*sinh(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2715, 2720} \[ \int (i \sinh (c+d x))^{7/2} \, dx=-\frac {10 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right ),2\right )}{21 d}+\frac {2 i (i \sinh (c+d x))^{5/2} \cosh (c+d x)}{7 d}+\frac {10 i \sqrt {i \sinh (c+d x)} \cosh (c+d x)}{21 d} \]

[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 2715

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
erQ[2*n]

Rule 2720

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

Rubi steps \begin{align*} \text {integral}& = \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 \operatorname {EllipticF}\left (\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\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] (verified)

Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.71 \[ \int (i \sinh (c+d x))^{7/2} \, dx=\frac {i \left (20 \operatorname {EllipticF}\left (\frac {1}{4} (-2 i c+\pi -2 i d x),2\right )+(23 \cosh (c+d x)-3 \cosh (3 (c+d x))) \sqrt {i \sinh (c+d x)}\right )}{42 d} \]

[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

Maple [A] (verified)

Time = 0.85 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.34

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

[In]

int((I*sinh(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

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*
x+c))^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.14 \[ \int (i \sinh (c+d x))^{7/2} \, dx=\frac {{\left (\sqrt {\frac {1}{2}} {\left (-3 i \, e^{\left (6 \, d x + 6 \, c\right )} + 23 i \, e^{\left (4 \, d x + 4 \, c\right )} + 23 i \, e^{\left (2 \, d x + 2 \, c\right )} - 3 i\right )} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} - 40 i \, \sqrt {2} \sqrt {i} e^{\left (3 \, d x + 3 \, c\right )} {\rm weierstrassPInverse}\left (4, 0, e^{\left (d x + c\right )}\right )\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{84 \, d} \]

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

Sympy [F(-1)]

Timed out. \[ \int (i \sinh (c+d x))^{7/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int (i \sinh (c+d x))^{7/2} \, dx=\int { \left (i \, \sinh \left (d x + c\right )\right )^{\frac {7}{2}} \,d x } \]

[In]

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

[Out]

integrate((I*sinh(d*x + c))^(7/2), x)

Giac [F]

\[ \int (i \sinh (c+d x))^{7/2} \, dx=\int { \left (i \, \sinh \left (d x + c\right )\right )^{\frac {7}{2}} \,d x } \]

[In]

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

[Out]

integrate((I*sinh(d*x + c))^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int (i \sinh (c+d x))^{7/2} \, dx=\int {\left (\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2} \,d x \]

[In]

int((sinh(c + d*x)*1i)^(7/2),x)

[Out]

int((sinh(c + d*x)*1i)^(7/2), x)