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

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

[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

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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 = {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*}

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Mathematica [A]  time = 0.17, size = 65, normalized size = 0.71 \[ \frac {i \left (20 F\left (\left .\frac {1}{4} (-2 i c-2 i d x+\pi )\right |2\right )+\sqrt {i \sinh (c+d x)} (23 \cosh (c+d x)-3 \cosh (3 (c+d x)))\right )}{42 d} \]

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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fricas [F]  time = 0.63, size = 0, normalized size = 0.00 \[ \frac {{\left (\sqrt {\frac {1}{2}} {\left (-3 i \, e^{\left (6 \, d x + 6 \, c\right )} + 23 i \, e^{\left (4 \, d x + 4 \, c\right )} + 23 i \, e^{\left (2 \, d x + 2 \, c\right )} - 3 i\right )} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} + 84 \, d e^{\left (3 \, d x + 3 \, c\right )} {\rm integral}\left (-\frac {10 i \, \sqrt {\frac {1}{2}} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{21 \, {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )}}, x\right )\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{84 \, d} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (i \, \sinh \left (d x + c\right )\right )^{\frac {7}{2}}\,{d x} \]

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)

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maple [A]  time = 0.07, size = 122, normalized size = 1.34 \[ -\frac {i \left (6 i \sinh \left (d x +c \right ) \left (\cosh ^{4}\left (d x +c \right )\right )-5 \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-16 i \sinh \left (d x +c \right ) \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{21 \cosh \left (d x +c \right ) \sqrt {i \sinh \left (d x +c \right )}\, d} \]

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*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 = 0.00, size = 0, normalized size = 0.00 \[ \int \left (i \, \sinh \left (d x + c\right )\right )^{\frac {7}{2}}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sinh(c + d*x)*1i)^(7/2),x)

[Out]

int((sinh(c + d*x)*1i)^(7/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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