3.1.30 \(\int \frac {1}{(i \sinh (c+d x))^{7/2}} \, dx\) [30]

Optimal. Leaf size=91 \[ \frac {6 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right )}{5 d}+\frac {2 i \cosh (c+d x)}{5 d (i \sinh (c+d x))^{5/2}}+\frac {6 i \cosh (c+d x)}{5 d \sqrt {i \sinh (c+d x)}} \]

[Out]

-6/5*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)*EllipticE(cos(1/2*I*c+1/4*Pi+1/2*
I*d*x),2^(1/2))/d+2/5*I*cosh(d*x+c)/d/(I*sinh(d*x+c))^(5/2)+6/5*I*cosh(d*x+c)/d/(I*sinh(d*x+c))^(1/2)

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Rubi [A]
time = 0.02, 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 = {2716, 2719} \begin {gather*} \frac {6 i E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{5 d}+\frac {6 i \cosh (c+d x)}{5 d \sqrt {i \sinh (c+d x)}}+\frac {2 i \cosh (c+d x)}{5 d (i \sinh (c+d x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(I*Sinh[c + d*x])^(-7/2),x]

[Out]

(((6*I)/5)*EllipticE[(I*c - Pi/2 + I*d*x)/2, 2])/d + (((2*I)/5)*Cosh[c + d*x])/(d*(I*Sinh[c + d*x])^(5/2)) + (
((6*I)/5)*Cosh[c + d*x])/(d*Sqrt[I*Sinh[c + d*x]])

Rule 2716

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

Rule 2719

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

Rubi steps

\begin {align*} \int \frac {1}{(i \sinh (c+d x))^{7/2}} \, dx &=\frac {2 i \cosh (c+d x)}{5 d (i \sinh (c+d x))^{5/2}}+\frac {3}{5} \int \frac {1}{(i \sinh (c+d x))^{3/2}} \, dx\\ &=\frac {2 i \cosh (c+d x)}{5 d (i \sinh (c+d x))^{5/2}}+\frac {6 i \cosh (c+d x)}{5 d \sqrt {i \sinh (c+d x)}}-\frac {3}{5} \int \sqrt {i \sinh (c+d x)} \, dx\\ &=\frac {6 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right )}{5 d}+\frac {2 i \cosh (c+d x)}{5 d (i \sinh (c+d x))^{5/2}}+\frac {6 i \cosh (c+d x)}{5 d \sqrt {i \sinh (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 80, normalized size = 0.88 \begin {gather*} -\frac {2 i \left (-3 \cosh (c+d x)+\coth (c+d x) \text {csch}(c+d x)+3 E\left (\left .\frac {1}{4} (-2 i c+\pi -2 i d x)\right |2\right ) \sqrt {i \sinh (c+d x)}\right )}{5 d \sqrt {i \sinh (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(I*Sinh[c + d*x])^(-7/2),x]

[Out]

(((-2*I)/5)*(-3*Cosh[c + d*x] + Coth[c + d*x]*Csch[c + d*x] + 3*EllipticE[((-2*I)*c + Pi - (2*I)*d*x)/4, 2]*Sq
rt[I*Sinh[c + d*x]]))/(d*Sqrt[I*Sinh[c + d*x]])

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Maple [A]
time = 0.74, size = 204, normalized size = 2.24

method	result	size
default	\(-\frac {i \left (6 \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (i-\sinh \left (d x +c \right )\right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \left (\sinh ^{2}\left (d x +c \right )\right ) \EllipticE \left (\sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (i-\sinh \left (d x +c \right )\right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \left (\sinh ^{2}\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-6 \left (\sinh ^{4}\left (d x +c \right )\right )-4 \left (\sinh ^{2}\left (d x +c \right )\right )+2\right )}{5 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right ) \sqrt {i \sinh \left (d x +c \right )}\, d}\)	\(204\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(I*sinh(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/5*I/sinh(d*x+c)^2*(6*(-I*(sinh(d*x+c)+I))^(1/2)*2^(1/2)*(-I*(I-sinh(d*x+c)))^(1/2)*(I*sinh(d*x+c))^(1/2)*si
nh(d*x+c)^2*EllipticE((-I*(sinh(d*x+c)+I))^(1/2),1/2*2^(1/2))-3*(-I*(sinh(d*x+c)+I))^(1/2)*2^(1/2)*(-I*(I-sinh
(d*x+c)))^(1/2)*(I*sinh(d*x+c))^(1/2)*sinh(d*x+c)^2*EllipticF((-I*(sinh(d*x+c)+I))^(1/2),1/2*2^(1/2))-6*sinh(d
*x+c)^4-4*sinh(d*x+c)^2+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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 178, normalized size = 1.96 \begin {gather*} \frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} {\left (3 \, e^{\left (6 \, d x + 6 \, c\right )} - 8 \, e^{\left (4 \, d x + 4 \, c\right )} + e^{\left (2 \, d x + 2 \, c\right )}\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 \, {\left (\sqrt {2} \sqrt {i} e^{\left (6 \, d x + 6 \, c\right )} - 3 \, \sqrt {2} \sqrt {i} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, \sqrt {2} \sqrt {i} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {2} \sqrt {i}\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, e^{\left (d x + c\right )}\right )\right )\right )}}{5 \, {\left (d e^{\left (6 \, d x + 6 \, c\right )} - 3 \, d e^{\left (4 \, d x + 4 \, c\right )} + 3 \, d e^{\left (2 \, d x + 2 \, c\right )} - d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(I*sinh(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

2/5*(2*sqrt(1/2)*(3*e^(6*d*x + 6*c) - 8*e^(4*d*x + 4*c) + e^(2*d*x + 2*c))*sqrt(I*e^(2*d*x + 2*c) - I)*e^(-1/2
*d*x - 1/2*c) + 3*(sqrt(2)*sqrt(I)*e^(6*d*x + 6*c) - 3*sqrt(2)*sqrt(I)*e^(4*d*x + 4*c) + 3*sqrt(2)*sqrt(I)*e^(
2*d*x + 2*c) - sqrt(2)*sqrt(I))*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, e^(d*x + c))))/(d*e^(6*d*x + 6
*c) - 3*d*e^(4*d*x + 4*c) + 3*d*e^(2*d*x + 2*c) - d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (i \sinh {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(I*sinh(d*x+c))**(7/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(I*sinh(d*x+c))^(7/2),x, algorithm="giac")

[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 \begin {gather*} \int \frac {1}{{\left (\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)*1i)^(7/2),x)

[Out]

int(1/(sinh(c + d*x)*1i)^(7/2), x)

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