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

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

[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]])

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Rubi [A]  time = 0.0333186, 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 = {2636, 2639} \[ \frac{6 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{5 d}+\frac{6 i \cosh (c+d x)}{5 d \sqrt{i \sinh (c+d x)}}+\frac{2 i \cosh (c+d x)}{5 d (i \sinh (c+d x))^{5/2}} \]

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 2636

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 2639

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

Mathematica [A]  time = 0.130367, size = 80, normalized size = 0.88 \[ -\frac{2 i \left (-3 \cosh (c+d x)+\coth (c+d x) \text{csch}(c+d x)+3 \sqrt{i \sinh (c+d x)} E\left (\left .\frac{1}{4} (-2 i c-2 i d x+\pi )\right |2\right )\right )}{5 d \sqrt{i \sinh (c+d x)}} \]

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 = 0.054, size = 204, normalized size = 2.2 \begin{align*}{\frac{-{\frac{i}{5}}}{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}\cosh \left ( dx+c \right ) d} \left ( 6\,\sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) }\sqrt{2}\sqrt{-i \left ( i-\sinh \left ( dx+c \right ) \right ) }\sqrt{i\sinh \left ( dx+c \right ) } \left ( \sinh \left ( dx+c \right ) \right ) ^{2}{\it EllipticE} \left ( \sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) },1/2\,\sqrt{2} \right ) -3\,\sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) }\sqrt{2}\sqrt{-i \left ( i-\sinh \left ( dx+c \right ) \right ) }\sqrt{i\sinh \left ( dx+c \right ) } \left ( \sinh \left ( dx+c \right ) \right ) ^{2}{\it EllipticF} \left ( \sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) },1/2\,\sqrt{2} \right ) -6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}-4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}+2 \right ){\frac{1}{\sqrt{i\sinh \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{4 \, \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 )} + 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 )}{\rm integral}\left (-\frac{6 \, \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 )}}{5 \,{\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )}}, x\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(4*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) + 5*(d*e^(6*d*x + 6*c) - 3*d*e^(4*d*x + 4*c) + 3*d*e^(2*d*x + 2*c) - d)*integral(-6/5*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))/(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(-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(1/(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 \frac{1}{\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(1/(I*sinh(d*x+c))^(7/2),x, algorithm="giac")

[Out]

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