Integrand size = 14, antiderivative size = 91 \[ \int \frac {1}{(i \sinh (c+d x))^{7/2}} \, dx=\frac {6 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right )}{5 d}+\frac {2 i \cosh (c+d x)}{5 d (i \sinh (c+d x))^{5/2}}+\frac {6 i \cosh (c+d x)}{5 d \sqrt {i \sinh (c+d x)}} \]
-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)
Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(i \sinh (c+d x))^{7/2}} \, dx=-\frac {2 i \left (-3 \cosh (c+d x)+\coth (c+d x) \text {csch}(c+d x)+3 E\left (\left .\frac {1}{4} (-2 i c+\pi -2 i d x)\right |2\right ) \sqrt {i \sinh (c+d x)}\right )}{5 d \sqrt {i \sinh (c+d x)}} \]
(((-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]*Sqrt[I*Sinh[c + d*x]]))/(d*Sqrt[I*Sinh[c + d*x]])
Time = 0.31 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3116, 3042, 3116, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(i \sinh (c+d x))^{7/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (i c+i d x)^{7/2}}dx\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle \frac {3}{5} \int \frac {1}{(i \sinh (c+d x))^{3/2}}dx+\frac {2 i \cosh (c+d x)}{5 d (i \sinh (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{5} \int \frac {1}{\sin (i c+i d x)^{3/2}}dx+\frac {2 i \cosh (c+d x)}{5 d (i \sinh (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle \frac {3}{5} \left (\frac {2 i \cosh (c+d x)}{d \sqrt {i \sinh (c+d x)}}-\int \sqrt {i \sinh (c+d x)}dx\right )+\frac {2 i \cosh (c+d x)}{5 d (i \sinh (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{5} \left (\frac {2 i \cosh (c+d x)}{d \sqrt {i \sinh (c+d x)}}-\int \sqrt {\sin (i c+i d x)}dx\right )+\frac {2 i \cosh (c+d x)}{5 d (i \sinh (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 i \cosh (c+d x)}{5 d (i \sinh (c+d x))^{5/2}}+\frac {3}{5} \left (\frac {2 i E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{d}+\frac {2 i \cosh (c+d x)}{d \sqrt {i \sinh (c+d x)}}\right )\) |
(3*(((2*I)*EllipticE[(I*c - Pi/2 + I*d*x)/2, 2])/d + ((2*I)*Cosh[c + d*x]) /(d*Sqrt[I*Sinh[c + d*x]])))/5 + (((2*I)/5)*Cosh[c + d*x])/(d*(I*Sinh[c + d*x])^(5/2))
3.1.30.3.1 Defintions of rubi rules used
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] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
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]
Time = 0.98 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.24
method | result | size |
default | \(-\frac {i \left (6 \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (i-\sinh \left (d x +c \right )\right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \sinh \left (d x +c \right )^{2} \operatorname {EllipticE}\left (\sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (i-\sinh \left (d x +c \right )\right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \sinh \left (d x +c \right )^{2} \operatorname {EllipticF}\left (\sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-6 \sinh \left (d x +c \right )^{4}-4 \sinh \left (d x +c \right )^{2}+2\right )}{5 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right ) \sqrt {i \sinh \left (d x +c \right )}\, d}\) | \(204\) |
-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)*sinh(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
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.96 \[ \int \frac {1}{(i \sinh (c+d x))^{7/2}} \, dx=\frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} {\left (3 \, e^{\left (6 \, d x + 6 \, c\right )} - 8 \, e^{\left (4 \, d x + 4 \, c\right )} + e^{\left (2 \, d x + 2 \, c\right )}\right )} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} + 3 \, {\left (\sqrt {2} \sqrt {i} e^{\left (6 \, d x + 6 \, c\right )} - 3 \, \sqrt {2} \sqrt {i} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, \sqrt {2} \sqrt {i} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {2} \sqrt {i}\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, e^{\left (d x + c\right )}\right )\right )\right )}}{5 \, {\left (d e^{\left (6 \, d x + 6 \, c\right )} - 3 \, d e^{\left (4 \, d x + 4 \, c\right )} + 3 \, d e^{\left (2 \, d x + 2 \, c\right )} - d\right )}} \]
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)
\[ \int \frac {1}{(i \sinh (c+d x))^{7/2}} \, dx=\int \frac {1}{\left (i \sinh {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {1}{(i \sinh (c+d x))^{7/2}} \, dx=\int { \frac {1}{\left (i \, \sinh \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {1}{(i \sinh (c+d x))^{7/2}} \, dx=\int { \frac {1}{\left (i \, \sinh \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(i \sinh (c+d x))^{7/2}} \, dx=\int \frac {1}{{\left (\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2}} \,d x \]