Integrand size = 12, antiderivative size = 77 \[ \int \frac {1}{(c \sin (a+b x))^{5/2}} \, dx=-\frac {2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{3 b c^2 \sqrt {c \sin (a+b x)}} \] Output:
-2/3*cos(b*x+a)/b/c/(c*sin(b*x+a))^(3/2)+2/3*InverseJacobiAM(1/2*a-1/4*Pi+ 1/2*b*x,2^(1/2))*sin(b*x+a)^(1/2)/b/c^2/(c*sin(b*x+a))^(1/2)
Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(c \sin (a+b x))^{5/2}} \, dx=-\frac {2 \left (\cos (a+b x)+\operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sin ^{\frac {3}{2}}(a+b x)\right )}{3 b c (c \sin (a+b x))^{3/2}} \] Input:
Integrate[(c*Sin[a + b*x])^(-5/2),x]
Output:
(-2*(Cos[a + b*x] + EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(3/2) ))/(3*b*c*(c*Sin[a + b*x])^(3/2))
Time = 0.52 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3116, 3042, 3121, 3042, 3120}
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.
\(\displaystyle \int \frac {1}{(c \sin (a+b x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(c \sin (a+b x))^{5/2}}dx\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {c \sin (a+b x)}}dx}{3 c^2}-\frac {2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {c \sin (a+b x)}}dx}{3 c^2}-\frac {2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {\sqrt {\sin (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}}dx}{3 c^2 \sqrt {c \sin (a+b x)}}-\frac {2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\sin (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}}dx}{3 c^2 \sqrt {c \sin (a+b x)}}-\frac {2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 \sqrt {\sin (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{3 b c^2 \sqrt {c \sin (a+b x)}}-\frac {2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}}\) |
Input:
Int[(c*Sin[a + b*x])^(-5/2),x]
Output:
(-2*Cos[a + b*x])/(3*b*c*(c*Sin[a + b*x])^(3/2)) + (2*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(3*b*c^2*Sqrt[c*Sin[a + b*x]])
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[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]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Time = 2.97 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.36
method | result | size |
default | \(-\frac {\sqrt {1-\sin \left (b x +a \right )}\, \sqrt {2+2 \sin \left (b x +a \right )}\, \sin \left (b x +a \right )^{\frac {5}{2}} \operatorname {EllipticF}\left (\sqrt {1-\sin \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (b x +a \right )^{3}+2 \sin \left (b x +a \right )}{3 c^{2} \sin \left (b x +a \right )^{2} \cos \left (b x +a \right ) \sqrt {c \sin \left (b x +a \right )}\, b}\) | \(105\) |
Input:
int(1/(c*sin(b*x+a))^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/3/c^2*((1-sin(b*x+a))^(1/2)*(2+2*sin(b*x+a))^(1/2)*sin(b*x+a)^(5/2)*Ell ipticF((1-sin(b*x+a))^(1/2),1/2*2^(1/2))-2*sin(b*x+a)^3+2*sin(b*x+a))/sin( b*x+a)^2/cos(b*x+a)/(c*sin(b*x+a))^(1/2)/b
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.43 \[ \int \frac {1}{(c \sin (a+b x))^{5/2}} \, dx=\frac {2 \, {\left ({\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt {-\frac {1}{2} i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + {\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt {\frac {1}{2} i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + \sqrt {c \sin \left (b x + a\right )} \cos \left (b x + a\right )\right )}}{3 \, {\left (b c^{3} \cos \left (b x + a\right )^{2} - b c^{3}\right )}} \] Input:
integrate(1/(c*sin(b*x+a))^(5/2),x, algorithm="fricas")
Output:
2/3*((cos(b*x + a)^2 - 1)*sqrt(-1/2*I*c)*weierstrassPInverse(4, 0, cos(b*x + a) + I*sin(b*x + a)) + (cos(b*x + a)^2 - 1)*sqrt(1/2*I*c)*weierstrassPI nverse(4, 0, cos(b*x + a) - I*sin(b*x + a)) + sqrt(c*sin(b*x + a))*cos(b*x + a))/(b*c^3*cos(b*x + a)^2 - b*c^3)
\[ \int \frac {1}{(c \sin (a+b x))^{5/2}} \, dx=\int \frac {1}{\left (c \sin {\left (a + b x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(c*sin(b*x+a))**(5/2),x)
Output:
Integral((c*sin(a + b*x))**(-5/2), x)
\[ \int \frac {1}{(c \sin (a+b x))^{5/2}} \, dx=\int { \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(c*sin(b*x+a))^(5/2),x, algorithm="maxima")
Output:
integrate((c*sin(b*x + a))^(-5/2), x)
\[ \int \frac {1}{(c \sin (a+b x))^{5/2}} \, dx=\int { \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(c*sin(b*x+a))^(5/2),x, algorithm="giac")
Output:
integrate((c*sin(b*x + a))^(-5/2), x)
Timed out. \[ \int \frac {1}{(c \sin (a+b x))^{5/2}} \, dx=\int \frac {1}{{\left (c\,\sin \left (a+b\,x\right )\right )}^{5/2}} \,d x \] Input:
int(1/(c*sin(a + b*x))^(5/2),x)
Output:
int(1/(c*sin(a + b*x))^(5/2), x)
\[ \int \frac {1}{(c \sin (a+b x))^{5/2}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {\sin \left (b x +a \right )}}{\sin \left (b x +a \right )^{3}}d x \right )}{c^{3}} \] Input:
int(1/(c*sin(b*x+a))^(5/2),x)
Output:
(sqrt(c)*int(sqrt(sin(a + b*x))/sin(a + b*x)**3,x))/c**3