\(\int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx\) [16]

Optimal result

Integrand size = 10, antiderivative size = 90 \[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}-\frac {10 \cos (a+b x)}{21 b \sqrt {\csc (a+b x)}}+\frac {10 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{21 b} \]

[Out]

-2/7*cos(b*x+a)/b/csc(b*x+a)^(5/2)-10/21*cos(b*x+a)/b/csc(b*x+a)^(1/2)-10/21*(sin(1/2*a+1/4*Pi+1/2*b*x)^2)^(1/
2)/sin(1/2*a+1/4*Pi+1/2*b*x)*EllipticF(cos(1/2*a+1/4*Pi+1/2*b*x),2^(1/2))*csc(b*x+a)^(1/2)*sin(b*x+a)^(1/2)/b

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3854, 3856, 2720} \[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}-\frac {10 \cos (a+b x)}{21 b \sqrt {\csc (a+b x)}}+\frac {10 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{21 b} \]

[In]

Int[Csc[a + b*x]^(-7/2),x]

[Out]

(-2*Cos[a + b*x])/(7*b*Csc[a + b*x]^(5/2)) - (10*Cos[a + b*x])/(21*b*Sqrt[Csc[a + b*x]]) + (10*Sqrt[Csc[a + b*
x]]*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(21*b)

Rule 2720

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]

Rule 3854

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

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}+\frac {5}{7} \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx \\ & = -\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}-\frac {10 \cos (a+b x)}{21 b \sqrt {\csc (a+b x)}}+\frac {5}{21} \int \sqrt {\csc (a+b x)} \, dx \\ & = -\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}-\frac {10 \cos (a+b x)}{21 b \sqrt {\csc (a+b x)}}+\frac {1}{21} \left (5 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx \\ & = -\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}-\frac {10 \cos (a+b x)}{21 b \sqrt {\csc (a+b x)}}+\frac {10 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{21 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {\sqrt {\csc (a+b x)} \left (40 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}+26 \sin (2 (a+b x))-3 \sin (4 (a+b x))\right )}{84 b} \]

[In]

Integrate[Csc[a + b*x]^(-7/2),x]

[Out]

-1/84*(Sqrt[Csc[a + b*x]]*(40*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + 26*Sin[2*(a + b*x)] - 3
*Sin[4*(a + b*x)]))/b

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.16

method	result	size
default	\(\frac {\frac {2 \cos \left (x b +a \right )^{4} \sin \left (x b +a \right )}{7}+\frac {5 \sqrt {\sin \left (x b +a \right )+1}\, \sqrt {-2 \sin \left (x b +a \right )+2}\, \sqrt {-\sin \left (x b +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{21}-\frac {16 \cos \left (x b +a \right )^{2} \sin \left (x b +a \right )}{21}}{\cos \left (x b +a \right ) \sqrt {\sin \left (x b +a \right )}\, b}\)	\(104\)

[In]

int(1/csc(b*x+a)^(7/2),x,method=_RETURNVERBOSE)

[Out]

(2/7*cos(b*x+a)^4*sin(b*x+a)+5/21*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*EllipticF((
sin(b*x+a)+1)^(1/2),1/2*2^(1/2))-16/21*cos(b*x+a)^2*sin(b*x+a))/cos(b*x+a)/sin(b*x+a)^(1/2)/b

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=\frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{3} - 8 \, \cos \left (b x + a\right )\right )} \sqrt {\sin \left (b x + a\right )} - 5 i \, \sqrt {2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 i \, \sqrt {-2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{21 \, b} \]

[In]

integrate(1/csc(b*x+a)^(7/2),x, algorithm="fricas")

[Out]

1/21*(2*(3*cos(b*x + a)^3 - 8*cos(b*x + a))*sqrt(sin(b*x + a)) - 5*I*sqrt(2*I)*weierstrassPInverse(4, 0, cos(b
*x + a) + I*sin(b*x + a)) + 5*I*sqrt(-2*I)*weierstrassPInverse(4, 0, cos(b*x + a) - I*sin(b*x + a)))/b

Sympy [F]

\[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=\int \frac {1}{\csc ^{\frac {7}{2}}{\left (a + b x \right )}}\, dx \]

[In]

integrate(1/csc(b*x+a)**(7/2),x)

[Out]

Integral(csc(a + b*x)**(-7/2), x)

Maxima [F]

\[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {1}{\csc \left (b x + a\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/csc(b*x+a)^(7/2),x, algorithm="maxima")

[Out]

integrate(csc(b*x + a)^(-7/2), x)

Giac [F]

\[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {1}{\csc \left (b x + a\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/csc(b*x+a)^(7/2),x, algorithm="giac")

[Out]

integrate(csc(b*x + a)^(-7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx=\int \frac {1}{{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{7/2}} \,d x \]

[In]

int(1/(1/sin(a + b*x))^(7/2),x)

[Out]

int(1/(1/sin(a + b*x))^(7/2), x)