\(\int \frac {1}{(c \csc (a+b x))^{7/2}} \, dx\) [24]

Optimal result

Integrand size = 12, antiderivative size = 105 \[ \int \frac {1}{(c \csc (a+b x))^{7/2}} \, dx=-\frac {2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}-\frac {10 \cos (a+b x)}{21 b c^3 \sqrt {c \csc (a+b x)}}+\frac {10 \sqrt {c \csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{21 b c^4} \]

[Out]

-2/7*cos(b*x+a)/b/c/(c*csc(b*x+a))^(5/2)-10/21*cos(b*x+a)/b/c^3/(c*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))*(c*csc(b*x+a))^(1/2)*s
in(b*x+a)^(1/2)/b/c^4

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3854, 3856, 2720} \[ \int \frac {1}{(c \csc (a+b x))^{7/2}} \, dx=\frac {10 \sqrt {\sin (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right ) \sqrt {c \csc (a+b x)}}{21 b c^4}-\frac {10 \cos (a+b x)}{21 b c^3 \sqrt {c \csc (a+b x)}}-\frac {2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}} \]

[In]

Int[(c*Csc[a + b*x])^(-7/2),x]

[Out]

(-2*Cos[a + b*x])/(7*b*c*(c*Csc[a + b*x])^(5/2)) - (10*Cos[a + b*x])/(21*b*c^3*Sqrt[c*Csc[a + b*x]]) + (10*Sqr
t[c*Csc[a + b*x]]*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(21*b*c^4)

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 c (c \csc (a+b x))^{5/2}}+\frac {5 \int \frac {1}{(c \csc (a+b x))^{3/2}} \, dx}{7 c^2} \\ & = -\frac {2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}-\frac {10 \cos (a+b x)}{21 b c^3 \sqrt {c \csc (a+b x)}}+\frac {5 \int \sqrt {c \csc (a+b x)} \, dx}{21 c^4} \\ & = -\frac {2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}-\frac {10 \cos (a+b x)}{21 b c^3 \sqrt {c \csc (a+b x)}}+\frac {\left (5 \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{21 c^4} \\ & = -\frac {2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}-\frac {10 \cos (a+b x)}{21 b c^3 \sqrt {c \csc (a+b x)}}+\frac {10 \sqrt {c \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 c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(c \csc (a+b x))^{7/2}} \, dx=-\frac {\sqrt {c \csc (a+b x)} \left (40 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}+26 \sin (2 (a+b x))-3 \sin (4 (a+b x))\right )}{84 b c^4} \]

[In]

Integrate[(c*Csc[a + b*x])^(-7/2),x]

[Out]

-1/84*(Sqrt[c*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*c^4)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.86 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.70

method	result	size
default	\(\frac {\sin \left (x b +a \right )^{3} \left (5 i \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x b +a \right )+3 \sqrt {2}\, \cos \left (x b +a \right )^{3} \sin \left (x b +a \right )+5 i \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )-8 \sqrt {2}\, \cos \left (x b +a \right ) \sin \left (x b +a \right )\right ) \sqrt {2}}{21 b \,c^{3} \sqrt {c \csc \left (x b +a \right )}\, \left (\cos \left (x b +a \right )-1\right )^{2} \left (1+\cos \left (x b +a \right )\right )^{2}}\)	\(283\)

[In]

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

[Out]

1/21/b*sin(b*x+a)^3*(5*I*(I*(-I-cot(b*x+a)+csc(b*x+a)))^(1/2)*(I*(csc(b*x+a)-cot(b*x+a)))^(1/2)*EllipticF((-I*
(I-cot(b*x+a)+csc(b*x+a)))^(1/2),1/2*2^(1/2))*(-I*(I-cot(b*x+a)+csc(b*x+a)))^(1/2)*cos(b*x+a)+3*2^(1/2)*cos(b*
x+a)^3*sin(b*x+a)+5*I*(I*(-I-cot(b*x+a)+csc(b*x+a)))^(1/2)*(I*(csc(b*x+a)-cot(b*x+a)))^(1/2)*EllipticF((-I*(I-
cot(b*x+a)+csc(b*x+a)))^(1/2),1/2*2^(1/2))*(-I*(I-cot(b*x+a)+csc(b*x+a)))^(1/2)-8*2^(1/2)*cos(b*x+a)*sin(b*x+a
))/c^3/(c*csc(b*x+a))^(1/2)/(cos(b*x+a)-1)^2/(1+cos(b*x+a))^2*2^(1/2)

Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(c \csc (a+b x))^{7/2}} \, dx=\frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{3} - 8 \, \cos \left (b x + a\right )\right )} \sqrt {\frac {c}{\sin \left (b x + a\right )}} \sin \left (b x + a\right ) - 5 i \, \sqrt {2 i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 i \, \sqrt {-2 i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{21 \, b c^{4}} \]

[In]

integrate(1/(c*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(c/sin(b*x + a))*sin(b*x + a) - 5*I*sqrt(2*I*c)*weierstrassPIn
verse(4, 0, cos(b*x + a) + I*sin(b*x + a)) + 5*I*sqrt(-2*I*c)*weierstrassPInverse(4, 0, cos(b*x + a) - I*sin(b
*x + a)))/(b*c^4)

Sympy [F]

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

[In]

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

[Out]

Integral((c*csc(a + b*x))**(-7/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((c*csc(b*x + a))^(-7/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((c*csc(b*x + a))^(-7/2), x)

Mupad [F(-1)]

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

[In]

int(1/(c/sin(a + b*x))^(7/2),x)

[Out]

int(1/(c/sin(a + b*x))^(7/2), x)