\(\int \csc ^{\frac {7}{2}}(a+b x) \, dx\) [9]

Optimal result

Integrand size = 10, antiderivative size = 90 \[ \int \csc ^{\frac {7}{2}}(a+b x) \, dx=-\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {6 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{5 b} \]

[Out]

-2/5*cos(b*x+a)*csc(b*x+a)^(5/2)/b-6/5*cos(b*x+a)*csc(b*x+a)^(1/2)/b+6/5*(sin(1/2*a+1/4*Pi+1/2*b*x)^2)^(1/2)/s
in(1/2*a+1/4*Pi+1/2*b*x)*EllipticE(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 = {3853, 3856, 2719} \[ \int \csc ^{\frac {7}{2}}(a+b x) \, dx=-\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {6 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{5 b} \]

[In]

Int[Csc[a + b*x]^(7/2),x]

[Out]

(-6*Cos[a + b*x]*Sqrt[Csc[a + b*x]])/(5*b) - (2*Cos[a + b*x]*Csc[a + b*x]^(5/2))/(5*b) - (6*Sqrt[Csc[a + b*x]]
*EllipticE[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(5*b)

Rule 2719

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]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[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) \csc ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {3}{5} \int \csc ^{\frac {3}{2}}(a+b x) \, dx \\ & = -\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {3}{5} \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx \\ & = -\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {1}{5} \left (3 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx \\ & = -\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {6 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{5 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.70 \[ \int \csc ^{\frac {7}{2}}(a+b x) \, dx=\frac {\csc ^{\frac {5}{2}}(a+b x) \left (-7 \cos (a+b x)+3 \cos (3 (a+b x))+12 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sin ^{\frac {5}{2}}(a+b x)\right )}{10 b} \]

[In]

Integrate[Csc[a + b*x]^(7/2),x]

[Out]

(Csc[a + b*x]^(5/2)*(-7*Cos[a + b*x] + 3*Cos[3*(a + b*x)] + 12*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x
]^(5/2)))/(10*b)

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.78

method	result	size
default	\(\frac {6 \sqrt {\sin \left (x b +a \right )+1}\, \sqrt {-2 \sin \left (x b +a \right )+2}\, \sqrt {-\sin \left (x b +a \right )}\, \sin \left (x b +a \right )^{2} \operatorname {EllipticE}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {\sin \left (x b +a \right )+1}\, \sqrt {-2 \sin \left (x b +a \right )+2}\, \sqrt {-\sin \left (x b +a \right )}\, \sin \left (x b +a \right )^{2} \operatorname {EllipticF}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )+6 \sin \left (x b +a \right )^{4}-4 \sin \left (x b +a \right )^{2}-2}{5 \sin \left (x b +a \right )^{\frac {5}{2}} \cos \left (x b +a \right ) b}\)	\(160\)

[In]

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

[Out]

1/5/sin(b*x+a)^(5/2)*(6*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*sin(b*x+a)^2*Elliptic
E((sin(b*x+a)+1)^(1/2),1/2*2^(1/2))-3*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*sin(b*x
+a)^2*EllipticF((sin(b*x+a)+1)^(1/2),1/2*2^(1/2))+6*sin(b*x+a)^4-4*sin(b*x+a)^2-2)/cos(b*x+a)/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 = 120, normalized size of antiderivative = 1.33 \[ \int \csc ^{\frac {7}{2}}(a+b x) \, dx=-\frac {3 \, \sqrt {2 i} {\left (\cos \left (b x + a\right )^{2} - 1\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 \, \sqrt {-2 i} {\left (\cos \left (b x + a\right )^{2} - 1\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + \frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{3} - 4 \, \cos \left (b x + a\right )\right )}}{\sqrt {\sin \left (b x + a\right )}}}{5 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(csc(a + b*x)**(7/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(csc(b*x + a)^(7/2), x)

Giac [F]

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

[In]

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

[Out]

integrate(csc(b*x + a)^(7/2), x)

Mupad [F(-1)]

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

[In]

int((1/sin(a + b*x))^(7/2),x)

[Out]

int((1/sin(a + b*x))^(7/2), x)