\(\int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx\) [241]

Optimal result

Integrand size = 21, antiderivative size = 126 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}-\frac {21 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b d^4 \sqrt {\cos (a+b x)}}+\frac {7 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}+\frac {21 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}} \] Output:

-csc(b*x+a)/b/d/(d*cos(b*x+a))^(5/2)-21/5*(d*cos(b*x+a))^(1/2)*EllipticE(s 
in(1/2*a+1/2*b*x),2^(1/2))/b/d^4/cos(b*x+a)^(1/2)+7/5*sin(b*x+a)/b/d/(d*co 
s(b*x+a))^(5/2)+21/5*sin(b*x+a)/b/d^3/(d*cos(b*x+a))^(1/2)
 

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.65 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {-5 \cos (a+b x) \cot (a+b x)-21 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+16 \sin (a+b x)+2 \sec (a+b x) \tan (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}} \] Input:

Integrate[Csc[a + b*x]^2/(d*Cos[a + b*x])^(7/2),x]
 

Output:

(-5*Cos[a + b*x]*Cot[a + b*x] - 21*Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/ 
2, 2] + 16*Sin[a + b*x] + 2*Sec[a + b*x]*Tan[a + b*x])/(5*b*d^3*Sqrt[d*Cos 
[a + b*x]])
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3050, 3042, 3116, 3042, 3116, 3042, 3121, 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.

\(\displaystyle \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sin (a+b x)^2 (d \cos (a+b x))^{7/2}}dx\)

\(\Big \downarrow \) 3050

\(\displaystyle \frac {7}{2} \int \frac {1}{(d \cos (a+b x))^{7/2}}dx-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{2} \int \frac {1}{\left (d \sin \left (a+b x+\frac {\pi }{2}\right )\right )^{7/2}}dx-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3116

\(\displaystyle \frac {7}{2} \left (\frac {3 \int \frac {1}{(d \cos (a+b x))^{3/2}}dx}{5 d^2}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{2} \left (\frac {3 \int \frac {1}{\left (d \sin \left (a+b x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{5 d^2}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3116

\(\displaystyle \frac {7}{2} \left (\frac {3 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\int \sqrt {d \cos (a+b x)}dx}{d^2}\right )}{5 d^2}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{2} \left (\frac {3 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\int \sqrt {d \sin \left (a+b x+\frac {\pi }{2}\right )}dx}{d^2}\right )}{5 d^2}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3121

\(\displaystyle \frac {7}{2} \left (\frac {3 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\sqrt {d \cos (a+b x)} \int \sqrt {\cos (a+b x)}dx}{d^2 \sqrt {\cos (a+b x)}}\right )}{5 d^2}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{2} \left (\frac {3 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\sqrt {d \cos (a+b x)} \int \sqrt {\sin \left (a+b x+\frac {\pi }{2}\right )}dx}{d^2 \sqrt {\cos (a+b x)}}\right )}{5 d^2}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {7}{2} \left (\frac {3 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{b d^2 \sqrt {\cos (a+b x)}}\right )}{5 d^2}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\)

Input:

Int[Csc[a + b*x]^2/(d*Cos[a + b*x])^(7/2),x]
 

Output:

-(Csc[a + b*x]/(b*d*(d*Cos[a + b*x])^(5/2))) + (7*((2*Sin[a + b*x])/(5*b*d 
*(d*Cos[a + b*x])^(5/2)) + (3*((-2*Sqrt[d*Cos[a + b*x]]*EllipticE[(a + b*x 
)/2, 2])/(b*d^2*Sqrt[Cos[a + b*x]]) + (2*Sin[a + b*x])/(b*d*Sqrt[d*Cos[a + 
 b*x]])))/(5*d^2)))/2
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3050

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m + 1)/(a 
*b*f*(m + 1))), x] + Simp[(m + n + 2)/(a^2*(m + 1))   Int[(b*Cos[e + f*x])^ 
n*(a*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, - 
1] && IntegersQ[2*m, 2*n]
 

rule 3116

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]
 

rule 3119

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 3121

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs. \(2(112)=224\).

Time = 6.37 (sec) , antiderivative size = 408, normalized size of antiderivative = 3.24

method	result	size
default	\(-\frac {\sqrt {d \left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \left (168 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \operatorname {EllipticE}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+336 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{8}-168 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \operatorname {EllipticE}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-672 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}+42 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}+448 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}-112 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+5\right ) \left (-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4} d +\sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d \right )^{\frac {3}{2}}}{10 d^{5} \left (2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{5} \left (8 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}-12 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+6 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \sqrt {d \left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}\, b}\)	\(408\)

Input:

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

Output:

-1/10*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)/d^5/(2*sin 
(1/2*b*x+1/2*a)^2-1)/cos(1/2*b*x+1/2*a)/sin(1/2*b*x+1/2*a)^5/(8*sin(1/2*b* 
x+1/2*a)^6-12*sin(1/2*b*x+1/2*a)^4+6*sin(1/2*b*x+1/2*a)^2-1)*(168*cos(1/2* 
b*x+1/2*a)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*(sin(1/2*b*x+1/2*a)^2)^(1 
/2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*sin(1/2*b*x+1/2*a)^4+336*sin(1/2*b*x+ 
1/2*a)^8-168*cos(1/2*b*x+1/2*a)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*(sin 
(1/2*b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*sin(1/2*b*x+1/2* 
a)^2-672*sin(1/2*b*x+1/2*a)^6+42*cos(1/2*b*x+1/2*a)*(sin(1/2*b*x+1/2*a)^2) 
^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*(2*sin(1/2*b*x+1/2*a)^2-1)^(1 
/2)+448*sin(1/2*b*x+1/2*a)^4-112*sin(1/2*b*x+1/2*a)^2+5)*(-2*sin(1/2*b*x+1 
/2*a)^4*d+sin(1/2*b*x+1/2*a)^2*d)^(3/2)/(d*(2*cos(1/2*b*x+1/2*a)^2-1))^(1/ 
2)/b
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.15 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {-21 i \, \sqrt {\frac {1}{2}} \sqrt {d} \cos \left (b x + a\right )^{3} \sin \left (b x + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 21 i \, \sqrt {\frac {1}{2}} \sqrt {d} \cos \left (b x + a\right )^{3} \sin \left (b x + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) - {\left (21 \, \cos \left (b x + a\right )^{4} - 14 \, \cos \left (b x + a\right )^{2} - 2\right )} \sqrt {d \cos \left (b x + a\right )}}{5 \, b d^{4} \cos \left (b x + a\right )^{3} \sin \left (b x + a\right )} \] Input:

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

Output:

1/5*(-21*I*sqrt(1/2)*sqrt(d)*cos(b*x + a)^3*sin(b*x + a)*weierstrassZeta(- 
4, 0, weierstrassPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a))) + 21*I*sq 
rt(1/2)*sqrt(d)*cos(b*x + a)^3*sin(b*x + a)*weierstrassZeta(-4, 0, weierst 
rassPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a))) - (21*cos(b*x + a)^4 - 
 14*cos(b*x + a)^2 - 2)*sqrt(d*cos(b*x + a)))/(b*d^4*cos(b*x + a)^3*sin(b* 
x + a))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int(1/(sin(a + b*x)^2*(d*cos(a + b*x))^(7/2)),x)
 

Output:

int(1/(sin(a + b*x)^2*(d*cos(a + b*x))^(7/2)), x)
 

Reduce [F]

\[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\cos \left (b x +a \right )}\, \csc \left (b x +a \right )^{2}}{\cos \left (b x +a \right )^{4}}d x \right )}{d^{4}} \] Input:

int(csc(b*x+a)^2/(d*cos(b*x+a))^(7/2),x)
 

Output:

(sqrt(d)*int((sqrt(cos(a + b*x))*csc(a + b*x)**2)/cos(a + b*x)**4,x))/d**4