\(\int \frac {\cot ^6(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx\) [408]

Optimal result

Integrand size = 26, antiderivative size = 96 \[ \int \frac {\cot ^6(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\cot (e+f x)}{f \sqrt {a \cos ^2(e+f x)}}+\frac {2 \cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^4(e+f x)}{5 f \sqrt {a \cos ^2(e+f x)}} \] Output:

-cot(f*x+e)/f/(a*cos(f*x+e)^2)^(1/2)+2/3*cot(f*x+e)*csc(f*x+e)^2/f/(a*cos( 
f*x+e)^2)^(1/2)-1/5*cot(f*x+e)*csc(f*x+e)^4/f/(a*cos(f*x+e)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.51 \[ \int \frac {\cot ^6(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\cot (e+f x) \left (15-10 \csc ^2(e+f x)+3 \csc ^4(e+f x)\right )}{15 f \sqrt {a \cos ^2(e+f x)}} \] Input:

Integrate[Cot[e + f*x]^6/Sqrt[a - a*Sin[e + f*x]^2],x]
 

Output:

-1/15*(Cot[e + f*x]*(15 - 10*Csc[e + f*x]^2 + 3*Csc[e + f*x]^4))/(f*Sqrt[a 
*Cos[e + f*x]^2])
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.58, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3042, 3655, 3042, 3686, 3042, 25, 3086, 210, 2009}

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.

\(\displaystyle \int \frac {\cot ^6(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (e+f x)^6 \sqrt {a-a \sin (e+f x)^2}}dx\)

\(\Big \downarrow \) 3655

\(\displaystyle \int \frac {\cot ^6(e+f x)}{\sqrt {a \cos ^2(e+f x)}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\tan \left (e+f x+\frac {\pi }{2}\right )^6}{\sqrt {a \sin \left (e+f x+\frac {\pi }{2}\right )^2}}dx\)

\(\Big \downarrow \) 3686

\(\displaystyle \frac {\cos (e+f x) \int \cot ^5(e+f x) \csc (e+f x)dx}{\sqrt {a \cos ^2(e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cos (e+f x) \int -\sec \left (e+f x-\frac {\pi }{2}\right ) \tan \left (e+f x-\frac {\pi }{2}\right )^5dx}{\sqrt {a \cos ^2(e+f x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\cos (e+f x) \int \sec \left (\frac {1}{2} (2 e-\pi )+f x\right ) \tan \left (\frac {1}{2} (2 e-\pi )+f x\right )^5dx}{\sqrt {a \cos ^2(e+f x)}}\)

\(\Big \downarrow \) 3086

\(\displaystyle -\frac {\cos (e+f x) \int \left (\csc ^2(e+f x)-1\right )^2d\csc (e+f x)}{f \sqrt {a \cos ^2(e+f x)}}\)

\(\Big \downarrow \) 210

\(\displaystyle -\frac {\cos (e+f x) \int \left (\csc ^4(e+f x)-2 \csc ^2(e+f x)+1\right )d\csc (e+f x)}{f \sqrt {a \cos ^2(e+f x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\cos (e+f x) \left (\frac {1}{5} \csc ^5(e+f x)-\frac {2}{3} \csc ^3(e+f x)+\csc (e+f x)\right )}{f \sqrt {a \cos ^2(e+f x)}}\)

Input:

Int[Cot[e + f*x]^6/Sqrt[a - a*Sin[e + f*x]^2],x]
 

Output:

-((Cos[e + f*x]*(Csc[e + f*x] - (2*Csc[e + f*x]^3)/3 + Csc[e + f*x]^5/5))/ 
(f*Sqrt[a*Cos[e + f*x]^2]))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 210

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 
)^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3086

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_.), x_Symbol] :> Simp[a/f   Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 
), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 
] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])
 

rule 3655

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A 
ctivateTrig[u*(a*cos[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ 
[a + b, 0]
 

rule 3686

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff 
= FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ 
n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p]))   Int[ActivateTrig[u]*(Si 
n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] 
 && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / 
; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
 

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.77

method	result	size
default	\(-\frac {\cos \left (f x +e \right ) \left (15 \cos \left (f x +e \right )^{4}-20 \cos \left (f x +e \right )^{2}+8\right )}{15 \left (\cos \left (f x +e \right )+1\right )^{2} \left (\cos \left (f x +e \right )-1\right )^{2} \sin \left (f x +e \right ) \sqrt {a \cos \left (f x +e \right )^{2}}\, f}\)	\(74\)
risch	\(-\frac {2 i \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left (15 \,{\mathrm e}^{8 i \left (f x +e \right )}-20 \,{\mathrm e}^{6 i \left (f x +e \right )}+58 \,{\mathrm e}^{4 i \left (f x +e \right )}-20 \,{\mathrm e}^{2 i \left (f x +e \right )}+15\right )}{15 \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{5}}\)	\(103\)

Input:

int(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/15*cos(f*x+e)*(15*cos(f*x+e)^4-20*cos(f*x+e)^2+8)/(cos(f*x+e)+1)^2/(cos 
(f*x+e)-1)^2/sin(f*x+e)/(a*cos(f*x+e)^2)^(1/2)/f
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.82 \[ \int \frac {\cot ^6(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {{\left (15 \, \cos \left (f x + e\right )^{4} - 20 \, \cos \left (f x + e\right )^{2} + 8\right )} \sqrt {a \cos \left (f x + e\right )^{2}}}{15 \, {\left (a f \cos \left (f x + e\right )^{5} - 2 \, a f \cos \left (f x + e\right )^{3} + a f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \] Input:

integrate(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="fricas")
 

Output:

-1/15*(15*cos(f*x + e)^4 - 20*cos(f*x + e)^2 + 8)*sqrt(a*cos(f*x + e)^2)/( 
(a*f*cos(f*x + e)^5 - 2*a*f*cos(f*x + e)^3 + a*f*cos(f*x + e))*sin(f*x + e 
))
 

Sympy [F]

\[ \int \frac {\cot ^6(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\int \frac {\cot ^{6}{\left (e + f x \right )}}{\sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \] Input:

integrate(cot(f*x+e)**6/(a-a*sin(f*x+e)**2)**(1/2),x)
 

Output:

Integral(cot(e + f*x)**6/sqrt(-a*(sin(e + f*x) - 1)*(sin(e + f*x) + 1)), x 
)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1236 vs. \(2 (86) = 172\).

Time = 0.17 (sec) , antiderivative size = 1236, normalized size of antiderivative = 12.88 \[ \int \frac {\cot ^6(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\text {Too large to display} \] Input:

integrate(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="maxima")
 

Output:

2/15*((15*sin(9*f*x + 9*e) - 20*sin(7*f*x + 7*e) + 58*sin(5*f*x + 5*e) - 2 
0*sin(3*f*x + 3*e) + 15*sin(f*x + e))*cos(10*f*x + 10*e) + 75*(sin(8*f*x + 
 8*e) - 2*sin(6*f*x + 6*e) + 2*sin(4*f*x + 4*e) - sin(2*f*x + 2*e))*cos(9* 
f*x + 9*e) + 5*(20*sin(7*f*x + 7*e) - 58*sin(5*f*x + 5*e) + 20*sin(3*f*x + 
 3*e) - 15*sin(f*x + e))*cos(8*f*x + 8*e) + 100*(2*sin(6*f*x + 6*e) - 2*si 
n(4*f*x + 4*e) + sin(2*f*x + 2*e))*cos(7*f*x + 7*e) + 10*(58*sin(5*f*x + 5 
*e) - 20*sin(3*f*x + 3*e) + 15*sin(f*x + e))*cos(6*f*x + 6*e) + 290*(2*sin 
(4*f*x + 4*e) - sin(2*f*x + 2*e))*cos(5*f*x + 5*e) + 50*(4*sin(3*f*x + 3*e 
) - 3*sin(f*x + e))*cos(4*f*x + 4*e) - (15*cos(9*f*x + 9*e) - 20*cos(7*f*x 
 + 7*e) + 58*cos(5*f*x + 5*e) - 20*cos(3*f*x + 3*e) + 15*cos(f*x + e))*sin 
(10*f*x + 10*e) - 15*(5*cos(8*f*x + 8*e) - 10*cos(6*f*x + 6*e) + 10*cos(4* 
f*x + 4*e) - 5*cos(2*f*x + 2*e) + 1)*sin(9*f*x + 9*e) - 5*(20*cos(7*f*x + 
7*e) - 58*cos(5*f*x + 5*e) + 20*cos(3*f*x + 3*e) - 15*cos(f*x + e))*sin(8* 
f*x + 8*e) - 20*(10*cos(6*f*x + 6*e) - 10*cos(4*f*x + 4*e) + 5*cos(2*f*x + 
 2*e) - 1)*sin(7*f*x + 7*e) - 10*(58*cos(5*f*x + 5*e) - 20*cos(3*f*x + 3*e 
) + 15*cos(f*x + e))*sin(6*f*x + 6*e) - 58*(10*cos(4*f*x + 4*e) - 5*cos(2* 
f*x + 2*e) + 1)*sin(5*f*x + 5*e) - 50*(4*cos(3*f*x + 3*e) - 3*cos(f*x + e) 
)*sin(4*f*x + 4*e) - 20*(5*cos(2*f*x + 2*e) - 1)*sin(3*f*x + 3*e) + 100*co 
s(3*f*x + 3*e)*sin(2*f*x + 2*e) - 75*cos(f*x + e)*sin(2*f*x + 2*e) + 75*co 
s(2*f*x + 2*e)*sin(f*x + e) - 15*sin(f*x + e))*sqrt(a)/((a*cos(10*f*x +...
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\cot ^6(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m operator + Error: 
Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 43.88 (sec) , antiderivative size = 491, normalized size of antiderivative = 5.11 \[ \int \frac {\cot ^6(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx =\text {Too large to display} \] Input:

int(cot(e + f*x)^6/(a - a*sin(e + f*x)^2)^(1/2),x)
 

Output:

- (exp(e*3i + f*x*3i)*(a - a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f* 
x*1i)*1i)/2)^2)^(1/2)*4i)/(a*f*(exp(e*2i + f*x*2i) - 1)*(exp(e*1i + f*x*1i 
) + exp(e*3i + f*x*3i))) - (exp(e*3i + f*x*3i)*(a - a*((exp(- e*1i - f*x*1 
i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2)*32i)/(3*a*f*(exp(e*2i + f*x 
*2i) - 1)^2*(exp(e*1i + f*x*1i) + exp(e*3i + f*x*3i))) - (exp(e*3i + f*x*3 
i)*(a - a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/ 
2)*352i)/(15*a*f*(exp(e*2i + f*x*2i) - 1)^3*(exp(e*1i + f*x*1i) + exp(e*3i 
 + f*x*3i))) - (exp(e*3i + f*x*3i)*(a - a*((exp(- e*1i - f*x*1i)*1i)/2 - ( 
exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2)*128i)/(5*a*f*(exp(e*2i + f*x*2i) - 1)^4 
*(exp(e*1i + f*x*1i) + exp(e*3i + f*x*3i))) - (exp(e*3i + f*x*3i)*(a - a*( 
(exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2)*64i)/(5* 
a*f*(exp(e*2i + f*x*2i) - 1)^5*(exp(e*1i + f*x*1i) + exp(e*3i + f*x*3i)))
 

Reduce [F]

\[ \int \frac {\cot ^6(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\sqrt {a}\, \left (\int \frac {\sqrt {-\sin \left (f x +e \right )^{2}+1}\, \cot \left (f x +e \right )^{6}}{\sin \left (f x +e \right )^{2}-1}d x \right )}{a} \] Input:

int(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(1/2),x)
 

Output:

( - sqrt(a)*int((sqrt( - sin(e + f*x)**2 + 1)*cot(e + f*x)**6)/(sin(e + f* 
x)**2 - 1),x))/a