\(\int \frac {\cot ^6(e+f x)}{(a-a \sin ^2(e+f x))^{3/2}} \, dx\) [486]

Optimal result

Integrand size = 26, antiderivative size = 77 \[ \int \frac {\cot ^6(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt {a \cos ^2(e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3255, 3286, 2686, 14} \[ \int \frac {\cot ^6(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt {a \cos ^2(e+f x)}} \]

[In]

Int[Cot[e + f*x]^6/(a - a*Sin[e + f*x]^2)^(3/2),x]

[Out]

(Cot[e + f*x]*Csc[e + f*x]^2)/(3*a*f*Sqrt[a*Cos[e + f*x]^2]) - (Cot[e + f*x]*Csc[e + f*x]^4)/(5*a*f*Sqrt[a*Cos
[e + f*x]^2])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[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 3255

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

Rule 3286

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Sin[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
]])

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^6(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx \\ & = \frac {\cos (e+f x) \int \cot ^3(e+f x) \csc ^3(e+f x) \, dx}{a \sqrt {a \cos ^2(e+f x)}} \\ & = -\frac {\cos (e+f x) \text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{a f \sqrt {a \cos ^2(e+f x)}} \\ & = -\frac {\cos (e+f x) \text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a f \sqrt {a \cos ^2(e+f x)}} \\ & = \frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt {a \cos ^2(e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.53 \[ \int \frac {\cot ^6(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\cot ^3(e+f x) \left (-5+3 \csc ^2(e+f x)\right )}{15 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]

[In]

Integrate[Cot[e + f*x]^6/(a - a*Sin[e + f*x]^2)^(3/2),x]

[Out]

-1/15*(Cot[e + f*x]^3*(-5 + 3*Csc[e + f*x]^2))/(f*(a*Cos[e + f*x]^2)^(3/2))

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.61

method	result	size
default	\(\frac {\cos \left (f x +e \right ) \left (5 \left (\sin ^{2}\left (f x +e \right )\right )-3\right )}{15 a \sin \left (f x +e \right )^{5} \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f}\)	\(47\)
risch	\(-\frac {8 i \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left (5 \,{\mathrm e}^{6 i \left (f x +e \right )}+2 \,{\mathrm e}^{4 i \left (f x +e \right )}+5 \,{\mathrm e}^{2 i \left (f x +e \right )}\right )}{15 \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{5} f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, a}\)	\(94\)

[In]

int(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^6(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left (5 \, \cos \left (f x + e\right )^{2} - 2\right )}}{15 \, {\left (a^{2} f \cos \left (f x + e\right )^{5} - 2 \, a^{2} f \cos \left (f x + e\right )^{3} + a^{2} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \]

[In]

integrate(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

-1/15*sqrt(a*cos(f*x + e)^2)*(5*cos(f*x + e)^2 - 2)/((a^2*f*cos(f*x + e)^5 - 2*a^2*f*cos(f*x + e)^3 + a^2*f*co
s(f*x + e))*sin(f*x + e))

Sympy [F]

\[ \int \frac {\cot ^6(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\cot ^{6}{\left (e + f x \right )}}{\left (- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(cot(f*x+e)**6/(a-a*sin(f*x+e)**2)**(3/2),x)

[Out]

Integral(cot(e + f*x)**6/(-a*(sin(e + f*x) - 1)*(sin(e + f*x) + 1))**(3/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1063 vs. \(2 (69) = 138\).

Time = 0.35 (sec) , antiderivative size = 1063, normalized size of antiderivative = 13.81 \[ \int \frac {\cot ^6(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

8/15*((5*sin(7*f*x + 7*e) + 2*sin(5*f*x + 5*e) + 5*sin(3*f*x + 3*e))*cos(10*f*x + 10*e) - 5*(5*sin(7*f*x + 7*e
) + 2*sin(5*f*x + 5*e) + 5*sin(3*f*x + 3*e))*cos(8*f*x + 8*e) - 25*(2*sin(6*f*x + 6*e) - 2*sin(4*f*x + 4*e) +
sin(2*f*x + 2*e))*cos(7*f*x + 7*e) + 10*(2*sin(5*f*x + 5*e) + 5*sin(3*f*x + 3*e))*cos(6*f*x + 6*e) + 10*(2*sin
(4*f*x + 4*e) - sin(2*f*x + 2*e))*cos(5*f*x + 5*e) - (5*cos(7*f*x + 7*e) + 2*cos(5*f*x + 5*e) + 5*cos(3*f*x +
3*e))*sin(10*f*x + 10*e) + 5*(5*cos(7*f*x + 7*e) + 2*cos(5*f*x + 5*e) + 5*cos(3*f*x + 3*e))*sin(8*f*x + 8*e) +
5*(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*(2*cos(5*f*x + 5
*e) + 5*cos(3*f*x + 3*e))*sin(6*f*x + 6*e) - 2*(10*cos(4*f*x + 4*e) - 5*cos(2*f*x + 2*e) + 1)*sin(5*f*x + 5*e)
+ 50*cos(3*f*x + 3*e)*sin(4*f*x + 4*e) + 5*(5*cos(2*f*x + 2*e) - 1)*sin(3*f*x + 3*e) - 50*cos(4*f*x + 4*e)*si
n(3*f*x + 3*e) - 25*cos(3*f*x + 3*e)*sin(2*f*x + 2*e))*sqrt(a)/((a^2*cos(10*f*x + 10*e)^2 + 25*a^2*cos(8*f*x +
8*e)^2 + 100*a^2*cos(6*f*x + 6*e)^2 + 100*a^2*cos(4*f*x + 4*e)^2 + 25*a^2*cos(2*f*x + 2*e)^2 + a^2*sin(10*f*x
+ 10*e)^2 + 25*a^2*sin(8*f*x + 8*e)^2 + 100*a^2*sin(6*f*x + 6*e)^2 + 100*a^2*sin(4*f*x + 4*e)^2 - 100*a^2*sin
(4*f*x + 4*e)*sin(2*f*x + 2*e) + 25*a^2*sin(2*f*x + 2*e)^2 - 10*a^2*cos(2*f*x + 2*e) + a^2 - 2*(5*a^2*cos(8*f*
x + 8*e) - 10*a^2*cos(6*f*x + 6*e) + 10*a^2*cos(4*f*x + 4*e) - 5*a^2*cos(2*f*x + 2*e) + a^2)*cos(10*f*x + 10*e
) - 10*(10*a^2*cos(6*f*x + 6*e) - 10*a^2*cos(4*f*x + 4*e) + 5*a^2*cos(2*f*x + 2*e) - a^2)*cos(8*f*x + 8*e) - 2
0*(10*a^2*cos(4*f*x + 4*e) - 5*a^2*cos(2*f*x + 2*e) + a^2)*cos(6*f*x + 6*e) - 20*(5*a^2*cos(2*f*x + 2*e) - a^2
)*cos(4*f*x + 4*e) - 10*(a^2*sin(8*f*x + 8*e) - 2*a^2*sin(6*f*x + 6*e) + 2*a^2*sin(4*f*x + 4*e) - a^2*sin(2*f*
x + 2*e))*sin(10*f*x + 10*e) - 50*(2*a^2*sin(6*f*x + 6*e) - 2*a^2*sin(4*f*x + 4*e) + a^2*sin(2*f*x + 2*e))*sin
(8*f*x + 8*e) - 100*(2*a^2*sin(4*f*x + 4*e) - a^2*sin(2*f*x + 2*e))*sin(6*f*x + 6*e))*f)

Giac [F(-2)]

Exception generated. \[ \int \frac {\cot ^6(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 19.60 (sec) , antiderivative size = 393, normalized size of antiderivative = 5.10 \[ \int \frac {\cot ^6(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,16{}\mathrm {i}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^2\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,272{}\mathrm {i}}{15\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^3\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,128{}\mathrm {i}}{5\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^4\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,64{}\mathrm {i}}{5\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^5\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )} \]

[In]

int(cot(e + f*x)^6/(a - a*sin(e + f*x)^2)^(3/2),x)

[Out]

- (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)*16i)/(3*a^2*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*3i)*(a - a*((exp(- e*1
i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2)*272i)/(15*a^2*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^2*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^2*f*(exp(e*
2i + f*x*2i) - 1)^5*(exp(e*1i + f*x*1i) + exp(e*3i + f*x*3i)))