\(\int \frac {\cot ^4(e+f x)}{(a-a \sin ^2(e+f x))^{3/2}} \, dx\) [485]

Optimal result

Integrand size = 26, antiderivative size = 38 \[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3255, 3286, 2686, 30} \[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}} \]

[In]

Int[Cot[e + f*x]^4/(a - a*Sin[e + f*x]^2)^(3/2),x]

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 ^4(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx \\ & = \frac {\cos (e+f x) \int \cot (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 \, 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)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\cot ^3(e+f x)}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]

[In]

Integrate[Cot[e + f*x]^4/(a - a*Sin[e + f*x]^2)^(3/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {\cos \left (f x +e \right )}{3 \sin \left (f x +e \right )^{3} a \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f}\)	\(35\)
risch	\(\frac {8 i \left ({\mathrm e}^{4 i \left (f x +e \right )}+{\mathrm e}^{2 i \left (f x +e \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3} f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, a}\)	\(68\)

[In]

int(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32 \[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {\sqrt {a \cos \left (f x + e\right )^{2}}}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{3} - a^{2} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \]

[In]

integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(cot(f*x+e)**4/(a-a*sin(f*x+e)**2)**(3/2),x)

[Out]

Integral(cot(e + f*x)**4/(-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. 382 vs. \(2 (34) = 68\).

Time = 0.34 (sec) , antiderivative size = 382, normalized size of antiderivative = 10.05 \[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {8 \, {\left (\cos \left (3 \, f x + 3 \, e\right ) \sin \left (6 \, f x + 6 \, e\right ) - 3 \, \cos \left (3 \, f x + 3 \, e\right ) \sin \left (4 \, f x + 4 \, e\right ) - {\left (3 \, \cos \left (2 \, f x + 2 \, e\right ) - 1\right )} \sin \left (3 \, f x + 3 \, e\right ) - \cos \left (6 \, f x + 6 \, e\right ) \sin \left (3 \, f x + 3 \, e\right ) + 3 \, \cos \left (4 \, f x + 4 \, e\right ) \sin \left (3 \, f x + 3 \, e\right ) + 3 \, \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right )\right )} \sqrt {a}}{3 \, {\left (a^{2} \cos \left (6 \, f x + 6 \, e\right )^{2} + 9 \, a^{2} \cos \left (4 \, f x + 4 \, e\right )^{2} + 9 \, a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + a^{2} \sin \left (6 \, f x + 6 \, e\right )^{2} + 9 \, a^{2} \sin \left (4 \, f x + 4 \, e\right )^{2} - 18 \, a^{2} \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 9 \, a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} - 6 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2} - 2 \, {\left (3 \, a^{2} \cos \left (4 \, f x + 4 \, e\right ) - 3 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2}\right )} \cos \left (6 \, f x + 6 \, e\right ) - 6 \, {\left (3 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) - a^{2}\right )} \cos \left (4 \, f x + 4 \, e\right ) - 6 \, {\left (a^{2} \sin \left (4 \, f x + 4 \, e\right ) - a^{2} \sin \left (2 \, f x + 2 \, e\right )\right )} \sin \left (6 \, f x + 6 \, e\right )\right )} f} \]

[In]

integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

8/3*(cos(3*f*x + 3*e)*sin(6*f*x + 6*e) - 3*cos(3*f*x + 3*e)*sin(4*f*x + 4*e) - (3*cos(2*f*x + 2*e) - 1)*sin(3*
f*x + 3*e) - cos(6*f*x + 6*e)*sin(3*f*x + 3*e) + 3*cos(4*f*x + 4*e)*sin(3*f*x + 3*e) + 3*cos(3*f*x + 3*e)*sin(
2*f*x + 2*e))*sqrt(a)/((a^2*cos(6*f*x + 6*e)^2 + 9*a^2*cos(4*f*x + 4*e)^2 + 9*a^2*cos(2*f*x + 2*e)^2 + a^2*sin
(6*f*x + 6*e)^2 + 9*a^2*sin(4*f*x + 4*e)^2 - 18*a^2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 9*a^2*sin(2*f*x + 2*e)
^2 - 6*a^2*cos(2*f*x + 2*e) + a^2 - 2*(3*a^2*cos(4*f*x + 4*e) - 3*a^2*cos(2*f*x + 2*e) + a^2)*cos(6*f*x + 6*e)
- 6*(3*a^2*cos(2*f*x + 2*e) - a^2)*cos(4*f*x + 4*e) - 6*(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. \[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(cot(f*x+e)^4/(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 = 17.72 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.32 \[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,16{}\mathrm {i}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^3\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )} \]

[In]

int(cot(e + f*x)^4/(a - a*sin(e + f*x)^2)^(3/2),x)

[Out]

(exp(e*4i + f*x*4i)*(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*(e
xp(e*2i + f*x*2i) - 1)^3*(exp(e*2i + f*x*2i) + 1))