3.476 \(\int \frac{\cot ^4(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx\)

Optimal. Leaf size=60 \[ \frac{\cot (e+f x)}{f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt{a \cos ^2(e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.117242, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3176, 3207, 2606} \[ \frac{\cot (e+f x)}{f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt{a \cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3176

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 3207

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
]])

Rule 2606

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])

Rubi steps

\begin{align*} \int \frac{\cot ^4(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx &=\int \frac{\cot ^4(e+f x)}{\sqrt{a \cos ^2(e+f x)}} \, dx\\ &=\frac{\cos (e+f x) \int \cot ^3(e+f x) \csc (e+f x) \, dx}{\sqrt{a \cos ^2(e+f x)}}\\ &=-\frac{\cos (e+f x) \operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{f \sqrt{a \cos ^2(e+f x)}}\\ &=\frac{\cot (e+f x)}{f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt{a \cos ^2(e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.0603917, size = 37, normalized size = 0.62 \[ -\frac{\cot (e+f x) \left (\csc ^2(e+f x)-3\right )}{3 f \sqrt{a \cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.717, size = 44, normalized size = 0.7 \begin{align*}{\frac{\cos \left ( fx+e \right ) \left ( 3\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}-1 \right ) }{3\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}f}{\frac{1}{\sqrt{a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(1/2),x)

[Out]

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

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Maxima [B]  time = 1.70911, size = 709, normalized size = 11.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="maxima")

[Out]

-2/3*((3*sin(5*f*x + 5*e) - 2*sin(3*f*x + 3*e) + 3*sin(f*x + e))*cos(6*f*x + 6*e) + 9*(sin(4*f*x + 4*e) - sin(
2*f*x + 2*e))*cos(5*f*x + 5*e) + 3*(2*sin(3*f*x + 3*e) - 3*sin(f*x + e))*cos(4*f*x + 4*e) - (3*cos(5*f*x + 5*e
) - 2*cos(3*f*x + 3*e) + 3*cos(f*x + e))*sin(6*f*x + 6*e) - 3*(3*cos(4*f*x + 4*e) - 3*cos(2*f*x + 2*e) + 1)*si
n(5*f*x + 5*e) - 3*(2*cos(3*f*x + 3*e) - 3*cos(f*x + e))*sin(4*f*x + 4*e) - 2*(3*cos(2*f*x + 2*e) - 1)*sin(3*f
*x + 3*e) + 6*cos(3*f*x + 3*e)*sin(2*f*x + 2*e) - 9*cos(f*x + e)*sin(2*f*x + 2*e) + 9*cos(2*f*x + 2*e)*sin(f*x
+ e) - 3*sin(f*x + e))*sqrt(a)/((a*cos(6*f*x + 6*e)^2 + 9*a*cos(4*f*x + 4*e)^2 + 9*a*cos(2*f*x + 2*e)^2 + a*s
in(6*f*x + 6*e)^2 + 9*a*sin(4*f*x + 4*e)^2 - 18*a*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 9*a*sin(2*f*x + 2*e)^2 -
2*(3*a*cos(4*f*x + 4*e) - 3*a*cos(2*f*x + 2*e) + a)*cos(6*f*x + 6*e) - 6*(3*a*cos(2*f*x + 2*e) - a)*cos(4*f*x
+ 4*e) - 6*a*cos(2*f*x + 2*e) - 6*(a*sin(4*f*x + 4*e) - a*sin(2*f*x + 2*e))*sin(6*f*x + 6*e) + a)*f)

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Fricas [A]  time = 1.58113, size = 143, normalized size = 2.38 \begin{align*} \frac{\sqrt{a \cos \left (f x + e\right )^{2}}{\left (3 \, \cos \left (f x + e\right )^{2} - 2\right )}}{3 \,{\left (a f \cos \left (f x + e\right )^{3} - a f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{4}{\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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**4/(a-a*sin(f*x+e)**2)**(1/2),x)

[Out]

Integral(cot(e + f*x)**4/sqrt(-a*(sin(e + f*x) - 1)*(sin(e + f*x) + 1)), x)

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Giac [A]  time = 1.34488, size = 134, normalized size = 2.23 \begin{align*} \frac{\frac{\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 9 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right )} - \frac{9 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3}}}{24 \, \sqrt{a} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="giac")

[Out]

1/24*((tan(1/2*f*x + 1/2*e)^3 - 9*tan(1/2*f*x + 1/2*e))/sgn(tan(1/2*f*x + 1/2*e)^4 - 1) - (9*tan(1/2*f*x + 1/2
*e)^2 - 1)/(sgn(tan(1/2*f*x + 1/2*e)^4 - 1)*tan(1/2*f*x + 1/2*e)^3))/(sqrt(a)*f)