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

Optimal. Leaf size=31 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right )}{\sqrt{a} f} \]

[Out]

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

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Rubi [A]  time = 0.0758756, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3176, 3205, 63, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right )}{\sqrt{a} f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3205

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFact
ors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(b*ff^(n/2)*x^(n/2))^p)/(1 - ff*x
)^((m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2
]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0430917, size = 49, normalized size = 1.58 \[ \frac{\cos (e+f x) \left (\log \left (\sin \left (\frac{1}{2} (e+f x)\right )\right )-\log \left (\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )}{f \sqrt{a \cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[e + f*x]*(-Log[Cos[(e + f*x)/2]] + Log[Sin[(e + f*x)/2]]))/(f*Sqrt[a*Cos[e + f*x]^2])

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Maple [A]  time = 0.797, size = 40, normalized size = 1.3 \begin{align*} -{\frac{1}{f}\ln \left ({\frac{1}{\sin \left ( fx+e \right ) } \left ( 2\,a+2\,\sqrt{a}\sqrt{a \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \right ) } \right ){\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.7319, size = 208, normalized size = 6.71 \begin{align*} \left [-\frac{\sqrt{a \cos \left (f x + e\right )^{2}} \log \left (-\frac{\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1}\right )}{2 \, a f \cos \left (f x + e\right )}, \frac{\sqrt{-a} \arctan \left (\frac{\sqrt{a \cos \left (f x + e\right )^{2}} \sqrt{-a}}{a}\right )}{a f}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(a*cos(f*x + e)^2)*log(-(cos(f*x + e) + 1)/(cos(f*x + e) - 1))/(a*f*cos(f*x + e)), sqrt(-a)*arctan(s
qrt(a*cos(f*x + e)^2)*sqrt(-a)/a)/(a*f)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\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)/(a-a*sin(f*x+e)**2)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.13288, size = 43, normalized size = 1.39 \begin{align*} \frac{\arctan \left (\frac{\sqrt{-a \sin \left (f x + e\right )^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sqrt(-a*sin(f*x + e)^2 + a)/sqrt(-a))/(sqrt(-a)*f)