\(\int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx\) [471]

Optimal result

Integrand size = 24, antiderivative size = 31 \[ \int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a} f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3255, 3284, 65, 212} \[ \int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a} f} \]

[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 65

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 212

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

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 3284

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
]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a} f} \]

[In]

Integrate[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))

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00

method	result	size
default	\(-\frac {\cos \left (f x +e \right ) \operatorname {arctanh}\left (\cos \left (f x +e \right )\right )}{\sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f}\)	\(31\)
risch	\(-\frac {2 \ln \left ({\mathrm e}^{i f x}+{\mathrm e}^{-i e}\right ) \cos \left (f x +e \right )}{f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}}+\frac {2 \ln \left ({\mathrm e}^{i f x}-{\mathrm e}^{-i e}\right ) \cos \left (f x +e \right )}{f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}}\)	\(104\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.71 \[ \int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\left [-\frac {\sqrt {a \cos \left (f x + e\right )^{2}} \log \left (-\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1}\right )}{2 \, a f \cos \left (f x + e\right )}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {a \cos \left (f x + e\right )^{2}} \sqrt {-a}}{a}\right )}{a f}\right ] \]

[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)]

Sympy [F]

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

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).

Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\log \left (\frac {2 \, \sqrt {-a \sin \left (f x + e\right )^{2} + a} \sqrt {a}}{{\left | \sin \left (f x + e\right ) \right |}} + \frac {2 \, a}{{\left | \sin \left (f x + e\right ) \right |}}\right )}{\sqrt {a} f} \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate(cot(f*x+e)/(a-a*sin(f*x+e)^2)^(1/2),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 i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\int \frac {\mathrm {cot}\left (e+f\,x\right )}{\sqrt {a-a\,{\sin \left (e+f\,x\right )}^2}} \,d x \]

[In]

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

[Out]

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