3.580 \(\int \frac {\cot (x)}{\sqrt {a+b \sin ^n(x)}} \, dx\)

Optimal. Leaf size=29 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^n(x)}}{\sqrt {a}}\right )}{\sqrt {a} n} \]

[Out]

-2*arctanh((a+b*sin(x)^n)^(1/2)/a^(1/2))/n/a^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3230, 266, 63, 208} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^n(x)}}{\sqrt {a}}\right )}{\sqrt {a} n} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]/Sqrt[a + b*Sin[x]^n],x]

[Out]

(-2*ArcTanh[Sqrt[a + b*Sin[x]^n]/Sqrt[a]])/(Sqrt[a]*n)

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 208

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 3230

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + b*(c*ff*x)^n)^p)/(1 - ff^2*x^2)^(
(m + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^n(x)}}{\sqrt {a}}\right )}{\sqrt {a} n} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]/Sqrt[a + b*Sin[x]^n],x]

[Out]

(-2*ArcTanh[Sqrt[a + b*Sin[x]^n]/Sqrt[a]])/(Sqrt[a]*n)

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fricas [A]  time = 0.48, size = 74, normalized size = 2.55 \[ \left [\frac {\log \left (\frac {b \sin \relax (x)^{n} - 2 \, \sqrt {b \sin \relax (x)^{n} + a} \sqrt {a} + 2 \, a}{\sin \relax (x)^{n}}\right )}{\sqrt {a} n}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b \sin \relax (x)^{n} + a} \sqrt {-a}}{a}\right )}{a n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+b*sin(x)^n)^(1/2),x, algorithm="fricas")

[Out]

[log((b*sin(x)^n - 2*sqrt(b*sin(x)^n + a)*sqrt(a) + 2*a)/sin(x)^n)/(sqrt(a)*n), 2*sqrt(-a)*arctan(sqrt(b*sin(x
)^n + a)*sqrt(-a)/a)/(a*n)]

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giac [A]  time = 0.13, size = 27, normalized size = 0.93 \[ \frac {2 \, \arctan \left (\frac {\sqrt {b \sin \relax (x)^{n} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+b*sin(x)^n)^(1/2),x, algorithm="giac")

[Out]

2*arctan(sqrt(b*sin(x)^n + a)/sqrt(-a))/(sqrt(-a)*n)

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maple [A]  time = 0.16, size = 24, normalized size = 0.83 \[ -\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (\sin ^{n}\relax (x )\right )}}{\sqrt {a}}\right )}{n \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(a+b*sin(x)^n)^(1/2),x)

[Out]

-2*arctanh((a+b*sin(x)^n)^(1/2)/a^(1/2))/n/a^(1/2)

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maxima [A]  time = 0.81, size = 41, normalized size = 1.41 \[ \frac {\log \left (\frac {\sqrt {b \sin \relax (x)^{n} + a} - \sqrt {a}}{\sqrt {b \sin \relax (x)^{n} + a} + \sqrt {a}}\right )}{\sqrt {a} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+b*sin(x)^n)^(1/2),x, algorithm="maxima")

[Out]

log((sqrt(b*sin(x)^n + a) - sqrt(a))/(sqrt(b*sin(x)^n + a) + sqrt(a)))/(sqrt(a)*n)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {cot}\relax (x)}{\sqrt {a+b\,{\sin \relax (x)}^n}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(a + b*sin(x)^n)^(1/2),x)

[Out]

int(cot(x)/(a + b*sin(x)^n)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot {\relax (x )}}{\sqrt {a + b \sin ^{n}{\relax (x )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+b*sin(x)**n)**(1/2),x)

[Out]

Integral(cot(x)/sqrt(a + b*sin(x)**n), x)

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