\(\int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx\) [554]

Optimal result

Integrand size = 15, antiderivative size = 28 \[ \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sin ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]

[Out]

-2/3*arctanh((a+b*sin(x)^3)^(1/2)/a^(1/2))/a^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3309, 272, 65, 214} \[ \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sin ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]

[In]

Int[Cot[x]/Sqrt[a + b*Sin[x]^3],x]

[Out]

(-2*ArcTanh[Sqrt[a + b*Sin[x]^3]/Sqrt[a]])/(3*Sqrt[a])

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 214

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

Rule 272

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 3309

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*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^3}} \, dx,x,\sin (x)\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^3(x)\right ) \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^3(x)}\right )}{3 b} \\ & = -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sin ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sin ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]

[In]

Integrate[Cot[x]/Sqrt[a + b*Sin[x]^3],x]

[Out]

(-2*ArcTanh[Sqrt[a + b*Sin[x]^3]/Sqrt[a]])/(3*Sqrt[a])

Maple [F]

\[\int \frac {\cot \left (x \right )}{\sqrt {a +b \left (\sin ^{3}\left (x \right )\right )}}d x\]

[In]

int(cot(x)/(a+b*sin(x)^3)^(1/2),x)

[Out]

int(cot(x)/(a+b*sin(x)^3)^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(cot(x)/(a+b*sin(x)^3)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   failed of mode Union(SparseUnivariatePol
ynomial(Expression(Complex(Integer))),failed) cannot be coerced to mode SparseUnivariatePolynomial(Expression(
Complex(Int

Sympy [F]

\[ \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx=\int \frac {\cot {\left (x \right )}}{\sqrt {a + b \sin ^{3}{\left (x \right )}}}\, dx \]

[In]

integrate(cot(x)/(a+b*sin(x)**3)**(1/2),x)

[Out]

Integral(cot(x)/sqrt(a + b*sin(x)**3), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx=\frac {\log \left (\frac {\sqrt {b \sin \left (x\right )^{3} + a} - \sqrt {a}}{\sqrt {b \sin \left (x\right )^{3} + a} + \sqrt {a}}\right )}{3 \, \sqrt {a}} \]

[In]

integrate(cot(x)/(a+b*sin(x)^3)^(1/2),x, algorithm="maxima")

[Out]

1/3*log((sqrt(b*sin(x)^3 + a) - sqrt(a))/(sqrt(b*sin(x)^3 + a) + sqrt(a)))/sqrt(a)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {b \sin \left (x\right )^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} \]

[In]

integrate(cot(x)/(a+b*sin(x)^3)^(1/2),x, algorithm="giac")

[Out]

2/3*arctan(sqrt(b*sin(x)^3 + a)/sqrt(-a))/sqrt(-a)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx=\int \frac {\mathrm {cot}\left (x\right )}{\sqrt {b\,{\sin \left (x\right )}^3+a}} \,d x \]

[In]

int(cot(x)/(a + b*sin(x)^3)^(1/2),x)

[Out]

int(cot(x)/(a + b*sin(x)^3)^(1/2), x)