\(\int \frac {\cot (x) \csc (x)}{\sqrt {1+\sin ^2(x)}} \, dx\) [739]

Optimal result

Integrand size = 15, antiderivative size = 14 \[ \int \frac {\cot (x) \csc (x)}{\sqrt {1+\sin ^2(x)}} \, dx=-\csc (x) \sqrt {1+\sin ^2(x)} \]

[Out]

-csc(x)*(1+sin(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ \int \frac {\cot (x) \csc (x)}{\sqrt {1+\sin ^2(x)}} \, dx=\sqrt {\sin ^2(x)+1} (-\csc (x)) \]

[In]

Int[(Cot[x]*Csc[x])/Sqrt[1 + Sin[x]^2],x]

[Out]

-(Csc[x]*Sqrt[1 + Sin[x]^2])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x^2}} \, dx,x,\sin (x)\right ) \\ & = -\csc (x) \sqrt {1+\sin ^2(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x) \csc (x)}{\sqrt {1+\sin ^2(x)}} \, dx=-\csc (x) \sqrt {1+\sin ^2(x)} \]

[In]

Integrate[(Cot[x]*Csc[x])/Sqrt[1 + Sin[x]^2],x]

[Out]

-(Csc[x]*Sqrt[1 + Sin[x]^2])

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07

method	result	size
default	\(-\frac {\sqrt {\sin \left (x \right )^{2}+1}}{\sin \left (x \right )}\)	\(15\)

[In]

int(cot(x)*csc(x)/(sin(x)^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/sin(x)*(sin(x)^2+1)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {\cot (x) \csc (x)}{\sqrt {1+\sin ^2(x)}} \, dx=-\frac {\sqrt {-\cos \left (x\right )^{2} + 2} - \sin \left (x\right )}{\sin \left (x\right )} \]

[In]

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

[Out]

-(sqrt(-cos(x)^2 + 2) - sin(x))/sin(x)

Sympy [F]

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

[In]

integrate(cot(x)*csc(x)/(1+sin(x)**2)**(1/2),x)

[Out]

Integral(cot(x)*csc(x)/sqrt(sin(x)**2 + 1), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x) \csc (x)}{\sqrt {1+\sin ^2(x)}} \, dx=-\frac {\sqrt {\sin \left (x\right )^{2} + 1}}{\sin \left (x\right )} \]

[In]

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

[Out]

-sqrt(sin(x)^2 + 1)/sin(x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {\cot (x) \csc (x)}{\sqrt {1+\sin ^2(x)}} \, dx=\frac {2}{{\left (\sqrt {\sin \left (x\right )^{2} + 1} - \sin \left (x\right )\right )}^{2} - 1} \]

[In]

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

[Out]

2/((sqrt(sin(x)^2 + 1) - sin(x))^2 - 1)

Mupad [B] (verification not implemented)

Time = 26.83 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.43 \[ \int \frac {\cot (x) \csc (x)}{\sqrt {1+\sin ^2(x)}} \, dx=-\frac {\sqrt {\frac {1}{{\sin \left (x\right )}^2}+1}}{\sin \left (x\right )\,\left (\sqrt {\frac {1}{{\sin \left (x\right )}^2}+1}+1\right )\,\sqrt {{\sin \left (x\right )}^2+1}} \]

[In]

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

[Out]

-(1/sin(x)^2 + 1)^(1/2)/(sin(x)*((1/sin(x)^2 + 1)^(1/2) + 1)*(sin(x)^2 + 1)^(1/2))