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

Optimal. Leaf size=14 \[ \sqrt{\sin ^2(x)+1} (-\csc (x)) \]

[Out]

-(Csc[x]*Sqrt[1 + Sin[x]^2])

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Rubi [A]  time = 0.0765542, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ \sqrt{\sin ^2(x)+1} (-\csc (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Csc[x]*Sqrt[1 + Sin[x]^2])

Rule 264

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*} \int \frac{\cot (x) \csc (x)}{\sqrt{1+\sin ^2(x)}} \, dx &=\operatorname{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]  time = 0.01771, size = 14, normalized size = 1. \[ \sqrt{\sin ^2(x)+1} (-\csc (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Csc[x]*Sqrt[1 + Sin[x]^2])

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Maple [A]  time = 0.463, size = 15, normalized size = 1.1 \begin{align*} -{\frac{1}{\sin \left ( x \right ) }\sqrt{1+ \left ( \sin \left ( x \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.44459, size = 19, normalized size = 1.36 \begin{align*} -\frac{\sqrt{\sin \left (x\right )^{2} + 1}}{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(sin(x)^2 + 1)/sin(x)

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Fricas [A]  time = 2.12135, size = 54, normalized size = 3.86 \begin{align*} -\frac{\sqrt{-\cos \left (x\right )^{2} + 2} - \sin \left (x\right )}{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (x \right )} \csc{\left (x \right )}}{\sqrt{\sin ^{2}{\left (x \right )} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.09895, size = 28, normalized size = 2. \begin{align*} \frac{2}{{\left (\sqrt{\sin \left (x\right )^{2} + 1} - \sin \left (x\right )\right )}^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)