3.1.10 \(\int \cot (x) \csc (x) \, dx\) [10]

Optimal. Leaf size=4 \[ -\csc (x) \]

[Out]

-csc(x)

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Rubi [A]
time = 0.00, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2686, 8} \begin {gather*} -\csc (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[x]*Csc[x],x]

[Out]

-Csc[x]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=-\text {Subst}(\int 1 \, dx,x,\csc (x))\\ &=-\csc (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 4, normalized size = 1.00 \begin {gather*} -\csc (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]*Csc[x],x]

[Out]

-Csc[x]

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Maple [A]
time = 0.02, size = 7, normalized size = 1.75

method	result	size
parallelrisch	\(-\csc \left (x \right )\)	\(5\)
derivativedivides	\(-\frac {1}{\sin \left (x \right )}\)	\(7\)
default	\(-\frac {1}{\sin \left (x \right )}\)	\(7\)
norman	\(\frac {-\frac {1}{2}-\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}}{\tan \left (\frac {x}{2}\right )}\)	\(18\)
risch	\(-\frac {2 i {\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/sin(x)^2,x,method=_RETURNVERBOSE)

[Out]

-1/sin(x)

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Maxima [A]
time = 0.34, size = 6, normalized size = 1.50 \begin {gather*} -\frac {1}{\sin \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/sin(x)^2,x, algorithm="maxima")

[Out]

-1/sin(x)

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Fricas [A]
time = 0.57, size = 6, normalized size = 1.50 \begin {gather*} -\frac {1}{\sin \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/sin(x)^2,x, algorithm="fricas")

[Out]

-1/sin(x)

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Sympy [A]
time = 0.03, size = 5, normalized size = 1.25 \begin {gather*} - \frac {1}{\sin {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/sin(x)**2,x)

[Out]

-1/sin(x)

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Giac [A]
time = 0.52, size = 6, normalized size = 1.50 \begin {gather*} -\frac {1}{\sin \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/sin(x)^2,x, algorithm="giac")

[Out]

-1/sin(x)

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Mupad [B]
time = 0.10, size = 6, normalized size = 1.50 \begin {gather*} -\frac {1}{\sin \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/sin(x)^2,x)

[Out]

-1/sin(x)

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