3.7 \(\int \cot (x) \csc (x) \, dx\)

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

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Rubi [A]  time = 0.01, 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 = {2606, 8} \[ -\csc (x) \]

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 2606

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 {align*} \int \cot (x) \csc (x) \, dx &=-\operatorname {Subst}(\int 1 \, dx,x,\csc (x))\\ &=-\csc (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 4, normalized size = 1.00 \[ -\csc (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Csc[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot (x) \csc (x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Cot[x]*Csc[x],x]

[Out]

Could not integrate

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fricas [A]  time = 1.02, size = 6, normalized size = 1.50 \[ -\frac {1}{\sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)*csc(x),x, algorithm="fricas")

[Out]

-1/sin(x)

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giac [A]  time = 0.99, size = 6, normalized size = 1.50 \[ -\frac {1}{\sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)*csc(x),x, algorithm="giac")

[Out]

-1/sin(x)

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maple [A]  time = 0.03, size = 5, normalized size = 1.25

method	result	size
derivativedivides	\(-\csc \relax (x )\)	\(5\)
default	\(-\csc \relax (x )\)	\(5\)
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(cot(x)*csc(x),x,method=_RETURNVERBOSE)

[Out]

-csc(x)

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maxima [A]  time = 0.44, size = 6, normalized size = 1.50 \[ -\frac {1}{\sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)*csc(x),x, algorithm="maxima")

[Out]

-1/sin(x)

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mupad [B]  time = 0.21, size = 6, normalized size = 1.50 \[ -\frac {1}{\sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/sin(x),x)

[Out]

-1/sin(x)

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sympy [A]  time = 0.07, size = 5, normalized size = 1.25 \[ - \frac {1}{\sin {\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)*csc(x),x)

[Out]

-1/sin(x)

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