3.36 \(\int \csc ^2(x) \sec ^2(x) \, dx\)

Optimal. Leaf size=7 \[ \tan (x)-\cot (x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2620, 14} \[ \tan (x)-\cot (x) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^2*Sec[x]^2,x]

[Out]

-Cot[x] + Tan[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2620

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rubi steps

\begin {align*} \int \csc ^2(x) \sec ^2(x) \, dx &=\operatorname {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,\tan (x)\right )\\ &=-\cot (x)+\tan (x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 6, normalized size = 0.86 \[ -2 \cot (2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^2*Sec[x]^2,x]

[Out]

-2*Cot[2*x]

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^2(x) \sec ^2(x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Csc[x]^2*Sec[x]^2,x]

[Out]

Could not integrate

________________________________________________________________________________________

fricas [B]  time = 1.34, size = 18, normalized size = 2.57 \[ -\frac {2 \, \cos \relax (x)^{2} - 1}{\cos \relax (x) \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^2/sin(x)^2,x, algorithm="fricas")

[Out]

-(2*cos(x)^2 - 1)/(cos(x)*sin(x))

________________________________________________________________________________________

giac [A]  time = 1.02, size = 9, normalized size = 1.29 \[ -\frac {1}{\tan \relax (x)} + \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^2/sin(x)^2,x, algorithm="giac")

[Out]

-1/tan(x) + tan(x)

________________________________________________________________________________________

maple [A]  time = 0.29, size = 15, normalized size = 2.14

method	result	size
default	\(\frac {1}{\sin \relax (x ) \cos \relax (x )}-2 \cot \relax (x )\)	\(15\)
risch	\(-\frac {4 i}{\left (1+{\mathrm e}^{2 i x}\right ) \left ({\mathrm e}^{2 i x}-1\right )}\)	\(22\)
norman	\(\frac {\frac {1}{2}-3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\frac {\left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2}}{\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(x)^2/sin(x)^2,x,method=_RETURNVERBOSE)

[Out]

1/sin(x)/cos(x)-2*cot(x)

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 9, normalized size = 1.29 \[ -\frac {1}{\tan \relax (x)} + \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^2/sin(x)^2,x, algorithm="maxima")

[Out]

-1/tan(x) + tan(x)

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 6, normalized size = 0.86 \[ -2\,\mathrm {cot}\left (2\,x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^2*sin(x)^2),x)

[Out]

-2*cot(2*x)

________________________________________________________________________________________

sympy [B]  time = 0.07, size = 12, normalized size = 1.71 \[ - \frac {2 \cos {\left (2 x \right )}}{\sin {\left (2 x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)**2/sin(x)**2,x)

[Out]

-2*cos(2*x)/sin(2*x)

________________________________________________________________________________________