3.3.53 \(\int \csc ^4(x) \sec ^4(x) \, dx\) [253]

Optimal. Leaf size=17 \[ -8 \cot (2 x)-\frac {8}{3} \cot ^3(2 x) \]

[Out]

-8*cot(2*x)-8/3*cot(2*x)^3

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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.47, 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 = {2700, 276} \begin {gather*} \frac {\tan ^3(x)}{3}+3 \tan (x)-\frac {1}{3} \cot ^3(x)-3 \cot (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[x]^4*Sec[x]^4,x]

[Out]

-3*Cot[x] - Cot[x]^3/3 + 3*Tan[x] + Tan[x]^3/3

Rule 276

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

Rule 2700

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 {gather*} \begin {aligned} \text {Integral} &=\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^4} \, dx,x,\tan (x)\right )\\ &=\text {Subst}\left (\int \left (3+\frac {1}{x^4}+\frac {3}{x^2}+x^2\right ) \, dx,x,\tan (x)\right )\\ &=-3 \cot (x)-\frac {\cot ^3(x)}{3}+3 \tan (x)+\frac {\tan ^3(x)}{3}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 33, normalized size = 1.94 \begin {gather*} -\frac {8 \cot (x)}{3}-\frac {1}{3} \cot (x) \csc ^2(x)+\frac {8 \tan (x)}{3}+\frac {1}{3} \sec ^2(x) \tan (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^4*Sec[x]^4,x]

[Out]

(-8*Cot[x])/3 - (Cot[x]*Csc[x]^2)/3 + (8*Tan[x])/3 + (Sec[x]^2*Tan[x])/3

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(35\) vs. \(2(15)=30\).
time = 0.22, size = 36, normalized size = 2.12

method	result	size
parallelrisch	\(-\frac {\left (-\cos \left (6 x \right )+3 \cos \left (2 x \right )\right ) \left (\sec ^{3}\left (x \right )\right ) \left (\csc ^{3}\left (x \right )\right )}{6}\)	\(24\)
risch	\(\frac {32 i \left (3 \,{\mathrm e}^{4 i x}-1\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{3} \left ({\mathrm e}^{2 i x}+1\right )^{3}}\)	\(31\)
default	\(\frac {1}{3 \sin \left (x \right )^{3} \cos \left (x \right )^{3}}-\frac {2}{3 \sin \left (x \right )^{3} \cos \left (x \right )}+\frac {8}{3 \cos \left (x \right ) \sin \left (x \right )}-\frac {16 \cot \left (x \right )}{3}\)	\(36\)
norman	\(\frac {\frac {1}{24}+\frac {5 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{4}-\frac {91 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8}+\frac {35 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{2}-\frac {91 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{8}+\frac {5 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{4}+\frac {\left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{24}}{\tan \left (\frac {x}{2}\right )^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{3}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(x)^4/cos(x)^4,x,method=_RETURNVERBOSE)

[Out]

1/3/sin(x)^3/cos(x)^3-2/3/sin(x)^3/cos(x)+8/3/cos(x)/sin(x)-16/3*cot(x)

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Maxima [A]
time = 0.38, size = 25, normalized size = 1.47 \begin {gather*} \frac {1}{3} \, \tan \left (x\right )^{3} - \frac {9 \, \tan \left (x\right )^{2} + 1}{3 \, \tan \left (x\right )^{3}} + 3 \, \tan \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)^4/cos(x)^4,x, algorithm="maxima")

[Out]

1/3*tan(x)^3 - 1/3*(9*tan(x)^2 + 1)/tan(x)^3 + 3*tan(x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (15) = 30\).
time = 0.60, size = 39, normalized size = 2.29 \begin {gather*} -\frac {16 \, \cos \left (x\right )^{6} - 24 \, \cos \left (x\right )^{4} + 6 \, \cos \left (x\right )^{2} + 1}{3 \, {\left (\cos \left (x\right )^{5} - \cos \left (x\right )^{3}\right )} \sin \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)^4/cos(x)^4,x, algorithm="fricas")

[Out]

-1/3*(16*cos(x)^6 - 24*cos(x)^4 + 6*cos(x)^2 + 1)/((cos(x)^5 - cos(x)^3)*sin(x))

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Sympy [A]
time = 0.02, size = 29, normalized size = 1.71 \begin {gather*} - \frac {16 \cos {\left (2 x \right )}}{3 \sin {\left (2 x \right )}} - \frac {8 \cos {\left (2 x \right )}}{3 \sin ^{3}{\left (2 x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)**4/cos(x)**4,x)

[Out]

-16*cos(2*x)/(3*sin(2*x)) - 8*cos(2*x)/(3*sin(2*x)**3)

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Giac [A]
time = 0.58, size = 18, normalized size = 1.06 \begin {gather*} -\frac {8 \, {\left (3 \, \tan \left (2 \, x\right )^{2} + 1\right )}}{3 \, \tan \left (2 \, x\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)^4/cos(x)^4,x, algorithm="giac")

[Out]

-8/3*(3*tan(2*x)^2 + 1)/tan(2*x)^3

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Mupad [B]
time = 0.21, size = 38, normalized size = 2.24 \begin {gather*} -\frac {24\,\cos \left (2\,x\right )-16\,{\cos \left (2\,x\right )}^3}{3\,\sin \left (2\,x\right )-3\,{\cos \left (2\,x\right )}^2\,\sin \left (2\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^4*sin(x)^4),x)

[Out]

-(24*cos(2*x) - 16*cos(2*x)^3)/(3*sin(2*x) - 3*cos(2*x)^2*sin(2*x))

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Chatgpt [F] Failed to verify
time = 1.00, size = 25, normalized size = 1.47 \begin {gather*} -\frac {\tan \left (x \right ) \cot \left (x \right )}{3}+\frac {\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )}{3}+\frac {\ln \left (\tan \left (x \right )-\sec \left (x \right )\right )}{3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(1/sin(x)^4/cos(x)^4,x)

[Out]

-1/3*tan(x)*cot(x)+1/3*ln(sec(x)+tan(x))+1/3*ln(tan(x)-sec(x))

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