3.347 \(\int \csc ^6(x) \sec ^6(x) \, dx\)

Optimal. Leaf size=41 \[ \frac{\tan ^5(x)}{5}+\frac{5 \tan ^3(x)}{3}+10 \tan (x)-\frac{1}{5} \cot ^5(x)-\frac{5 \cot ^3(x)}{3}-10 \cot (x) \]

[Out]

-10*Cot[x] - (5*Cot[x]^3)/3 - Cot[x]^5/5 + 10*Tan[x] + (5*Tan[x]^3)/3 + Tan[x]^5/5

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Rubi [A]  time = 0.0338857, antiderivative size = 41, normalized size of antiderivative = 1., 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, 270} \[ \frac{\tan ^5(x)}{5}+\frac{5 \tan ^3(x)}{3}+10 \tan (x)-\frac{1}{5} \cot ^5(x)-\frac{5 \cot ^3(x)}{3}-10 \cot (x) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^6*Sec[x]^6,x]

[Out]

-10*Cot[x] - (5*Cot[x]^3)/3 - Cot[x]^5/5 + 10*Tan[x] + (5*Tan[x]^3)/3 + Tan[x]^5/5

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]

Rule 270

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]

Rubi steps

\begin{align*} \int \csc ^6(x) \sec ^6(x) \, dx &=\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^5}{x^6} \, dx,x,\tan (x)\right )\\ &=\operatorname{Subst}\left (\int \left (10+\frac{1}{x^6}+\frac{5}{x^4}+\frac{10}{x^2}+5 x^2+x^4\right ) \, dx,x,\tan (x)\right )\\ &=-10 \cot (x)-\frac{5 \cot ^3(x)}{3}-\frac{\cot ^5(x)}{5}+10 \tan (x)+\frac{5 \tan ^3(x)}{3}+\frac{\tan ^5(x)}{5}\\ \end{align*}

Mathematica [A]  time = 0.0363721, size = 53, normalized size = 1.29 \[ \frac{128 \tan (x)}{15}-\frac{128 \cot (x)}{15}-\frac{1}{5} \cot (x) \csc ^4(x)-\frac{19}{15} \cot (x) \csc ^2(x)+\frac{1}{5} \tan (x) \sec ^4(x)+\frac{19}{15} \tan (x) \sec ^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^6*Sec[x]^6,x]

[Out]

(-128*Cot[x])/15 - (19*Cot[x]*Csc[x]^2)/15 - (Cot[x]*Csc[x]^4)/5 + (128*Tan[x])/15 + (19*Sec[x]^2*Tan[x])/15 +
(Sec[x]^4*Tan[x])/5

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Maple [A]  time = 0.011, size = 56, normalized size = 1.4 \begin{align*}{\frac{1}{5\, \left ( \sin \left ( x \right ) \right ) ^{5} \left ( \cos \left ( x \right ) \right ) ^{5}}}-{\frac{2}{5\, \left ( \sin \left ( x \right ) \right ) ^{5} \left ( \cos \left ( x \right ) \right ) ^{3}}}+{\frac{16}{15\, \left ( \cos \left ( x \right ) \right ) ^{3} \left ( \sin \left ( x \right ) \right ) ^{3}}}-{\frac{32}{15\,\cos \left ( x \right ) \left ( \sin \left ( x \right ) \right ) ^{3}}}+{\frac{128}{15\,\cos \left ( x \right ) \sin \left ( x \right ) }}-{\frac{256\,\cot \left ( x \right ) }{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(x)^6/sin(x)^6,x)

[Out]

1/5/sin(x)^5/cos(x)^5-2/5/sin(x)^5/cos(x)^3+16/15/sin(x)^3/cos(x)^3-32/15/sin(x)^3/cos(x)+128/15/cos(x)/sin(x)
-256/15*cot(x)

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Maxima [A]  time = 0.943402, size = 50, normalized size = 1.22 \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} + \frac{5}{3} \, \tan \left (x\right )^{3} - \frac{150 \, \tan \left (x\right )^{4} + 25 \, \tan \left (x\right )^{2} + 3}{15 \, \tan \left (x\right )^{5}} + 10 \, \tan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^6/sin(x)^6,x, algorithm="maxima")

[Out]

1/5*tan(x)^5 + 5/3*tan(x)^3 - 1/15*(150*tan(x)^4 + 25*tan(x)^2 + 3)/tan(x)^5 + 10*tan(x)

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Fricas [A]  time = 1.93181, size = 174, normalized size = 4.24 \begin{align*} -\frac{256 \, \cos \left (x\right )^{10} - 640 \, \cos \left (x\right )^{8} + 480 \, \cos \left (x\right )^{6} - 80 \, \cos \left (x\right )^{4} - 10 \, \cos \left (x\right )^{2} - 3}{15 \,{\left (\cos \left (x\right )^{9} - 2 \, \cos \left (x\right )^{7} + \cos \left (x\right )^{5}\right )} \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^6/sin(x)^6,x, algorithm="fricas")

[Out]

-1/15*(256*cos(x)^10 - 640*cos(x)^8 + 480*cos(x)^6 - 80*cos(x)^4 - 10*cos(x)^2 - 3)/((cos(x)^9 - 2*cos(x)^7 +
cos(x)^5)*sin(x))

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Sympy [A]  time = 0.067319, size = 44, normalized size = 1.07 \begin{align*} - \frac{256 \cos{\left (2 x \right )}}{15 \sin{\left (2 x \right )}} - \frac{128 \cos{\left (2 x \right )}}{15 \sin ^{3}{\left (2 x \right )}} - \frac{32 \cos{\left (2 x \right )}}{5 \sin ^{5}{\left (2 x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)**6/sin(x)**6,x)

[Out]

-256*cos(2*x)/(15*sin(2*x)) - 128*cos(2*x)/(15*sin(2*x)**3) - 32*cos(2*x)/(5*sin(2*x)**5)

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Giac [A]  time = 1.06296, size = 35, normalized size = 0.85 \begin{align*} -\frac{32 \,{\left (15 \, \tan \left (2 \, x\right )^{4} + 10 \, \tan \left (2 \, x\right )^{2} + 3\right )}}{15 \, \tan \left (2 \, x\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^6/sin(x)^6,x, algorithm="giac")

[Out]

-32/15*(15*tan(2*x)^4 + 10*tan(2*x)^2 + 3)/tan(2*x)^5