\(\int \csc ^6(x) \sec ^6(x) \, dx\) [347]

Optimal result

Integrand size = 9, antiderivative size = 41 \[ \int \csc ^6(x) \sec ^6(x) \, dx=-10 \cot (x)-\frac {5 \cot ^3(x)}{3}-\frac {\cot ^5(x)}{5}+10 \tan (x)+\frac {5 \tan ^3(x)}{3}+\frac {\tan ^5(x)}{5} \]

[Out]

-10*cot(x)-5/3*cot(x)^3-1/5*cot(x)^5+10*tan(x)+5/3*tan(x)^3+1/5*tan(x)^5

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2700, 276} \[ \int \csc ^6(x) \sec ^6(x) \, dx=\frac {\tan ^5(x)}{5}+\frac {5 \tan ^3(x)}{3}+10 \tan (x)-\frac {1}{5} \cot ^5(x)-\frac {5 \cot ^3(x)}{3}-10 \cot (x) \]

[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 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{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (1+x^2\right )^5}{x^6} \, dx,x,\tan (x)\right ) \\ & = \text {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] (verified)

Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.29 \[ \int \csc ^6(x) \sec ^6(x) \, dx=-\frac {128 \cot (x)}{15}-\frac {19}{15} \cot (x) \csc ^2(x)-\frac {1}{5} \cot (x) \csc ^4(x)+\frac {128 \tan (x)}{15}+\frac {19}{15} \sec ^2(x) \tan (x)+\frac {1}{5} \sec ^4(x) \tan (x) \]

[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

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.68

method	result	size
parallelrisch	\(-\frac {\left (\sec ^{5}\left (x \right )\right ) \left (\csc ^{5}\left (x \right )\right ) \left (\cos \left (10 x \right )-5 \cos \left (6 x \right )+10 \cos \left (2 x \right )\right )}{30}\)	\(28\)
risch	\(-\frac {512 i \left (10 \,{\mathrm e}^{8 i x}-5 \,{\mathrm e}^{4 i x}+1\right )}{15 \left ({\mathrm e}^{2 i x}-1\right )^{5} \left ({\mathrm e}^{2 i x}+1\right )^{5}}\)	\(38\)
default	\(\frac {1}{5 \sin \left (x \right )^{5} \cos \left (x \right )^{5}}-\frac {2}{5 \sin \left (x \right )^{5} \cos \left (x \right )^{3}}+\frac {16}{15 \sin \left (x \right )^{3} \cos \left (x \right )^{3}}-\frac {32}{15 \sin \left (x \right )^{3} \cos \left (x \right )}+\frac {128}{15 \cos \left (x \right ) \sin \left (x \right )}-\frac {256 \cot \left (x \right )}{15}\)	\(56\)

[In]

int(1/cos(x)^6/sin(x)^6,x,method=_RETURNVERBOSE)

[Out]

-1/30*sec(x)^5*csc(x)^5*(cos(10*x)-5*cos(6*x)+10*cos(2*x))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.34 \[ \int \csc ^6(x) \sec ^6(x) \, dx=-\frac {256 \, \cos \left (x\right )^{10} - 640 \, \cos \left (x\right )^{8} + 480 \, \cos \left (x\right )^{6} - 80 \, \cos \left (x\right )^{4} - 10 \, \cos \left (x\right )^{2} - 3}{15 \, {\left (\cos \left (x\right )^{9} - 2 \, \cos \left (x\right )^{7} + \cos \left (x\right )^{5}\right )} \sin \left (x\right )} \]

[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))

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.07 \[ \int \csc ^6(x) \sec ^6(x) \, dx=- \frac {256 \cos {\left (2 x \right )}}{15 \sin {\left (2 x \right )}} - \frac {128 \cos {\left (2 x \right )}}{15 \sin ^{3}{\left (2 x \right )}} - \frac {32 \cos {\left (2 x \right )}}{5 \sin ^{5}{\left (2 x \right )}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90 \[ \int \csc ^6(x) \sec ^6(x) \, dx=\frac {1}{5} \, \tan \left (x\right )^{5} + \frac {5}{3} \, \tan \left (x\right )^{3} - \frac {150 \, \tan \left (x\right )^{4} + 25 \, \tan \left (x\right )^{2} + 3}{15 \, \tan \left (x\right )^{5}} + 10 \, \tan \left (x\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.63 \[ \int \csc ^6(x) \sec ^6(x) \, dx=-\frac {32 \, {\left (15 \, \tan \left (2 \, x\right )^{4} + 10 \, \tan \left (2 \, x\right )^{2} + 3\right )}}{15 \, \tan \left (2 \, x\right )^{5}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66 \[ \int \csc ^6(x) \sec ^6(x) \, dx=-\frac {32\,\left (\frac {\cos \left (2\,x\right )}{3}-\frac {\cos \left (6\,x\right )}{6}+\frac {\cos \left (10\,x\right )}{30}\right )}{{\sin \left (2\,x\right )}^5} \]

[In]

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

[Out]

-(32*(cos(2*x)/3 - cos(6*x)/6 + cos(10*x)/30))/sin(2*x)^5