3.312 \(\int \frac {1}{(\csc (x)-\sin (x))^6} \, dx\)

Optimal. Leaf size=25 \[ \frac {\tan ^{11}(x)}{11}+\frac {2 \tan ^9(x)}{9}+\frac {\tan ^7(x)}{7} \]

[Out]

1/7*tan(x)^7+2/9*tan(x)^9+1/11*tan(x)^11

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Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {270} \[ \frac {\tan ^{11}(x)}{11}+\frac {2 \tan ^9(x)}{9}+\frac {\tan ^7(x)}{7} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x] - Sin[x])^(-6),x]

[Out]

Tan[x]^7/7 + (2*Tan[x]^9)/9 + Tan[x]^11/11

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 \frac {1}{(\csc (x)-\sin (x))^6} \, dx &=\operatorname {Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,\tan (x)\right )\\ &=\frac {\tan ^7(x)}{7}+\frac {2 \tan ^9(x)}{9}+\frac {\tan ^{11}(x)}{11}\\ \end {align*}

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Mathematica [B]  time = 0.02, size = 57, normalized size = 2.28 \[ -\frac {8 \tan (x)}{693}+\frac {1}{11} \tan (x) \sec ^{10}(x)-\frac {23}{99} \tan (x) \sec ^8(x)+\frac {113}{693} \tan (x) \sec ^6(x)-\frac {1}{231} \tan (x) \sec ^4(x)-\frac {4}{693} \tan (x) \sec ^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x] - Sin[x])^(-6),x]

[Out]

(-8*Tan[x])/693 - (4*Sec[x]^2*Tan[x])/693 - (Sec[x]^4*Tan[x])/231 + (113*Sec[x]^6*Tan[x])/693 - (23*Sec[x]^8*T
an[x])/99 + (Sec[x]^10*Tan[x])/11

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fricas [B]  time = 0.86, size = 40, normalized size = 1.60 \[ -\frac {{\left (8 \, \cos \relax (x)^{10} + 4 \, \cos \relax (x)^{8} + 3 \, \cos \relax (x)^{6} - 113 \, \cos \relax (x)^{4} + 161 \, \cos \relax (x)^{2} - 63\right )} \sin \relax (x)}{693 \, \cos \relax (x)^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/693*(8*cos(x)^10 + 4*cos(x)^8 + 3*cos(x)^6 - 113*cos(x)^4 + 161*cos(x)^2 - 63)*sin(x)/cos(x)^11

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giac [A]  time = 0.13, size = 19, normalized size = 0.76 \[ \frac {1}{11} \, \tan \relax (x)^{11} + \frac {2}{9} \, \tan \relax (x)^{9} + \frac {1}{7} \, \tan \relax (x)^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*tan(x)^11 + 2/9*tan(x)^9 + 1/7*tan(x)^7

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maple [A]  time = 0.12, size = 20, normalized size = 0.80 \[ \frac {\left (\tan ^{7}\relax (x )\right )}{7}+\frac {2 \left (\tan ^{9}\relax (x )\right )}{9}+\frac {\left (\tan ^{11}\relax (x )\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(csc(x)-sin(x))^6,x)

[Out]

1/7*tan(x)^7+2/9*tan(x)^9+1/11*tan(x)^11

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maxima [A]  time = 0.35, size = 19, normalized size = 0.76 \[ \frac {1}{11} \, \tan \relax (x)^{11} + \frac {2}{9} \, \tan \relax (x)^{9} + \frac {1}{7} \, \tan \relax (x)^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*tan(x)^11 + 2/9*tan(x)^9 + 1/7*tan(x)^7

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mupad [B]  time = 2.91, size = 33, normalized size = 1.32 \[ \frac {8\,{\cos \relax (x)}^4\,{\sin \relax (x)}^7+28\,{\cos \relax (x)}^2\,{\sin \relax (x)}^7+63\,{\sin \relax (x)}^7}{693\,{\cos \relax (x)}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x) - 1/sin(x))^6,x)

[Out]

(63*sin(x)^7 + 28*cos(x)^2*sin(x)^7 + 8*cos(x)^4*sin(x)^7)/(693*cos(x)^11)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \sin {\relax (x )} + \csc {\relax (x )}\right )^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))**6,x)

[Out]

Integral((-sin(x) + csc(x))**(-6), x)

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