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]

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

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Rubi [A]  time = 0.0219173, antiderivative size = 25, normalized size of antiderivative = 1., 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*}

Mathematica [B]  time = 0.017864, 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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Maple [A]  time = 0.053, size = 20, normalized size = 0.8 \begin{align*}{\frac{ \left ( \tan \left ( x \right ) \right ) ^{7}}{7}}+{\frac{2\, \left ( \tan \left ( x \right ) \right ) ^{9}}{9}}+{\frac{ \left ( \tan \left ( x \right ) \right ) ^{11}}{11}} \end{align*}

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 = 0.984245, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{11} \, \tan \left (x\right )^{11} + \frac{2}{9} \, \tan \left (x\right )^{9} + \frac{1}{7} \, \tan \left (x\right )^{7} \end{align*}

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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Fricas [B]  time = 2.07, size = 135, normalized size = 5.4 \begin{align*} -\frac{{\left (8 \, \cos \left (x\right )^{10} + 4 \, \cos \left (x\right )^{8} + 3 \, \cos \left (x\right )^{6} - 113 \, \cos \left (x\right )^{4} + 161 \, \cos \left (x\right )^{2} - 63\right )} \sin \left (x\right )}{693 \, \cos \left (x\right )^{11}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.15698, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{11} \, \tan \left (x\right )^{11} + \frac{2}{9} \, \tan \left (x\right )^{9} + \frac{1}{7} \, \tan \left (x\right )^{7} \end{align*}

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