3.313 \(\int \frac{1}{(\csc (x)-\sin (x))^7} \, dx\)

Optimal. Leaf size=33 \[ \frac{\sec ^{13}(x)}{13}-\frac{3 \sec ^{11}(x)}{11}+\frac{\sec ^9(x)}{3}-\frac{\sec ^7(x)}{7} \]

[Out]

-Sec[x]^7/7 + Sec[x]^9/3 - (3*Sec[x]^11)/11 + Sec[x]^13/13

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Rubi [A]  time = 0.0426652, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4397, 2606, 270} \[ \frac{\sec ^{13}(x)}{13}-\frac{3 \sec ^{11}(x)}{11}+\frac{\sec ^9(x)}{3}-\frac{\sec ^7(x)}{7} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x] - Sin[x])^(-7),x]

[Out]

-Sec[x]^7/7 + Sec[x]^9/3 - (3*Sec[x]^11)/11 + Sec[x]^13/13

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

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))^7} \, dx &=\int \sec ^7(x) \tan ^7(x) \, dx\\ &=\operatorname{Subst}\left (\int x^6 \left (-1+x^2\right )^3 \, dx,x,\sec (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-x^6+3 x^8-3 x^{10}+x^{12}\right ) \, dx,x,\sec (x)\right )\\ &=-\frac{1}{7} \sec ^7(x)+\frac{\sec ^9(x)}{3}-\frac{3 \sec ^{11}(x)}{11}+\frac{\sec ^{13}(x)}{13}\\ \end{align*}

Mathematica [A]  time = 0.0173609, size = 33, normalized size = 1. \[ \frac{\sec ^{13}(x)}{13}-\frac{3 \sec ^{11}(x)}{11}+\frac{\sec ^9(x)}{3}-\frac{\sec ^7(x)}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x] - Sin[x])^(-7),x]

[Out]

-Sec[x]^7/7 + Sec[x]^9/3 - (3*Sec[x]^11)/11 + Sec[x]^13/13

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Maple [A]  time = 0.058, size = 26, normalized size = 0.8 \begin{align*} -{\frac{1}{7\, \left ( \cos \left ( x \right ) \right ) ^{7}}}+{\frac{1}{3\, \left ( \cos \left ( x \right ) \right ) ^{9}}}+{\frac{1}{13\, \left ( \cos \left ( x \right ) \right ) ^{13}}}-{\frac{3}{11\, \left ( \cos \left ( x \right ) \right ) ^{11}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(csc(x)-sin(x))^7,x)

[Out]

-1/7/cos(x)^7+1/3/cos(x)^9+1/13/cos(x)^13-3/11/cos(x)^11

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Maxima [B]  time = 1.05678, size = 366, normalized size = 11.09 \begin{align*} -\frac{32 \,{\left (\frac{13 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{78 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{286 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac{2288 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac{10296 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} + \frac{16302 \, \sin \left (x\right )^{12}}{{\left (\cos \left (x\right ) + 1\right )}^{12}} + \frac{18018 \, \sin \left (x\right )^{14}}{{\left (\cos \left (x\right ) + 1\right )}^{14}} + \frac{9009 \, \sin \left (x\right )^{16}}{{\left (\cos \left (x\right ) + 1\right )}^{16}} + \frac{3003 \, \sin \left (x\right )^{18}}{{\left (\cos \left (x\right ) + 1\right )}^{18}} - 1\right )}}{3003 \,{\left (\frac{13 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{78 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{286 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac{715 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac{1287 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} - \frac{1716 \, \sin \left (x\right )^{12}}{{\left (\cos \left (x\right ) + 1\right )}^{12}} + \frac{1716 \, \sin \left (x\right )^{14}}{{\left (\cos \left (x\right ) + 1\right )}^{14}} - \frac{1287 \, \sin \left (x\right )^{16}}{{\left (\cos \left (x\right ) + 1\right )}^{16}} + \frac{715 \, \sin \left (x\right )^{18}}{{\left (\cos \left (x\right ) + 1\right )}^{18}} - \frac{286 \, \sin \left (x\right )^{20}}{{\left (\cos \left (x\right ) + 1\right )}^{20}} + \frac{78 \, \sin \left (x\right )^{22}}{{\left (\cos \left (x\right ) + 1\right )}^{22}} - \frac{13 \, \sin \left (x\right )^{24}}{{\left (\cos \left (x\right ) + 1\right )}^{24}} + \frac{\sin \left (x\right )^{26}}{{\left (\cos \left (x\right ) + 1\right )}^{26}} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^7,x, algorithm="maxima")

[Out]

-32/3003*(13*sin(x)^2/(cos(x) + 1)^2 - 78*sin(x)^4/(cos(x) + 1)^4 + 286*sin(x)^6/(cos(x) + 1)^6 + 2288*sin(x)^
8/(cos(x) + 1)^8 + 10296*sin(x)^10/(cos(x) + 1)^10 + 16302*sin(x)^12/(cos(x) + 1)^12 + 18018*sin(x)^14/(cos(x)
+ 1)^14 + 9009*sin(x)^16/(cos(x) + 1)^16 + 3003*sin(x)^18/(cos(x) + 1)^18 - 1)/(13*sin(x)^2/(cos(x) + 1)^2 -
78*sin(x)^4/(cos(x) + 1)^4 + 286*sin(x)^6/(cos(x) + 1)^6 - 715*sin(x)^8/(cos(x) + 1)^8 + 1287*sin(x)^10/(cos(x
) + 1)^10 - 1716*sin(x)^12/(cos(x) + 1)^12 + 1716*sin(x)^14/(cos(x) + 1)^14 - 1287*sin(x)^16/(cos(x) + 1)^16 +
715*sin(x)^18/(cos(x) + 1)^18 - 286*sin(x)^20/(cos(x) + 1)^20 + 78*sin(x)^22/(cos(x) + 1)^22 - 13*sin(x)^24/(
cos(x) + 1)^24 + sin(x)^26/(cos(x) + 1)^26 - 1)

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Fricas [A]  time = 2.21782, size = 96, normalized size = 2.91 \begin{align*} -\frac{429 \, \cos \left (x\right )^{6} - 1001 \, \cos \left (x\right )^{4} + 819 \, \cos \left (x\right )^{2} - 231}{3003 \, \cos \left (x\right )^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^7,x, algorithm="fricas")

[Out]

-1/3003*(429*cos(x)^6 - 1001*cos(x)^4 + 819*cos(x)^2 - 231)/cos(x)^13

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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))**7,x)

[Out]

Timed out

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Giac [B]  time = 1.19071, size = 193, normalized size = 5.85 \begin{align*} -\frac{32 \,{\left (\frac{13 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac{78 \,{\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{286 \,{\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac{2288 \,{\left (\cos \left (x\right ) - 1\right )}^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{10296 \,{\left (\cos \left (x\right ) - 1\right )}^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac{16302 \,{\left (\cos \left (x\right ) - 1\right )}^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac{18018 \,{\left (\cos \left (x\right ) - 1\right )}^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} - \frac{9009 \,{\left (\cos \left (x\right ) - 1\right )}^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac{3003 \,{\left (\cos \left (x\right ) - 1\right )}^{9}}{{\left (\cos \left (x\right ) + 1\right )}^{9}} + 1\right )}}{3003 \,{\left (\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right )}^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^7,x, algorithm="giac")

[Out]

-32/3003*(13*(cos(x) - 1)/(cos(x) + 1) + 78*(cos(x) - 1)^2/(cos(x) + 1)^2 + 286*(cos(x) - 1)^3/(cos(x) + 1)^3
- 2288*(cos(x) - 1)^4/(cos(x) + 1)^4 + 10296*(cos(x) - 1)^5/(cos(x) + 1)^5 - 16302*(cos(x) - 1)^6/(cos(x) + 1)
^6 + 18018*(cos(x) - 1)^7/(cos(x) + 1)^7 - 9009*(cos(x) - 1)^8/(cos(x) + 1)^8 + 3003*(cos(x) - 1)^9/(cos(x) +
1)^9 + 1)/((cos(x) - 1)/(cos(x) + 1) + 1)^13