3.310 \(\int \frac{1}{(\csc (x)-\sin (x))^4} \, dx\)

Optimal. Leaf size=17 \[ \frac{\tan ^7(x)}{7}+\frac{\tan ^5(x)}{5} \]

[Out]

Tan[x]^5/5 + Tan[x]^7/7

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Rubi [A]  time = 0.0177155, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 0, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \frac{\tan ^7(x)}{7}+\frac{\tan ^5(x)}{5} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x] - Sin[x])^(-4),x]

[Out]

Tan[x]^5/5 + Tan[x]^7/7

Rubi steps

\begin{align*} \int \frac{1}{(\csc (x)-\sin (x))^4} \, dx &=\operatorname{Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,\tan (x)\right )\\ &=\frac{\tan ^5(x)}{5}+\frac{\tan ^7(x)}{7}\\ \end{align*}

Mathematica [B]  time = 0.0173619, size = 37, normalized size = 2.18 \[ \frac{2 \tan (x)}{35}+\frac{1}{7} \tan (x) \sec ^6(x)-\frac{8}{35} \tan (x) \sec ^4(x)+\frac{1}{35} \tan (x) \sec ^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x] - Sin[x])^(-4),x]

[Out]

(2*Tan[x])/35 + (Sec[x]^2*Tan[x])/35 - (8*Sec[x]^4*Tan[x])/35 + (Sec[x]^6*Tan[x])/7

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Maple [A]  time = 0.043, size = 14, normalized size = 0.8 \begin{align*}{\frac{ \left ( \tan \left ( x \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \tan \left ( x \right ) \right ) ^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(csc(x)-sin(x))^4,x)

[Out]

1/5*tan(x)^5+1/7*tan(x)^7

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Maxima [A]  time = 0.988165, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{7} \, \tan \left (x\right )^{7} + \frac{1}{5} \, \tan \left (x\right )^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^4,x, algorithm="maxima")

[Out]

1/7*tan(x)^7 + 1/5*tan(x)^5

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Fricas [A]  time = 2.13188, size = 85, normalized size = 5. \begin{align*} \frac{{\left (2 \, \cos \left (x\right )^{6} + \cos \left (x\right )^{4} - 8 \, \cos \left (x\right )^{2} + 5\right )} \sin \left (x\right )}{35 \, \cos \left (x\right )^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^4,x, algorithm="fricas")

[Out]

1/35*(2*cos(x)^6 + cos(x)^4 - 8*cos(x)^2 + 5)*sin(x)/cos(x)^7

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- \sin{\left (x \right )} + \csc{\left (x \right )}\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))**4,x)

[Out]

Integral((-sin(x) + csc(x))**(-4), x)

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Giac [A]  time = 1.14984, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{7} \, \tan \left (x\right )^{7} + \frac{1}{5} \, \tan \left (x\right )^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^4,x, algorithm="giac")

[Out]

1/7*tan(x)^7 + 1/5*tan(x)^5