3.329 \(\int \frac {1}{(-\cos (x)+\sec (x))^4} \, dx\)

Optimal. Leaf size=17 \[ -\frac {1}{7} \cot ^7(x)-\frac {\cot ^5(x)}{5} \]

[Out]

-1/5*cot(x)^5-1/7*cot(x)^7

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Rubi [A]  time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 0, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ -\frac {1}{7} \cot ^7(x)-\frac {\cot ^5(x)}{5} \]

Antiderivative was successfully verified.

[In]

Int[(-Cos[x] + Sec[x])^(-4),x]

[Out]

-Cot[x]^5/5 - Cot[x]^7/7

Rubi steps

\begin {align*} \int \frac {1}{(-\cos (x)+\sec (x))^4} \, dx &=\operatorname {Subst}\left (\int \left (\frac {1}{x^8}+\frac {1}{x^6}\right ) \, dx,x,\tan (x)\right )\\ &=-\frac {1}{5} \cot ^5(x)-\frac {\cot ^7(x)}{7}\\ \end {align*}

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Mathematica [B]  time = 0.02, size = 37, normalized size = 2.18 \[ -\frac {2 \cot (x)}{35}-\frac {1}{7} \cot (x) \csc ^6(x)+\frac {8}{35} \cot (x) \csc ^4(x)-\frac {1}{35} \cot (x) \csc ^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-Cos[x] + Sec[x])^(-4),x]

[Out]

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

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fricas [B]  time = 0.65, size = 39, normalized size = 2.29 \[ -\frac {2 \, \cos \relax (x)^{7} - 7 \, \cos \relax (x)^{5}}{35 \, {\left (\cos \relax (x)^{6} - 3 \, \cos \relax (x)^{4} + 3 \, \cos \relax (x)^{2} - 1\right )} \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-cos(x)+sec(x))^4,x, algorithm="fricas")

[Out]

-1/35*(2*cos(x)^7 - 7*cos(x)^5)/((cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*sin(x))

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giac [A]  time = 0.14, size = 14, normalized size = 0.82 \[ -\frac {7 \, \tan \relax (x)^{2} + 5}{35 \, \tan \relax (x)^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-cos(x)+sec(x))^4,x, algorithm="giac")

[Out]

-1/35*(7*tan(x)^2 + 5)/tan(x)^7

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maple [A]  time = 0.11, size = 14, normalized size = 0.82 \[ -\frac {1}{7 \tan \relax (x )^{7}}-\frac {1}{5 \tan \relax (x )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-cos(x)+sec(x))^4,x)

[Out]

-1/7/tan(x)^7-1/5/tan(x)^5

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maxima [A]  time = 0.32, size = 14, normalized size = 0.82 \[ -\frac {7 \, \tan \relax (x)^{2} + 5}{35 \, \tan \relax (x)^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-cos(x)+sec(x))^4,x, algorithm="maxima")

[Out]

-1/35*(7*tan(x)^2 + 5)/tan(x)^7

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mupad [B]  time = 2.43, size = 16, normalized size = 0.94 \[ \frac {{\cos \relax (x)}^5\,\left (\cos \left (2\,x\right )-6\right )}{35\,{\sin \relax (x)}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x) - 1/cos(x))^4,x)

[Out]

(cos(x)^5*(cos(2*x) - 6))/(35*sin(x)^7)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \cos {\relax (x )} + \sec {\relax (x )}\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-cos(x)+sec(x))**4,x)

[Out]

Integral((-cos(x) + sec(x))**(-4), x)

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