3.281 \(\int \frac{1}{(\sec (x)+\tan (x))^4} \, dx\)

Optimal. Leaf size=26 \[ x-\frac{2 \cos ^3(x)}{3 (\sin (x)+1)^3}+\frac{2 \cos (x)}{\sin (x)+1} \]

[Out]

x - (2*Cos[x]^3)/(3*(1 + Sin[x])^3) + (2*Cos[x])/(1 + Sin[x])

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Rubi [A]  time = 0.0689964, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4391, 2680, 8} \[ x-\frac{2 \cos ^3(x)}{3 (\sin (x)+1)^3}+\frac{2 \cos (x)}{\sin (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[x] + Tan[x])^(-4),x]

[Out]

x - (2*Cos[x]^3)/(3*(1 + Sin[x])^3) + (2*Cos[x])/(1 + Sin[x])

Rule 4391

Int[(u_.)*((b_.)*sec[(c_.) + (d_.)*(x_)]^(n_.) + (a_.)*tan[(c_.) + (d_.)*(x_)]^(n_.))^(p_), x_Symbol] :> Int[A
ctivateTrig[u]*Sec[c + d*x]^(n*p)*(b + a*Sin[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rule 2680

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(2*g*(
g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(2*m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(2*m +
p + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{(\sec (x)+\tan (x))^4} \, dx &=\int \frac{\cos ^4(x)}{(1+\sin (x))^4} \, dx\\ &=-\frac{2 \cos ^3(x)}{3 (1+\sin (x))^3}-\int \frac{\cos ^2(x)}{(1+\sin (x))^2} \, dx\\ &=-\frac{2 \cos ^3(x)}{3 (1+\sin (x))^3}+\frac{2 \cos (x)}{1+\sin (x)}+\int 1 \, dx\\ &=x-\frac{2 \cos ^3(x)}{3 (1+\sin (x))^3}+\frac{2 \cos (x)}{1+\sin (x)}\\ \end{align*}

Mathematica [B]  time = 0.0747828, size = 62, normalized size = 2.38 \[ \frac{3 (3 x-8) \cos \left (\frac{x}{2}\right )+(16-3 x) \cos \left (\frac{3 x}{2}\right )+6 \sin \left (\frac{x}{2}\right ) (2 x+x \cos (x)-4)}{6 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[x] + Tan[x])^(-4),x]

[Out]

(3*(-8 + 3*x)*Cos[x/2] + (16 - 3*x)*Cos[(3*x)/2] + 6*(-4 + 2*x + x*Cos[x])*Sin[x/2])/(6*(Cos[x/2] + Sin[x/2])^
3)

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Maple [A]  time = 0.072, size = 23, normalized size = 0.9 \begin{align*} -{\frac{16}{3} \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) ^{-3}}+8\, \left ( 1+\tan \left ( x/2 \right ) \right ) ^{-2}+x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sec(x)+tan(x))^4,x)

[Out]

-16/3/(1+tan(1/2*x))^3+8/(1+tan(1/2*x))^2+x

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Maxima [B]  time = 1.51288, size = 86, normalized size = 3.31 \begin{align*} \frac{8 \,{\left (\frac{3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}}{3 \,{\left (\frac{3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{3 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{\sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + 1\right )}} + 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/3*(3*sin(x)/(cos(x) + 1) + 1)/(3*sin(x)/(cos(x) + 1) + 3*sin(x)^2/(cos(x) + 1)^2 + sin(x)^3/(cos(x) + 1)^3 +
1) + 2*arctan(sin(x)/(cos(x) + 1))

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Fricas [B]  time = 2.149, size = 188, normalized size = 7.23 \begin{align*} \frac{{\left (3 \, x - 8\right )} \cos \left (x\right )^{2} -{\left (3 \, x + 4\right )} \cos \left (x\right ) -{\left ({\left (3 \, x + 8\right )} \cos \left (x\right ) + 6 \, x + 4\right )} \sin \left (x\right ) - 6 \, x + 4}{3 \,{\left (\cos \left (x\right )^{2} -{\left (\cos \left (x\right ) + 2\right )} \sin \left (x\right ) - \cos \left (x\right ) - 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((3*x - 8)*cos(x)^2 - (3*x + 4)*cos(x) - ((3*x + 8)*cos(x) + 6*x + 4)*sin(x) - 6*x + 4)/(cos(x)^2 - (cos(x
) + 2)*sin(x) - cos(x) - 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sec(x)+tan(x))**4,x)

[Out]

Integral((tan(x) + sec(x))**(-4), x)

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Giac [A]  time = 1.17076, size = 27, normalized size = 1.04 \begin{align*} x + \frac{8 \,{\left (3 \, \tan \left (\frac{1}{2} \, x\right ) + 1\right )}}{3 \,{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 8/3*(3*tan(1/2*x) + 1)/(tan(1/2*x) + 1)^3