3.437 \(\int \frac{(-3+\cos (2 x)) \sec ^4(x)}{\sqrt{4-\cot ^2(x)}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{1}{3} \tan ^3(x) \sqrt{4-\cot ^2(x)}-\frac{2}{3} \tan (x) \sqrt{4-\cot ^2(x)} \]

[Out]

(-2*Sqrt[4 - Cot[x]^2]*Tan[x])/3 - (Sqrt[4 - Cot[x]^2]*Tan[x]^3)/3

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Rubi [A]  time = 0.149407, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {12, 434, 453, 191} \[ -\frac{1}{3} \tan ^3(x) \sqrt{4-\cot ^2(x)}-\frac{2}{3} \tan (x) \sqrt{4-\cot ^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[((-3 + Cos[2*x])*Sec[x]^4)/Sqrt[4 - Cot[x]^2],x]

[Out]

(-2*Sqrt[4 - Cot[x]^2]*Tan[x])/3 - (Sqrt[4 - Cot[x]^2]*Tan[x]^3)/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 434

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[((a + b*x^n)^p*(d + c*x
^n)^q)/x^(n*q), x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] ||  !IntegerQ[p])

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{(-3+\cos (2 x)) \sec ^4(x)}{\sqrt{4-\cot ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{2 \left (-1-2 x^2\right )}{\sqrt{4-\frac{1}{x^2}}} \, dx,x,\tan (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{-1-2 x^2}{\sqrt{4-\frac{1}{x^2}}} \, dx,x,\tan (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{\left (-2-\frac{1}{x^2}\right ) x^2}{\sqrt{4-\frac{1}{x^2}}} \, dx,x,\tan (x)\right )\\ &=-\frac{1}{3} \sqrt{4-\cot ^2(x)} \tan ^3(x)-\frac{8}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{4-\frac{1}{x^2}}} \, dx,x,\tan (x)\right )\\ &=-\frac{2}{3} \sqrt{4-\cot ^2(x)} \tan (x)-\frac{1}{3} \sqrt{4-\cot ^2(x)} \tan ^3(x)\\ \end{align*}

Mathematica [A]  time = 0.0784498, size = 36, normalized size = 0.92 \[ \frac{(\cos (2 x)+3) (5 \cos (2 x)-3) \csc (x) \sec ^3(x)}{12 \sqrt{4-\cot ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[((-3 + Cos[2*x])*Sec[x]^4)/Sqrt[4 - Cot[x]^2],x]

[Out]

((3 + Cos[2*x])*(-3 + 5*Cos[2*x])*Csc[x]*Sec[x]^3)/(12*Sqrt[4 - Cot[x]^2])

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Maple [A]  time = 0.457, size = 61, normalized size = 1.6 \begin{align*} -{\frac{ \left ( 5\, \left ( \cos \left ( x \right ) \right ) ^{2}+2 \right ) \sin \left ( x \right ) \sqrt{4}}{12\, \left ( \cos \left ( x \right ) \right ) ^{3}}\sqrt{-{\frac{-4+5\, \left ( \cos \left ( x \right ) \right ) ^{2}}{ \left ( \sin \left ( x \right ) \right ) ^{2}}}}}+{\frac{\sin \left ( x \right ) }{2\,\cos \left ( x \right ) }\sqrt{-{\frac{-4+5\, \left ( \cos \left ( x \right ) \right ) ^{2}}{ \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3+cos(2*x))/cos(x)^4/(4-cot(x)^2)^(1/2),x)

[Out]

-1/12*(5*cos(x)^2+2)*sin(x)*(-(-4+5*cos(x)^2)/sin(x)^2)^(1/2)*4^(1/2)/cos(x)^3+1/2*(-(-4+5*cos(x)^2)/sin(x)^2)
^(1/2)*sin(x)/cos(x)

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Maxima [B]  time = 0.975043, size = 85, normalized size = 2.18 \begin{align*} -\frac{1}{48} \,{\left (-\frac{1}{\tan \left (x\right )^{2}} + 4\right )}^{\frac{3}{2}} \tan \left (x\right )^{3} + \frac{3}{16} \, \sqrt{-\frac{1}{\tan \left (x\right )^{2}} + 4} \tan \left (x\right ) - \frac{8 \, \tan \left (x\right )^{4} + 26 \, \tan \left (x\right )^{2} - 7}{8 \, \sqrt{2 \, \tan \left (x\right ) + 1} \sqrt{2 \, \tan \left (x\right ) - 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+cos(2*x))/cos(x)^4/(4-cot(x)^2)^(1/2),x, algorithm="maxima")

[Out]

-1/48*(-1/tan(x)^2 + 4)^(3/2)*tan(x)^3 + 3/16*sqrt(-1/tan(x)^2 + 4)*tan(x) - 1/8*(8*tan(x)^4 + 26*tan(x)^2 - 7
)/(sqrt(2*tan(x) + 1)*sqrt(2*tan(x) - 1))

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Fricas [A]  time = 3.44099, size = 101, normalized size = 2.59 \begin{align*} -\frac{{\left (\cos \left (x\right )^{2} + 1\right )} \sqrt{\frac{5 \, \cos \left (x\right )^{2} - 4}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right )}{3 \, \cos \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+cos(2*x))/cos(x)^4/(4-cot(x)^2)^(1/2),x, algorithm="fricas")

[Out]

-1/3*(cos(x)^2 + 1)*sqrt((5*cos(x)^2 - 4)/(cos(x)^2 - 1))*sin(x)/cos(x)^3

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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((-3+cos(2*x))/cos(x)**4/(4-cot(x)**2)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (2 \, x\right ) - 3}{\sqrt{-\cot \left (x\right )^{2} + 4} \cos \left (x\right )^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+cos(2*x))/cos(x)^4/(4-cot(x)^2)^(1/2),x, algorithm="giac")

[Out]

integrate((cos(2*x) - 3)/(sqrt(-cot(x)^2 + 4)*cos(x)^4), x)