3.374 \(\int \cos (x) \cot (x) \sec (3 x) \, dx\)

Optimal. Leaf size=11 \[ -\frac{1}{2} \log \left (\csc ^2(x)-4\right ) \]

[Out]

-Log[-4 + Csc[x]^2]/2

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Rubi [A]  time = 0.0332537, antiderivative size = 17, normalized size of antiderivative = 1.55, number of steps used = 5, number of rules used = 5, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {4356, 266, 36, 31, 29} \[ \log (\sin (x))-\frac{1}{2} \log \left (1-4 \sin ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Cot[x]*Sec[3*x],x]

[Out]

Log[Sin[x]] - Log[1 - 4*Sin[x]^2]/2

Rule 4356

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0130305, size = 17, normalized size = 1.55 \[ \log (\sin (x))-\frac{1}{2} \log \left (1-4 \sin ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Cot[x]*Sec[3*x],x]

[Out]

Log[Sin[x]] - Log[1 - 4*Sin[x]^2]/2

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Maple [B]  time = 0.044, size = 27, normalized size = 2.5 \begin{align*}{\frac{\ln \left ( \cos \left ( x \right ) +1 \right ) }{2}}+{\frac{\ln \left ( \cos \left ( x \right ) -1 \right ) }{2}}-{\frac{\ln \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}-3 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2/cos(3*x)/sin(x),x)

[Out]

1/2*ln(cos(x)+1)+1/2*ln(cos(x)-1)-1/2*ln(4*cos(x)^2-3)

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Maxima [B]  time = 0.955627, size = 124, normalized size = 11.27 \begin{align*} -\frac{1}{4} \, \log \left (-2 \,{\left (\cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} - 2 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right ) + \frac{1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2/cos(3*x)/sin(x),x, algorithm="maxima")

[Out]

-1/4*log(-2*(cos(2*x) - 1)*cos(4*x) + cos(4*x)^2 + cos(2*x)^2 + sin(4*x)^2 - 2*sin(4*x)*sin(2*x) + sin(2*x)^2
- 2*cos(2*x) + 1) + 1/2*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 1/2*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)

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Fricas [A]  time = 2.21157, size = 59, normalized size = 5.36 \begin{align*} -\frac{1}{2} \, \log \left (4 \, \cos \left (x\right )^{2} - 3\right ) + \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2/cos(3*x)/sin(x),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{2}{\left (x \right )}}{\sin{\left (x \right )} \cos{\left (3 x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**2/cos(3*x)/sin(x),x)

[Out]

Integral(cos(x)**2/(sin(x)*cos(3*x)), x)

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Giac [B]  time = 1.11134, size = 32, normalized size = 2.91 \begin{align*} \frac{1}{2} \, \log \left (-\cos \left (x\right )^{2} + 1\right ) - \frac{1}{2} \, \log \left ({\left | 4 \, \cos \left (x\right )^{2} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2/cos(3*x)/sin(x),x, algorithm="giac")

[Out]

1/2*log(-cos(x)^2 + 1) - 1/2*log(abs(4*cos(x)^2 - 3))