3.4.74 \(\int \cos (x) \cot (x) \sec (3 x) \, dx\) [374]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {4441, 272, 36, 31, 29} \begin {gather*} \log (\sin (x))-\frac {1}{2} \log \left (1-4 \sin ^2(x)\right ) \end {gather*}

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 29

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

Rule 31

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

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 272

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 4441

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])

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.55 \begin {gather*} \log (\sin (x))-\frac {1}{2} \log \left (1-4 \sin ^2(x)\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(26\) vs. \(2(9)=18\).
time = 0.09, size = 27, normalized size = 2.45

method	result	size
default	\(\frac {\ln \left (1+\cos \left (x \right )\right )}{2}-\frac {\ln \left (4 \left (\cos ^{2}\left (x \right )\right )-3\right )}{2}+\frac {\ln \left (\cos \left (x \right )-1\right )}{2}\)	\(27\)
risch	\(\ln \left ({\mathrm e}^{2 i x}-1\right )-\frac {\ln \left ({\mathrm e}^{4 i x}-{\mathrm e}^{2 i x}+1\right )}{2}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (9) = 18\).
time = 2.09, size = 92, normalized size = 8.36 \begin {gather*} -\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 {gather*}

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 = 1.41, size = 17, normalized size = 1.55 \begin {gather*} -\frac {1}{2} \, \log \left (4 \, \cos \left (x\right )^{2} - 3\right ) + \log \left (\frac {1}{2} \, \sin \left (x\right )\right ) \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{2}{\left (x \right )}}{\sin {\left (x \right )} \cos {\left (3 x \right )}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (9) = 18\).
time = 0.66, size = 24, normalized size = 2.18 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 0.62, size = 25, normalized size = 2.27 \begin {gather*} \ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-\frac {\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^4-14\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(tan(x/2)) - log(tan(x/2)^4 - 14*tan(x/2)^2 + 1)/2

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