3.124 \(\int \cos (x) \csc (3 x) \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{3} \log (\sin (x))-\frac{1}{6} \log \left (3-4 \sin ^2(x)\right ) \]

[Out]

Log[Sin[x]]/3 - Log[3 - 4*Sin[x]^2]/6

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Rubi [A]  time = 0.0257293, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {4356, 266, 36, 31, 29} \[ \frac{1}{3} \log (\sin (x))-\frac{1}{6} \log \left (3-4 \sin ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Csc[3*x],x]

[Out]

Log[Sin[x]]/3 - Log[3 - 4*Sin[x]^2]/6

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) \csc (3 x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{x \left (3-4 x^2\right )} \, dx,x,\sin (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(3-4 x) x} \, dx,x,\sin ^2(x)\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\sin ^2(x)\right )+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{3-4 x} \, dx,x,\sin ^2(x)\right )\\ &=\frac{1}{3} \log (\sin (x))-\frac{1}{6} \log \left (3-4 \sin ^2(x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0084318, size = 21, normalized size = 1. \[ \frac{1}{3} \log (\sin (x))-\frac{1}{6} \log \left (3-4 \sin ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Csc[3*x],x]

[Out]

Log[Sin[x]]/3 - Log[3 - 4*Sin[x]^2]/6

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \text{hanged} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*csc(3*x),x)

[Out]

int(cos(x)*csc(3*x),x)

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Maxima [B]  time = 1.52045, size = 174, normalized size = 8.29 \begin{align*} -\frac{1}{12} \, \log \left (2 \,{\left (\cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - \frac{1}{12} \, \log \left (-2 \,{\left (\cos \left (x\right ) - 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{6} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{6} \, \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)*csc(3*x),x, algorithm="maxima")

[Out]

-1/12*log(2*(cos(x) + 1)*cos(2*x) + cos(2*x)^2 + cos(x)^2 + sin(2*x)^2 + 2*sin(2*x)*sin(x) + sin(x)^2 + 2*cos(
x) + 1) - 1/12*log(-2*(cos(x) - 1)*cos(2*x) + cos(2*x)^2 + cos(x)^2 + sin(2*x)^2 - 2*sin(2*x)*sin(x) + sin(x)^
2 - 2*cos(x) + 1) + 1/6*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 1/6*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)

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Fricas [A]  time = 2.53787, size = 65, normalized size = 3.1 \begin{align*} -\frac{1}{6} \, \log \left (4 \, \cos \left (x\right )^{2} - 1\right ) + \frac{1}{3} \, \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)*csc(3*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.88602, size = 17, normalized size = 0.81 \begin{align*} - \frac{\log{\left (4 \sin ^{2}{\left (x \right )} - 3 \right )}}{6} + \frac{\log{\left (\sin{\left (x \right )} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(3*x),x)

[Out]

-log(4*sin(x)**2 - 3)/6 + log(sin(x))/3

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Giac [A]  time = 1.11439, size = 41, normalized size = 1.95 \begin{align*} -\frac{1}{6} \, \log \left ({\left | -\frac{3 \,{\left (\cos \left (x\right ) + 1\right )}}{\cos \left (x\right ) - 1} - \frac{3 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 10 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(3*x),x, algorithm="giac")

[Out]

-1/6*log(abs(-3*(cos(x) + 1)/(cos(x) - 1) - 3*(cos(x) - 1)/(cos(x) + 1) - 10))