3.125 \(\int \cos (x) \csc (4 x) \, dx\)

Optimal. Leaf size=26 \[ \frac {\tanh ^{-1}\left (\sqrt {2} \cos (x)\right )}{2 \sqrt {2}}-\frac {1}{4} \tanh ^{-1}(\cos (x)) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1093, 206} \[ \frac {\tanh ^{-1}\left (\sqrt {2} \cos (x)\right )}{2 \sqrt {2}}-\frac {1}{4} \tanh ^{-1}(\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Csc[4*x],x]

[Out]

-ArcTanh[Cos[x]]/4 + ArcTanh[Sqrt[2]*Cos[x]]/(2*Sqrt[2])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 1093

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \cos (x) \csc (4 x) \, dx &=-\operatorname {Subst}\left (\int \frac {1}{-4+12 x^2-8 x^4} \, dx,x,\cos (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{4-8 x^2} \, dx,x,\cos (x)\right )-2 \operatorname {Subst}\left (\int \frac {1}{8-8 x^2} \, dx,x,\cos (x)\right )\\ &=-\frac {1}{4} \tanh ^{-1}(\cos (x))+\frac {\tanh ^{-1}\left (\sqrt {2} \cos (x)\right )}{2 \sqrt {2}}\\ \end {align*}

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Mathematica [C]  time = 0.06, size = 66, normalized size = 2.54 \[ \frac {1}{4} \left (\log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )\right )+(1+i) (-1)^{3/4} \tanh ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )-1}{\sqrt {2}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )+1}{\sqrt {2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Csc[4*x],x]

[Out]

((1 + I)*(-1)^(3/4)*ArcTanh[(-1 + Tan[x/2])/Sqrt[2]] + Sqrt[2]*ArcTanh[(1 + Tan[x/2])/Sqrt[2]] - Log[Cos[x/2]]
+ Log[Sin[x/2]])/4

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fricas [B]  time = 0.83, size = 52, normalized size = 2.00 \[ \frac {1}{8} \, \sqrt {2} \log \left (-\frac {2 \, \cos \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) + 1}{2 \, \cos \relax (x)^{2} - 1}\right ) - \frac {1}{8} \, \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{8} \, \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(4*x),x, algorithm="fricas")

[Out]

1/8*sqrt(2)*log(-(2*cos(x)^2 + 2*sqrt(2)*cos(x) + 1)/(2*cos(x)^2 - 1)) - 1/8*log(1/2*cos(x) + 1/2) + 1/8*log(-
1/2*cos(x) + 1/2)

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giac [B]  time = 0.13, size = 48, normalized size = 1.85 \[ -\frac {1}{8} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \cos \relax (x) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \cos \relax (x) \right |}}\right ) - \frac {1}{8} \, \log \left (\cos \relax (x) + 1\right ) + \frac {1}{8} \, \log \left (-\cos \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(4*x),x, algorithm="giac")

[Out]

-1/8*sqrt(2)*log(abs(-2*sqrt(2) + 4*cos(x))/abs(2*sqrt(2) + 4*cos(x))) - 1/8*log(cos(x) + 1) + 1/8*log(-cos(x)
+ 1)

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maple [A]  time = 0.14, size = 28, normalized size = 1.08 \[ \frac {\ln \left (-1+\cos \relax (x )\right )}{8}+\frac {\arctanh \left (\cos \relax (x ) \sqrt {2}\right ) \sqrt {2}}{4}-\frac {\ln \left (1+\cos \relax (x )\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*csc(4*x),x)

[Out]

1/8*ln(-1+cos(x))+1/4*arctanh(cos(x)*2^(1/2))*2^(1/2)-1/8*ln(1+cos(x))

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maxima [B]  time = 0.44, size = 163, normalized size = 6.27 \[ \frac {1}{16} \, \sqrt {2} \log \left (2 \, \sqrt {2} \sin \left (2 \, x\right ) \sin \relax (x) + 2 \, {\left (\sqrt {2} \cos \relax (x) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 2 \, \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) + 1\right ) - \frac {1}{16} \, \sqrt {2} \log \left (-2 \, \sqrt {2} \sin \left (2 \, x\right ) \sin \relax (x) - 2 \, {\left (\sqrt {2} \cos \relax (x) - 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 2 \, \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) + 1\right ) - \frac {1}{8} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \frac {1}{8} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(4*x),x, algorithm="maxima")

[Out]

1/16*sqrt(2)*log(2*sqrt(2)*sin(2*x)*sin(x) + 2*(sqrt(2)*cos(x) + 1)*cos(2*x) + cos(2*x)^2 + 2*cos(x)^2 + sin(2
*x)^2 + 2*sin(x)^2 + 2*sqrt(2)*cos(x) + 1) - 1/16*sqrt(2)*log(-2*sqrt(2)*sin(2*x)*sin(x) - 2*(sqrt(2)*cos(x) -
1)*cos(2*x) + cos(2*x)^2 + 2*cos(x)^2 + sin(2*x)^2 + 2*sin(x)^2 - 2*sqrt(2)*cos(x) + 1) - 1/8*log(cos(x)^2 +
sin(x)^2 + 2*cos(x) + 1) + 1/8*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)

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mupad [B]  time = 2.31, size = 55, normalized size = 2.12 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{4}+\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {41\,\sqrt {2}}{8\,\left (\frac {169\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4}-\frac {29}{4}\right )}-\frac {239\,\sqrt {2}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,\left (\frac {169\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4}-\frac {29}{4}\right )}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/sin(4*x),x)

[Out]

log(tan(x/2))/4 + (2^(1/2)*atanh((41*2^(1/2))/(8*((169*tan(x/2)^2)/4 - 29/4)) - (239*2^(1/2)*tan(x/2)^2)/(8*((
169*tan(x/2)^2)/4 - 29/4))))/4

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sympy [B]  time = 6.14, size = 248, normalized size = 9.54 \[ - \frac {19601 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{110880 \sqrt {2} + 156808} - \frac {27720 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{110880 \sqrt {2} + 156808} + \frac {27720 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{110880 \sqrt {2} + 156808} + \frac {19601 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{110880 \sqrt {2} + 156808} + \frac {27720 \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{110880 \sqrt {2} + 156808} + \frac {19601 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{110880 \sqrt {2} + 156808} - \frac {19601 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{110880 \sqrt {2} + 156808} - \frac {27720 \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{110880 \sqrt {2} + 156808} + \frac {27720 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} \right )}}{110880 \sqrt {2} + 156808} + \frac {39202 \log {\left (\tan {\left (\frac {x}{2} \right )} \right )}}{110880 \sqrt {2} + 156808} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(4*x),x)

[Out]

-19601*sqrt(2)*log(tan(x/2) - 1 + sqrt(2))/(110880*sqrt(2) + 156808) - 27720*log(tan(x/2) - 1 + sqrt(2))/(1108
80*sqrt(2) + 156808) + 27720*log(tan(x/2) + 1 + sqrt(2))/(110880*sqrt(2) + 156808) + 19601*sqrt(2)*log(tan(x/2
) + 1 + sqrt(2))/(110880*sqrt(2) + 156808) + 27720*log(tan(x/2) - sqrt(2) - 1)/(110880*sqrt(2) + 156808) + 196
01*sqrt(2)*log(tan(x/2) - sqrt(2) - 1)/(110880*sqrt(2) + 156808) - 19601*sqrt(2)*log(tan(x/2) - sqrt(2) + 1)/(
110880*sqrt(2) + 156808) - 27720*log(tan(x/2) - sqrt(2) + 1)/(110880*sqrt(2) + 156808) + 27720*sqrt(2)*log(tan
(x/2))/(110880*sqrt(2) + 156808) + 39202*log(tan(x/2))/(110880*sqrt(2) + 156808)

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