Integrand size = 7, antiderivative size = 36 \[ \int \cos (x) \csc (6 x) \, dx=-\frac {1}{6} \text {arctanh}(\cos (x))-\frac {1}{6} \text {arctanh}(2 \cos (x))+\frac {\text {arctanh}\left (\frac {2 \cos (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \] Output:
-1/6*arctanh(cos(x))-1/6*arctanh(2*cos(x))+1/6*arctanh(2/3*3^(1/2)*cos(x)) *3^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(36)=72\).
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.31 \[ \int \cos (x) \csc (6 x) \, dx=\frac {1}{12} \left (-2 \sqrt {3} \text {arctanh}\left (\frac {-2+\tan \left (\frac {x}{2}\right )}{\sqrt {3}}\right )+2 \sqrt {3} \text {arctanh}\left (\frac {2+\tan \left (\frac {x}{2}\right )}{\sqrt {3}}\right )-2 \log \left (\cos \left (\frac {x}{2}\right )\right )+\log (1-2 \cos (x))-\log (1+2 \cos (x))+2 \log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \] Input:
Integrate[Cos[x]*Csc[6*x],x]
Output:
(-2*Sqrt[3]*ArcTanh[(-2 + Tan[x/2])/Sqrt[3]] + 2*Sqrt[3]*ArcTanh[(2 + Tan[ x/2])/Sqrt[3]] - 2*Log[Cos[x/2]] + Log[1 - 2*Cos[x]] - Log[1 + 2*Cos[x]] + 2*Log[Sin[x/2]])/12
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 4879, 27, 2460, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cos (x) \csc (6 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (x)}{\sin (6 x)}dx\) |
\(\Big \downarrow \) 4879 |
\(\displaystyle -\int \frac {1}{2 \left (-16 \cos ^6(x)+32 \cos ^4(x)-19 \cos ^2(x)+3\right )}d\cos (x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{-16 \cos ^6(x)+32 \cos ^4(x)-19 \cos ^2(x)+3}d\cos (x)\) |
\(\Big \downarrow \) 2460 |
\(\displaystyle -\frac {1}{2} \int \left (\frac {2}{4 \cos ^2(x)-3}-\frac {2}{3 \left (4 \cos ^2(x)-1\right )}-\frac {1}{3 \left (\cos ^2(x)-1\right )}\right )d\cos (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{3} \text {arctanh}(\cos (x))-\frac {1}{3} \text {arctanh}(2 \cos (x))+\frac {\text {arctanh}\left (\frac {2 \cos (x)}{\sqrt {3}}\right )}{\sqrt {3}}\right )\) |
Input:
Int[Cos[x]*Csc[6*x],x]
Output:
(-1/3*ArcTanh[Cos[x]] - ArcTanh[2*Cos[x]]/3 + ArcTanh[(2*Cos[x])/Sqrt[3]]/ Sqrt[3])/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, Int[ExpandIntegrand[u*(Qx /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[Q x, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0] && RationalFunctionQ[u, x]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa ctors[Cos[v], x]}, -d/Coefficient[v, x, 1] Subst[Int[SubstFor[1, Cos[v]/d , u/Sin[v], x], x], x, Cos[v]/d]], x] /; !FalseQ[v] && FunctionOfQ[Nonfree Factors[Cos[v], x], u/Sin[v], x]]
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.31
method | result | size |
default | \(-\frac {\ln \left (2 \cos \left (x \right )+1\right )}{12}-\frac {\ln \left (1+\cos \left (x \right )\right )}{12}+\frac {\ln \left (\cos \left (x \right )-1\right )}{12}+\frac {\operatorname {arctanh}\left (\frac {2 \sqrt {3}\, \cos \left (x \right )}{3}\right ) \sqrt {3}}{6}+\frac {\ln \left (2 \cos \left (x \right )-1\right )}{12}\) | \(47\) |
risch | \(\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{6}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{6}-\frac {\sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}-\sqrt {3}\, {\mathrm e}^{i x}+1\right )}{12}+\frac {\sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}+\sqrt {3}\, {\mathrm e}^{i x}+1\right )}{12}+\frac {\ln \left ({\mathrm e}^{2 i x}-{\mathrm e}^{i x}+1\right )}{12}-\frac {\ln \left ({\mathrm e}^{2 i x}+{\mathrm e}^{i x}+1\right )}{12}\) | \(99\) |
Input:
int(cos(x)*csc(6*x),x,method=_RETURNVERBOSE)
Output:
-1/12*ln(2*cos(x)+1)-1/12*ln(1+cos(x))+1/12*ln(cos(x)-1)+1/6*arctanh(2/3*3 ^(1/2)*cos(x))*3^(1/2)+1/12*ln(2*cos(x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (26) = 52\).
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.94 \[ \int \cos (x) \csc (6 x) \, dx=\frac {1}{12} \, \sqrt {3} \log \left (-\frac {4 \, \cos \left (x\right )^{2} + 4 \, \sqrt {3} \cos \left (x\right ) + 3}{4 \, \cos \left (x\right )^{2} - 3}\right ) - \frac {1}{12} \, \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + \frac {1}{12} \, \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + \frac {1}{12} \, \log \left (-2 \, \cos \left (x\right ) + 1\right ) - \frac {1}{12} \, \log \left (-2 \, \cos \left (x\right ) - 1\right ) \] Input:
integrate(cos(x)*csc(6*x),x, algorithm="fricas")
Output:
1/12*sqrt(3)*log(-(4*cos(x)^2 + 4*sqrt(3)*cos(x) + 3)/(4*cos(x)^2 - 3)) - 1/12*log(1/2*cos(x) + 1/2) + 1/12*log(-1/2*cos(x) + 1/2) + 1/12*log(-2*cos (x) + 1) - 1/12*log(-2*cos(x) - 1)
\[ \int \cos (x) \csc (6 x) \, dx=\int \cos {\left (x \right )} \csc {\left (6 x \right )}\, dx \] Input:
integrate(cos(x)*csc(6*x),x)
Output:
Integral(cos(x)*csc(6*x), x)
\[ \int \cos (x) \csc (6 x) \, dx=\int { \cos \left (x\right ) \csc \left (6 \, x\right ) \,d x } \] Input:
integrate(cos(x)*csc(6*x),x, algorithm="maxima")
Output:
-integrate(1/2*((sin(3*x) - sin(x))*cos(4*x) - (cos(3*x) - cos(x))*sin(4*x ) - (cos(2*x) - 1)*sin(3*x) + cos(3*x)*sin(2*x) - cos(x)*sin(2*x) + cos(2* x)*sin(x) - sin(x))/(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), x) - 1/2 4*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) + 1/24*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(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 1/12*log(cos( x)^2 + sin(x)^2 - 2*cos(x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (26) = 52\).
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.89 \[ \int \cos (x) \csc (6 x) \, dx=-\frac {1}{12} \, \sqrt {3} \log \left (\frac {{\left | -4 \, \sqrt {3} + 8 \, \cos \left (x\right ) \right |}}{{\left | 4 \, \sqrt {3} + 8 \, \cos \left (x\right ) \right |}}\right ) - \frac {1}{12} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac {1}{12} \, \log \left (-\cos \left (x\right ) + 1\right ) - \frac {1}{12} \, \log \left ({\left | 2 \, \cos \left (x\right ) + 1 \right |}\right ) + \frac {1}{12} \, \log \left ({\left | 2 \, \cos \left (x\right ) - 1 \right |}\right ) \] Input:
integrate(cos(x)*csc(6*x),x, algorithm="giac")
Output:
-1/12*sqrt(3)*log(abs(-4*sqrt(3) + 8*cos(x))/abs(4*sqrt(3) + 8*cos(x))) - 1/12*log(cos(x) + 1) + 1/12*log(-cos(x) + 1) - 1/12*log(abs(2*cos(x) + 1)) + 1/12*log(abs(2*cos(x) - 1))
Time = 16.12 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.06 \[ \int \cos (x) \csc (6 x) \, dx=\frac {\mathrm {atanh}\left (\frac {1073741824}{10761687\,\left (\frac {427973089951744\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{14348907}-\frac {47552804159488}{4782969}\right )}+\frac {797161}{797162}\right )}{6}+\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{6}+\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {4222769432625152\,\sqrt {3}}{4782969\,\left (\frac {101871591633190912\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4782969}-\frac {7314051205955584}{4782969}\right )}-\frac {19605196950732800\,\sqrt {3}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{1594323\,\left (\frac {101871591633190912\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4782969}-\frac {7314051205955584}{4782969}\right )}\right )}{6} \] Input:
int(cos(x)/sin(6*x),x)
Output:
atanh(1073741824/(10761687*((427973089951744*tan(x/2)^2)/14348907 - 475528 04159488/4782969)) + 797161/797162)/6 + log(tan(x/2))/6 + (3^(1/2)*atanh(( 4222769432625152*3^(1/2))/(4782969*((101871591633190912*tan(x/2)^2)/478296 9 - 7314051205955584/4782969)) - (19605196950732800*3^(1/2)*tan(x/2)^2)/(1 594323*((101871591633190912*tan(x/2)^2)/4782969 - 7314051205955584/4782969 ))))/6
\[ \int \cos (x) \csc (6 x) \, dx=\int \cos \left (x \right ) \csc \left (6 x \right )d x \] Input:
int(cos(x)*csc(6*x),x)
Output:
int(cos(x)*csc(6*x),x)