Integrand size = 7, antiderivative size = 81 \[ \int \cot (5 x) \sin (x) \, dx=-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{5} \left (5-\sqrt {5}\right )} \sin (x)\right )-\sqrt {\frac {2}{5 \left (5+\sqrt {5}\right )}} \text {arctanh}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sin (x)\right )+\sin (x) \] Output:
-1/10*(10+2*5^(1/2))^(1/2)*arctanh(1/5*(50-10*5^(1/2))^(1/2)*sin(x))-2^(1/ 2)/(25+5*5^(1/2))^(1/2)*arctanh(1/5*(50+10*5^(1/2))^(1/2)*sin(x))+sin(x)
Time = 0.17 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.94 \[ \int \cot (5 x) \sin (x) \, dx=\frac {1}{10} \left (-\sqrt {10-2 \sqrt {5}} \text {arctanh}\left (\sqrt {2+\frac {2}{\sqrt {5}}} \sin (x)\right )-\sqrt {2 \left (5+\sqrt {5}\right )} \text {arctanh}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \sin (x)\right )+10 \sin (x)\right ) \] Input:
Integrate[Cot[5*x]*Sin[x],x]
Output:
(-(Sqrt[10 - 2*Sqrt[5]]*ArcTanh[Sqrt[2 + 2/Sqrt[5]]*Sin[x]]) - Sqrt[2*(5 + Sqrt[5])]*ArcTanh[2*Sqrt[2/(5 + Sqrt[5])]*Sin[x]] + 10*Sin[x])/10
Time = 0.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4878, 2205, 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.
\(\displaystyle \int \sin (x) \cot (5 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cot (5 x)}{\csc (x)}dx\) |
\(\Big \downarrow \) 4878 |
\(\displaystyle \int \frac {16 \sin ^4(x)-12 \sin ^2(x)+1}{16 \sin ^4(x)-20 \sin ^2(x)+5}d\sin (x)\) |
\(\Big \downarrow \) 2205 |
\(\displaystyle \int \left (1-\frac {4 \left (1-2 \sin ^2(x)\right )}{16 \sin ^4(x)-20 \sin ^2(x)+5}\right )d\sin (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {arctanh}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \sin (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sin (x)\right )+\sin (x)\) |
Input:
Int[Cot[5*x]*Sin[x],x]
Output:
-1/5*(Sqrt[(5 + Sqrt[5])/2]*ArcTanh[2*Sqrt[2/(5 + Sqrt[5])]*Sin[x]]) - (Sq rt[(5 - Sqrt[5])/2]*ArcTanh[Sqrt[(2*(5 + Sqrt[5]))/5]*Sin[x]])/5 + Sin[x]
Int[(Px_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandInte grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 2] && Expon[Px, x^2] > 1
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa ctors[Sin[v], x]}, d/Coefficient[v, x, 1] Subst[Int[SubstFor[1, Sin[v]/d, u/Cos[v], x], x], x, Sin[v]/d]], x] /; !FalseQ[v] && FunctionOfQ[NonfreeF actors[Sin[v], x], u/Cos[v], x]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.68
method | result | size |
risch | \(-\frac {i {\mathrm e}^{i x}}{2}+\frac {i {\mathrm e}^{-i x}}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (125 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}-5 i \textit {\_R} \,{\mathrm e}^{i x}-1\right )\right )}{2}\) | \(55\) |
Input:
int(cot(5*x)*sin(x),x,method=_RETURNVERBOSE)
Output:
-1/2*I*exp(I*x)+1/2*I*exp(-I*x)+1/2*sum(_R*ln(exp(2*I*x)-5*I*_R*exp(I*x)-1 ),_R=RootOf(125*_Z^4-25*_Z^2+1))
Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.32 \[ \int \cot (5 x) \sin (x) \, dx=-\frac {1}{10} \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + 2 \, \sin \left (x\right )\right ) + \frac {1}{10} \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} - 2 \, \sin \left (x\right )\right ) - \frac {1}{10} \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} \log \left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + 2 \, \sin \left (x\right )\right ) + \frac {1}{10} \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} \log \left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} - 2 \, \sin \left (x\right )\right ) + \sin \left (x\right ) \] Input:
integrate(cot(5*x)*sin(x),x, algorithm="fricas")
Output:
-1/10*sqrt(1/2*sqrt(5) + 5/2)*log(sqrt(1/2*sqrt(5) + 5/2) + 2*sin(x)) + 1/ 10*sqrt(1/2*sqrt(5) + 5/2)*log(sqrt(1/2*sqrt(5) + 5/2) - 2*sin(x)) - 1/10* sqrt(-1/2*sqrt(5) + 5/2)*log(sqrt(-1/2*sqrt(5) + 5/2) + 2*sin(x)) + 1/10*s qrt(-1/2*sqrt(5) + 5/2)*log(sqrt(-1/2*sqrt(5) + 5/2) - 2*sin(x)) + sin(x)
\[ \int \cot (5 x) \sin (x) \, dx=\int \sin {\left (x \right )} \cot {\left (5 x \right )}\, dx \] Input:
integrate(cot(5*x)*sin(x),x)
Output:
Integral(sin(x)*cot(5*x), x)
\[ \int \cot (5 x) \sin (x) \, dx=\int { \cot \left (5 \, x\right ) \sin \left (x\right ) \,d x } \] Input:
integrate(cot(5*x)*sin(x),x, algorithm="maxima")
Output:
-integrate(1/2*((cos(3*x) + cos(2*x) + cos(x))*cos(4*x) + (2*cos(2*x) + 2* cos(x) + 1)*cos(3*x) + cos(3*x)^2 + (2*cos(x) + 1)*cos(2*x) + cos(2*x)^2 + cos(x)^2 + (sin(3*x) + sin(2*x) + sin(x))*sin(4*x) + 2*(sin(2*x) + sin(x) )*sin(3*x) + sin(3*x)^2 + sin(2*x)^2 + 2*sin(2*x)*sin(x) + sin(x)^2 + cos( x))/(2*(cos(3*x) + cos(2*x) + cos(x) + 1)*cos(4*x) + cos(4*x)^2 + 2*(cos(2 *x) + cos(x) + 1)*cos(3*x) + cos(3*x)^2 + 2*(cos(x) + 1)*cos(2*x) + cos(2* x)^2 + cos(x)^2 + 2*(sin(3*x) + sin(2*x) + sin(x))*sin(4*x) + sin(4*x)^2 + 2*(sin(2*x) + sin(x))*sin(3*x) + sin(3*x)^2 + sin(2*x)^2 + 2*sin(2*x)*sin (x) + sin(x)^2 + 2*cos(x) + 1), x) - integrate(-1/2*((cos(3*x) - cos(2*x) + cos(x))*cos(4*x) + (2*cos(2*x) - 2*cos(x) + 1)*cos(3*x) - cos(3*x)^2 + ( 2*cos(x) - 1)*cos(2*x) - cos(2*x)^2 - cos(x)^2 + (sin(3*x) - sin(2*x) + si n(x))*sin(4*x) + 2*(sin(2*x) - sin(x))*sin(3*x) - sin(3*x)^2 - sin(2*x)^2 + 2*sin(2*x)*sin(x) - sin(x)^2 + cos(x))/(2*(cos(3*x) - cos(2*x) + cos(x) - 1)*cos(4*x) - cos(4*x)^2 + 2*(cos(2*x) - cos(x) + 1)*cos(3*x) - cos(3*x) ^2 + 2*(cos(x) - 1)*cos(2*x) - cos(2*x)^2 - cos(x)^2 + 2*(sin(3*x) - sin(2 *x) + sin(x))*sin(4*x) - sin(4*x)^2 + 2*(sin(2*x) - sin(x))*sin(3*x) - sin (3*x)^2 - sin(2*x)^2 + 2*sin(2*x)*sin(x) - sin(x)^2 + 2*cos(x) - 1), x) + sin(x)
Time = 0.22 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.37 \[ \int \cot (5 x) \sin (x) \, dx=-\frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \log \left ({\left | \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} + 5} + \sin \left (x\right ) \right |}\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \log \left ({\left | -\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} + 5} + \sin \left (x\right ) \right |}\right ) - \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left ({\left | \sqrt {-\frac {1}{8} \, \sqrt {5} + \frac {5}{8}} + \sin \left (x\right ) \right |}\right ) + \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left ({\left | -\sqrt {-\frac {1}{8} \, \sqrt {5} + \frac {5}{8}} + \sin \left (x\right ) \right |}\right ) + \sin \left (x\right ) \] Input:
integrate(cot(5*x)*sin(x),x, algorithm="giac")
Output:
-1/20*sqrt(2*sqrt(5) + 10)*log(abs(1/2*sqrt(1/2)*sqrt(sqrt(5) + 5) + sin(x ))) + 1/20*sqrt(2*sqrt(5) + 10)*log(abs(-1/2*sqrt(1/2)*sqrt(sqrt(5) + 5) + sin(x))) - 1/20*sqrt(-2*sqrt(5) + 10)*log(abs(sqrt(-1/8*sqrt(5) + 5/8) + sin(x))) + 1/20*sqrt(-2*sqrt(5) + 10)*log(abs(-sqrt(-1/8*sqrt(5) + 5/8) + sin(x))) + sin(x)
Time = 17.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.47 \[ \int \cot (5 x) \sin (x) \, dx=\sin \left (x\right )-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\frac {25\,\sqrt {2}\,\sin \left (x\right )\,\sqrt {\sqrt {5}+5}}{2}+\frac {11\,\sqrt {2}\,\sqrt {5}\,\sin \left (x\right )\,\sqrt {\sqrt {5}+5}}{2}}{20\,\sqrt {5}+45}\right )\,\sqrt {\sqrt {5}+5}}{10}+\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\frac {25\,\sqrt {2}\,\sin \left (x\right )\,\sqrt {5-\sqrt {5}}}{2}-\frac {11\,\sqrt {2}\,\sqrt {5}\,\sin \left (x\right )\,\sqrt {5-\sqrt {5}}}{2}}{20\,\sqrt {5}-45}\right )\,\sqrt {5-\sqrt {5}}}{10} \] Input:
int(cot(5*x)*sin(x),x)
Output:
sin(x) - (2^(1/2)*atanh(((25*2^(1/2)*sin(x)*(5^(1/2) + 5)^(1/2))/2 + (11*2 ^(1/2)*5^(1/2)*sin(x)*(5^(1/2) + 5)^(1/2))/2)/(20*5^(1/2) + 45))*(5^(1/2) + 5)^(1/2))/10 + (2^(1/2)*atanh(((25*2^(1/2)*sin(x)*(5 - 5^(1/2))^(1/2))/2 - (11*2^(1/2)*5^(1/2)*sin(x)*(5 - 5^(1/2))^(1/2))/2)/(20*5^(1/2) - 45))*( 5 - 5^(1/2))^(1/2))/10
\[ \int \cot (5 x) \sin (x) \, dx=\int \cot \left (5 x \right ) \sin \left (x \right )d x \] Input:
int(cot(5*x)*sin(x),x)
Output:
int(cot(5*x)*sin(x),x)