Integrand size = 7, antiderivative size = 84 \[ \int \cos (x) \tan (5 x) \, dx=\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {arctanh}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \cos (x)\right )+\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cos (x)\right )-\cos (x) \]
-cos(x)+1/10*arctanh(1/5*cos(x)*(50+10*5^(1/2))^(1/2))*(10-2*5^(1/2))^(1/2 )+1/10*arctanh(2*cos(x)*2^(1/2)/(5+5^(1/2))^(1/2))*(10+2*5^(1/2))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(215\) vs. \(2(84)=168\).
Time = 1.04 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.56 \[ \int \cos (x) \tan (5 x) \, dx=\frac {\left (1+\sqrt {5}\right ) \text {arctanh}\left (\frac {4-\left (-1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )}{\sqrt {10 \left (5+\sqrt {5}\right )}}+\frac {\left (1+\sqrt {5}\right ) \text {arctanh}\left (\frac {4+\left (-1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )}{\sqrt {10 \left (5+\sqrt {5}\right )}}+\frac {\left (-1+\sqrt {5}\right ) \text {arctanh}\left (\frac {4-\left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {10-2 \sqrt {5}}}\right )}{\sqrt {50-10 \sqrt {5}}}+\frac {\left (-1+\sqrt {5}\right ) \text {arctanh}\left (\frac {4+\left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {10-2 \sqrt {5}}}\right )}{\sqrt {50-10 \sqrt {5}}}-\cos (x) \]
((1 + Sqrt[5])*ArcTanh[(4 - (-1 + Sqrt[5])*Tan[x/2])/Sqrt[2*(5 + Sqrt[5])] ])/Sqrt[10*(5 + Sqrt[5])] + ((1 + Sqrt[5])*ArcTanh[(4 + (-1 + Sqrt[5])*Tan [x/2])/Sqrt[2*(5 + Sqrt[5])]])/Sqrt[10*(5 + Sqrt[5])] + ((-1 + Sqrt[5])*Ar cTanh[(4 - (1 + Sqrt[5])*Tan[x/2])/Sqrt[10 - 2*Sqrt[5]]])/Sqrt[50 - 10*Sqr t[5]] + ((-1 + Sqrt[5])*ArcTanh[(4 + (1 + Sqrt[5])*Tan[x/2])/Sqrt[10 - 2*S qrt[5]]])/Sqrt[50 - 10*Sqrt[5]] - Cos[x]
Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4879, 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.
\(\displaystyle \int \cos (x) \tan (5 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan (5 x)}{\sec (x)}dx\) |
\(\Big \downarrow \) 4879 |
\(\displaystyle -\int \frac {16 \cos ^4(x)-12 \cos ^2(x)+1}{16 \cos ^4(x)-20 \cos ^2(x)+5}d\cos (x)\) |
\(\Big \downarrow \) 2205 |
\(\displaystyle -\int \left (1-\frac {4 \left (1-2 \cos ^2(x)\right )}{16 \cos ^4(x)-20 \cos ^2(x)+5}\right )d\cos (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {arctanh}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \cos (x)\right )+\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cos (x)\right )-\cos (x)\) |
(Sqrt[(5 + Sqrt[5])/2]*ArcTanh[2*Sqrt[2/(5 + Sqrt[5])]*Cos[x]])/5 + (Sqrt[ (5 - Sqrt[5])/2]*ArcTanh[Sqrt[(2*(5 + Sqrt[5]))/5]*Cos[x]])/5 - Cos[x]
3.2.8.3.1 Defintions of rubi rules used
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[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]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.64
method | result | size |
risch | \(-\frac {{\mathrm e}^{i x}}{2}-\frac {{\mathrm e}^{-i x}}{2}-i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2000 \textit {\_Z}^{4}+100 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}-10 i \textit {\_R} \,{\mathrm e}^{i x}+1\right )\right )\) | \(54\) |
-1/2*exp(I*x)-1/2*exp(-I*x)-I*sum(_R*ln(exp(2*I*x)-10*I*_R*exp(I*x)+1),_R= RootOf(2000*_Z^4+100*_Z^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (56) = 112\).
Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.54 \[ \int \cos (x) \tan (5 x) \, dx=\frac {1}{20} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {\sqrt {5} + 5} + 4 \, \cos \left (x\right )\right ) - \frac {1}{20} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {\sqrt {5} + 5} - 4 \, \cos \left (x\right )\right ) + \frac {1}{20} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {-\sqrt {5} + 5} + 4 \, \cos \left (x\right )\right ) - \frac {1}{20} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {-\sqrt {5} + 5} - 4 \, \cos \left (x\right )\right ) - \cos \left (x\right ) \]
1/20*sqrt(2)*sqrt(sqrt(5) + 5)*log(sqrt(2)*sqrt(sqrt(5) + 5) + 4*cos(x)) - 1/20*sqrt(2)*sqrt(sqrt(5) + 5)*log(sqrt(2)*sqrt(sqrt(5) + 5) - 4*cos(x)) + 1/20*sqrt(2)*sqrt(-sqrt(5) + 5)*log(sqrt(2)*sqrt(-sqrt(5) + 5) + 4*cos(x )) - 1/20*sqrt(2)*sqrt(-sqrt(5) + 5)*log(sqrt(2)*sqrt(-sqrt(5) + 5) - 4*co s(x)) - cos(x)
\[ \int \cos (x) \tan (5 x) \, dx=\int \cos {\left (x \right )} \tan {\left (5 x \right )}\, dx \]
\[ \int \cos (x) \tan (5 x) \, dx=\int { \cos \left (x\right ) \tan \left (5 \, x\right ) \,d x } \]
-cos(x) - integrate(((sin(7*x) - sin(5*x) + sin(3*x) - sin(x))*cos(8*x) + (sin(6*x) - sin(4*x) + sin(2*x))*cos(7*x) + (sin(5*x) - sin(3*x) + sin(x)) *cos(6*x) + (sin(4*x) - sin(2*x))*cos(5*x) + (sin(3*x) - sin(x))*cos(4*x) - (cos(7*x) - cos(5*x) + cos(3*x) - cos(x))*sin(8*x) - (cos(6*x) - cos(4*x ) + cos(2*x) - 1)*sin(7*x) - (cos(5*x) - cos(3*x) + cos(x))*sin(6*x) - (co s(4*x) - cos(2*x) + 1)*sin(5*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) - s in(x))/(2*(cos(6*x) - cos(4*x) + cos(2*x) - 1)*cos(8*x) - cos(8*x)^2 + 2*( cos(4*x) - cos(2*x) + 1)*cos(6*x) - cos(6*x)^2 + 2*(cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - cos(2*x)^2 + 2*(sin(6*x) - sin(4*x) + sin(2*x))*sin(8*x) - sin(8*x)^2 + 2*(sin(4*x) - sin(2*x))*sin(6*x) - sin(6*x)^2 - sin(4*x)^2 + 2*sin(4*x)*sin(2*x) - sin(2*x)^2 + 2*cos(2*x) - 1), x)
\[ \int \cos (x) \tan (5 x) \, dx=\int { \cos \left (x\right ) \tan \left (5 \, x\right ) \,d x } \]
Time = 27.90 (sec) , antiderivative size = 407, normalized size of antiderivative = 4.85 \[ \int \cos (x) \tan (5 x) \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {18032420192256\,\sqrt {2}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\sqrt {\sqrt {5}+5}}{\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}-\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}+2398739234816}-\frac {867583393792\,\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}+5}}{25\,\left (\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}-\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}+2398739234816\right )}-\frac {3805341024256\,\sqrt {2}\,\sqrt {\sqrt {5}+5}}{5\,\left (\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}-\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}+2398739234816\right )}+\frac {6886980059136\,\sqrt {2}\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\sqrt {\sqrt {5}+5}}{\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}-\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}+2398739234816}\right )\,\sqrt {\sqrt {5}+5}}{10}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {867583393792\,\sqrt {2}\,\sqrt {5}\,\sqrt {5-\sqrt {5}}}{25\,\left (\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}+\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}-2398739234816\right )}-\frac {3805341024256\,\sqrt {2}\,\sqrt {5-\sqrt {5}}}{5\,\left (\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}+\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}-2398739234816\right )}+\frac {18032420192256\,\sqrt {2}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\sqrt {5-\sqrt {5}}}{\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}+\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}-2398739234816}-\frac {6886980059136\,\sqrt {2}\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\sqrt {5-\sqrt {5}}}{\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}+\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}-2398739234816}\right )\,\sqrt {5-\sqrt {5}}}{10}-\frac {2}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1} \]
(2^(1/2)*atanh((18032420192256*2^(1/2)*tan(x/2)^2*(5^(1/2) + 5)^(1/2))/((8 851927597056*5^(1/2))/25 - (676375744741376*5^(1/2)*tan(x/2)^2)/25 - (3334 33343574016*tan(x/2)^2)/5 + 2398739234816) - (867583393792*2^(1/2)*5^(1/2) *(5^(1/2) + 5)^(1/2))/(25*((8851927597056*5^(1/2))/25 - (676375744741376*5 ^(1/2)*tan(x/2)^2)/25 - (333433343574016*tan(x/2)^2)/5 + 2398739234816)) - (3805341024256*2^(1/2)*(5^(1/2) + 5)^(1/2))/(5*((8851927597056*5^(1/2))/2 5 - (676375744741376*5^(1/2)*tan(x/2)^2)/25 - (333433343574016*tan(x/2)^2) /5 + 2398739234816)) + (6886980059136*2^(1/2)*5^(1/2)*tan(x/2)^2*(5^(1/2) + 5)^(1/2))/((8851927597056*5^(1/2))/25 - (676375744741376*5^(1/2)*tan(x/2 )^2)/25 - (333433343574016*tan(x/2)^2)/5 + 2398739234816))*(5^(1/2) + 5)^( 1/2))/10 - (2^(1/2)*atanh((867583393792*2^(1/2)*5^(1/2)*(5 - 5^(1/2))^(1/2 ))/(25*((8851927597056*5^(1/2))/25 - (676375744741376*5^(1/2)*tan(x/2)^2)/ 25 + (333433343574016*tan(x/2)^2)/5 - 2398739234816)) - (3805341024256*2^( 1/2)*(5 - 5^(1/2))^(1/2))/(5*((8851927597056*5^(1/2))/25 - (67637574474137 6*5^(1/2)*tan(x/2)^2)/25 + (333433343574016*tan(x/2)^2)/5 - 2398739234816) ) + (18032420192256*2^(1/2)*tan(x/2)^2*(5 - 5^(1/2))^(1/2))/((885192759705 6*5^(1/2))/25 - (676375744741376*5^(1/2)*tan(x/2)^2)/25 + (333433343574016 *tan(x/2)^2)/5 - 2398739234816) - (6886980059136*2^(1/2)*5^(1/2)*tan(x/2)^ 2*(5 - 5^(1/2))^(1/2))/((8851927597056*5^(1/2))/25 - (676375744741376*5^(1 /2)*tan(x/2)^2)/25 + (333433343574016*tan(x/2)^2)/5 - 2398739234816))*(...