Integrand size = 7, antiderivative size = 82 \[ \int \cos (x) \tan (5 x) \, dx=\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{5} \left (5-\sqrt {5}\right )} \cos (x)\right )+\sqrt {\frac {2}{5 \left (5+\sqrt {5}\right )}} \text {arctanh}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cos (x)\right )-\cos (x) \] Output:
1/10*(10+2*5^(1/2))^(1/2)*arctanh(1/5*(50-10*5^(1/2))^(1/2)*cos(x))+2^(1/2 )/(25+5*5^(1/2))^(1/2)*arctanh(1/5*(50+10*5^(1/2))^(1/2)*cos(x))-cos(x)
Leaf count is larger than twice the leaf count of optimal. \(215\) vs. \(2(82)=164\).
Time = 0.48 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.62 \[ \int \cos (x) \tan (5 x) \, dx=\frac {\left (1+\sqrt {5}\right ) \text {arctanh}\left (\frac {4-\left (-1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )}{\sqrt {10 \left (5+\sqrt {5}\right )}}+\frac {\left (1+\sqrt {5}\right ) \text {arctanh}\left (\frac {4+\left (-1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )}{\sqrt {10 \left (5+\sqrt {5}\right )}}+\frac {\left (-1+\sqrt {5}\right ) \text {arctanh}\left (\frac {4-\left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {10-2 \sqrt {5}}}\right )}{\sqrt {50-10 \sqrt {5}}}+\frac {\left (-1+\sqrt {5}\right ) \text {arctanh}\left (\frac {4+\left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {10-2 \sqrt {5}}}\right )}{\sqrt {50-10 \sqrt {5}}}-\cos (x) \] Input:
Integrate[Cos[x]*Tan[5*x],x]
Output:
((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 = 0.31 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.02, 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 |
\(\displaystyle \frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {arctanh}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \cos (x)\right )+\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cos (x)\right )-\cos (x)\) |
Input:
Int[Cos[x]*Tan[5*x],x]
Output:
(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]
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 = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {{\mathrm e}^{i x}}{2}-\frac {{\mathrm e}^{-i x}}{2}-i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2000 \textit {\_Z}^{4}+100 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}-10 i \textit {\_R} \,{\mathrm e}^{i x}+1\right )\right )\) | \(54\) |
Input:
int(cos(x)*tan(5*x),x,method=_RETURNVERBOSE)
Output:
-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))
Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.33 \[ \int \cos (x) \tan (5 x) \, dx=\frac {1}{10} \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + 2 \, \cos \left (x\right )\right ) - \frac {1}{10} \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} - 2 \, \cos \left (x\right )\right ) + \frac {1}{10} \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} \log \left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + 2 \, \cos \left (x\right )\right ) - \frac {1}{10} \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} \log \left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} - 2 \, \cos \left (x\right )\right ) - \cos \left (x\right ) \] Input:
integrate(cos(x)*tan(5*x),x, algorithm="fricas")
Output:
1/10*sqrt(1/2*sqrt(5) + 5/2)*log(sqrt(1/2*sqrt(5) + 5/2) + 2*cos(x)) - 1/1 0*sqrt(1/2*sqrt(5) + 5/2)*log(sqrt(1/2*sqrt(5) + 5/2) - 2*cos(x)) + 1/10*s qrt(-1/2*sqrt(5) + 5/2)*log(sqrt(-1/2*sqrt(5) + 5/2) + 2*cos(x)) - 1/10*sq rt(-1/2*sqrt(5) + 5/2)*log(sqrt(-1/2*sqrt(5) + 5/2) - 2*cos(x)) - cos(x)
\[ \int \cos (x) \tan (5 x) \, dx=\int \cos {\left (x \right )} \tan {\left (5 x \right )}\, dx \] Input:
integrate(cos(x)*tan(5*x),x)
Output:
Integral(cos(x)*tan(5*x), x)
\[ \int \cos (x) \tan (5 x) \, dx=\int { \cos \left (x\right ) \tan \left (5 \, x\right ) \,d x } \] Input:
integrate(cos(x)*tan(5*x),x, algorithm="maxima")
Output:
-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)
\[ \int \cos (x) \tan (5 x) \, dx=\int { \cos \left (x\right ) \tan \left (5 \, x\right ) \,d x } \] Input:
integrate(cos(x)*tan(5*x),x, algorithm="giac")
Output:
integrate(cos(x)*tan(5*x), x)
Time = 17.01 (sec) , antiderivative size = 407, normalized size of antiderivative = 4.96 \[ \int \cos (x) \tan (5 x) \, dx =\text {Too large to display} \] Input:
int(tan(5*x)*cos(x),x)
Output:
(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))*(...
\[ \int \cos (x) \tan (5 x) \, dx=\int \cos \left (x \right ) \tan \left (5 x \right )d x \] Input:
int(cos(x)*tan(5*x),x)
Output:
int(cos(x)*tan(5*x),x)