Integrand size = 7, antiderivative size = 89 \[ \int \cos (x) \tan (6 x) \, dx=\frac {\text {arctanh}\left (\sqrt {2} \cos (x)\right )}{3 \sqrt {2}}+\frac {1}{6} \sqrt {2-\sqrt {3}} \text {arctanh}\left (\frac {2 \cos (x)}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{6} \sqrt {2+\sqrt {3}} \text {arctanh}\left (\frac {2 \cos (x)}{\sqrt {2+\sqrt {3}}}\right )-\cos (x) \]
-cos(x)+1/6*arctanh(cos(x)*2^(1/2))*2^(1/2)+1/6*arctanh(2*cos(x)/(1/2*6^(1 /2)-1/2*2^(1/2)))*(1/2*6^(1/2)-1/2*2^(1/2))+1/6*arctanh(2*cos(x)/(1/2*6^(1 /2)+1/2*2^(1/2)))*(1/2*6^(1/2)+1/2*2^(1/2))
Result contains complex when optimal does not.
Time = 8.21 (sec) , antiderivative size = 628, normalized size of antiderivative = 7.06 \[ \int \cos (x) \tan (6 x) \, dx=\frac {1}{24} \left ((4+4 i) (-1)^{3/4} \text {arctanh}\left (\frac {-1+\tan \left (\frac {x}{2}\right )}{\sqrt {2}}\right )+(4-4 i) \sqrt [4]{-1} \text {arctanh}\left (\frac {1+\tan \left (\frac {x}{2}\right )}{\sqrt {2}}\right )-24 \cos (x)-\frac {2 \left (1+\sqrt {2}\right ) \left (x-2 \sqrt {3} \text {arctanh}\left (\frac {2+\left (2+\sqrt {2}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {6}}\right )-\log \left (\sec ^2\left (\frac {x}{2}\right )\right )+\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (\sqrt {2}-2 \cos (x)+2 \sin (x)\right )\right )\right )}{2+\sqrt {2}}+\sqrt {2} \left (x+2 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {2}+\left (-1+\sqrt {2}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {3}}\right )-\log \left (\sec ^2\left (\frac {x}{2}\right )\right )+\log \left (\sec ^2\left (\frac {x}{2}\right ) \left (1+\sqrt {2} \cos (x)-\sqrt {2} \sin (x)\right )\right )\right )-\frac {2 \left (2 \left (-2+\sqrt {6}\right ) \text {arctanh}\left (\sqrt {2}+\left (\sqrt {2}-\sqrt {3}\right ) \tan \left (\frac {x}{2}\right )\right )+\left (3 \sqrt {2}-2 \sqrt {3}\right ) \left (x-\log \left (\sec ^2\left (\frac {x}{2}\right )\right )+\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (\sqrt {3}+\sqrt {2} \cos (x)-\sqrt {2} \sin (x)\right )\right )\right )\right ) \left (\sqrt {2}-\sqrt {3} \sin (x)\right ) \left (-3+\sqrt {6}-\left (-2+\sqrt {6}\right ) \cos (x)+\left (-2+\sqrt {6}\right ) \sin (x)\right )}{-36+15 \sqrt {6}+\left (20-8 \sqrt {6}\right ) \cos (x)+\left (12-5 \sqrt {6}\right ) \cos (2 x)-50 \sin (x)+20 \sqrt {6} \sin (x)+12 \sin (2 x)-5 \sqrt {6} \sin (2 x)}+\frac {2 \left (-2 \left (\sqrt {2}+\sqrt {3}\right ) \text {arctanh}\left (\frac {2+\left (2+\sqrt {6}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2}}\right )+\left (3+\sqrt {6}\right ) \left (x-\log \left (\sec ^2\left (\frac {x}{2}\right )\right )+\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (\sqrt {6}-2 \cos (x)+2 \sin (x)\right )\right )\right )\right ) \left (2+\sqrt {6} \sin (x)\right ) \left (3+\sqrt {6}-\left (2+\sqrt {6}\right ) \cos (x)+\left (2+\sqrt {6}\right ) \sin (x)\right )}{-36-15 \sqrt {6}+4 \left (5+2 \sqrt {6}\right ) \cos (x)+\left (12+5 \sqrt {6}\right ) \cos (2 x)-50 \sin (x)-20 \sqrt {6} \sin (x)+12 \sin (2 x)+5 \sqrt {6} \sin (2 x)}\right ) \]
((4 + 4*I)*(-1)^(3/4)*ArcTanh[(-1 + Tan[x/2])/Sqrt[2]] + (4 - 4*I)*(-1)^(1 /4)*ArcTanh[(1 + Tan[x/2])/Sqrt[2]] - 24*Cos[x] - (2*(1 + Sqrt[2])*(x - 2* Sqrt[3]*ArcTanh[(2 + (2 + Sqrt[2])*Tan[x/2])/Sqrt[6]] - Log[Sec[x/2]^2] + Log[-(Sec[x/2]^2*(Sqrt[2] - 2*Cos[x] + 2*Sin[x]))]))/(2 + Sqrt[2]) + Sqrt[ 2]*(x + 2*Sqrt[3]*ArcTanh[(Sqrt[2] + (-1 + Sqrt[2])*Tan[x/2])/Sqrt[3]] - L og[Sec[x/2]^2] + Log[Sec[x/2]^2*(1 + Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x])]) - (2*(2*(-2 + Sqrt[6])*ArcTanh[Sqrt[2] + (Sqrt[2] - Sqrt[3])*Tan[x/2]] + (3* Sqrt[2] - 2*Sqrt[3])*(x - Log[Sec[x/2]^2] + Log[-(Sec[x/2]^2*(Sqrt[3] + Sq rt[2]*Cos[x] - Sqrt[2]*Sin[x]))]))*(Sqrt[2] - Sqrt[3]*Sin[x])*(-3 + Sqrt[6 ] - (-2 + Sqrt[6])*Cos[x] + (-2 + Sqrt[6])*Sin[x]))/(-36 + 15*Sqrt[6] + (2 0 - 8*Sqrt[6])*Cos[x] + (12 - 5*Sqrt[6])*Cos[2*x] - 50*Sin[x] + 20*Sqrt[6] *Sin[x] + 12*Sin[2*x] - 5*Sqrt[6]*Sin[2*x]) + (2*(-2*(Sqrt[2] + Sqrt[3])*A rcTanh[(2 + (2 + Sqrt[6])*Tan[x/2])/Sqrt[2]] + (3 + Sqrt[6])*(x - Log[Sec[ x/2]^2] + Log[-(Sec[x/2]^2*(Sqrt[6] - 2*Cos[x] + 2*Sin[x]))]))*(2 + Sqrt[6 ]*Sin[x])*(3 + Sqrt[6] - (2 + Sqrt[6])*Cos[x] + (2 + Sqrt[6])*Sin[x]))/(-3 6 - 15*Sqrt[6] + 4*(5 + 2*Sqrt[6])*Cos[x] + (12 + 5*Sqrt[6])*Cos[2*x] - 50 *Sin[x] - 20*Sqrt[6]*Sin[x] + 12*Sin[2*x] + 5*Sqrt[6]*Sin[2*x]))/24
Time = 0.32 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.04, 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.
\(\displaystyle \int \cos (x) \tan (6 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan (6 x)}{\sec (x)}dx\) |
\(\Big \downarrow \) 4879 |
\(\displaystyle -\int -\frac {2 \cos ^2(x) \left (16 \cos ^4(x)-16 \cos ^2(x)+3\right )}{-32 \cos ^6(x)+48 \cos ^4(x)-18 \cos ^2(x)+1}d\cos (x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {\cos ^2(x) \left (16 \cos ^4(x)-16 \cos ^2(x)+3\right )}{-32 \cos ^6(x)+48 \cos ^4(x)-18 \cos ^2(x)+1}d\cos (x)\) |
\(\Big \downarrow \) 2460 |
\(\displaystyle 2 \int \left (\frac {1-8 \cos ^2(x)}{3 \left (16 \cos ^4(x)-16 \cos ^2(x)+1\right )}-\frac {1}{6 \left (2 \cos ^2(x)-1\right )}-\frac {1}{2}\right )d\cos (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {\text {arctanh}\left (\sqrt {2} \cos (x)\right )}{6 \sqrt {2}}+\frac {1}{12} \sqrt {2-\sqrt {3}} \text {arctanh}\left (\frac {2 \cos (x)}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{12} \sqrt {2+\sqrt {3}} \text {arctanh}\left (\frac {2 \cos (x)}{\sqrt {2+\sqrt {3}}}\right )-\frac {\cos (x)}{2}\right )\) |
2*(ArcTanh[Sqrt[2]*Cos[x]]/(6*Sqrt[2]) + (Sqrt[2 - Sqrt[3]]*ArcTanh[(2*Cos [x])/Sqrt[2 - Sqrt[3]]])/12 + (Sqrt[2 + Sqrt[3]]*ArcTanh[(2*Cos[x])/Sqrt[2 + Sqrt[3]]])/12 - Cos[x]/2)
3.2.9.3.1 Defintions of rubi rules used
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]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.11
method | result | size |
risch | \(-\frac {{\mathrm e}^{i x}}{2}-\frac {{\mathrm e}^{-i x}}{2}-i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (20736 \textit {\_Z}^{4}+576 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}-12 i \textit {\_R} \,{\mathrm e}^{i x}+1\right )\right )+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+\sqrt {2}\, {\mathrm e}^{i x}+1\right )}{12}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-\sqrt {2}\, {\mathrm e}^{i x}+1\right )}{12}\) | \(99\) |
-1/2*exp(I*x)-1/2*exp(-I*x)-I*sum(_R*ln(exp(2*I*x)-12*I*_R*exp(I*x)+1),_R= RootOf(20736*_Z^4+576*_Z^2+1))+1/12*2^(1/2)*ln(exp(2*I*x)+2^(1/2)*exp(I*x) +1)-1/12*2^(1/2)*ln(exp(2*I*x)-2^(1/2)*exp(I*x)+1)
Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.51 \[ \int \cos (x) \tan (6 x) \, dx=\frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\sqrt {\sqrt {3} + 2} + 2 \, \cos \left (x\right )\right ) - \frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\sqrt {\sqrt {3} + 2} - 2 \, \cos \left (x\right )\right ) + \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left (\sqrt {-\sqrt {3} + 2} + 2 \, \cos \left (x\right )\right ) - \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left (\sqrt {-\sqrt {3} + 2} - 2 \, \cos \left (x\right )\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\frac {2 \, \cos \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) + 1}{2 \, \cos \left (x\right )^{2} - 1}\right ) - \cos \left (x\right ) \]
1/12*sqrt(sqrt(3) + 2)*log(sqrt(sqrt(3) + 2) + 2*cos(x)) - 1/12*sqrt(sqrt( 3) + 2)*log(sqrt(sqrt(3) + 2) - 2*cos(x)) + 1/12*sqrt(-sqrt(3) + 2)*log(sq rt(-sqrt(3) + 2) + 2*cos(x)) - 1/12*sqrt(-sqrt(3) + 2)*log(sqrt(-sqrt(3) + 2) - 2*cos(x)) + 1/12*sqrt(2)*log(-(2*cos(x)^2 + 2*sqrt(2)*cos(x) + 1)/(2 *cos(x)^2 - 1)) - cos(x)
\[ \int \cos (x) \tan (6 x) \, dx=\int \cos {\left (x \right )} \tan {\left (6 x \right )}\, dx \]
\[ \int \cos (x) \tan (6 x) \, dx=\int { \cos \left (x\right ) \tan \left (6 \, x\right ) \,d x } \]
1/24*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) - 1/24*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) - cos(x) - integrate(1/3*((2*sin(7*x) + sin(5*x) - sin(3*x) - 2*sin(x))*cos(8*x) + (sin(3*x) + 2*sin(x))*cos(4*x) - (2*cos(7*x) + cos(5 *x) - cos(3*x) - 2*cos(x))*sin(8*x) - 2*(cos(4*x) - 1)*sin(7*x) - (cos(4*x ) - 1)*sin(5*x) - (cos(3*x) + 2*cos(x))*sin(4*x) + 2*cos(7*x)*sin(4*x) + c os(5*x)*sin(4*x) - sin(3*x) - 2*sin(x))/(2*(cos(4*x) - 1)*cos(8*x) - cos(8 *x)^2 - cos(4*x)^2 - sin(8*x)^2 + 2*sin(8*x)*sin(4*x) - sin(4*x)^2 + 2*cos (4*x) - 1), x)
\[ \int \cos (x) \tan (6 x) \, dx=\int { \cos \left (x\right ) \tan \left (6 \, x\right ) \,d x } \]
Time = 29.06 (sec) , antiderivative size = 787, normalized size of antiderivative = 8.84 \[ \int \cos (x) \tan (6 x) \, dx=\text {Too large to display} \]
(6^(1/2)*(atan((2^(1/2)*321030945816576i)/(213254896304333030400*tan(x/2)^ 4 - 129275829262795438080*tan(x/2)^2 + 2176593611144037376) + (6^(1/2)*888 405273481134080i)/(213254896304333030400*tan(x/2)^4 - 12927582926279543808 0*tan(x/2)^2 + 2176593611144037376) - (2^(1/2)*tan(x/2)^2*1871105472480256 0i)/(213254896304333030400*tan(x/2)^4 - 129275829262795438080*tan(x/2)^2 + 2176593611144037376) + (2^(1/2)*tan(x/2)^4*10905601889064960i)/(213254896 304333030400*tan(x/2)^4 - 129275829262795438080*tan(x/2)^2 + 2176593611144 037376) - (6^(1/2)*tan(x/2)^2*52765833462352287744i)/(21325489630433303040 0*tan(x/2)^4 - 129275829262795438080*tan(x/2)^2 + 2176593611144037376) + ( 6^(1/2)*tan(x/2)^4*87054650497106012160i)/(213254896304333030400*tan(x/2)^ 4 - 129275829262795438080*tan(x/2)^2 + 2176593611144037376)) + atan((2^(1/ 2)*1443325504589801788190484332544i)/(589232404262260650654553866240*2^(1/ 2)*6^(1/2) + 119129717169909888440949339586560*tan(x/2)^2 - 34367271726987 959946466862039040*2^(1/2)*6^(1/2)*tan(x/2)^2 - 20870903094507989978345572 92544) - (6^(1/2)*852047139771204346616741888000i)/(5892324042622606506545 53866240*2^(1/2)*6^(1/2) + 119129717169909888440949339586560*tan(x/2)^2 - 34367271726987959946466862039040*2^(1/2)*6^(1/2)*tan(x/2)^2 - 208709030945 0798997834557292544) - (2^(1/2)*tan(x/2)^2*8418228357130530454356858241024 0i)/(589232404262260650654553866240*2^(1/2)*6^(1/2) + 11912971716990988844 0949339586560*tan(x/2)^2 - 34367271726987959946466862039040*2^(1/2)*6^(...