\(\int \cos (x) \sec (6 x) \, dx\) [61]

Optimal result

Integrand size = 7, antiderivative size = 85 \[ \int \cos (x) \sec (6 x) \, dx=-\frac {\text {arctanh}\left (\sqrt {2} \sin (x)\right )}{3 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}+\frac {\text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}} \] Output:

-1/6*arctanh(sin(x)*2^(1/2))*2^(1/2)+1/6*arctanh(2*sin(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*sin(x)/(1/2*6^(1/2)+1/2 
*2^(1/2)))/(1/2*6^(1/2)+1/2*2^(1/2))
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int \cos (x) \sec (6 x) \, dx=\frac {1}{6} \left (-\sqrt {2} \text {arctanh}\left (\sqrt {2} \sin (x)\right )+\sqrt {2+\sqrt {3}} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2-\sqrt {3}}}\right )+\sqrt {2-\sqrt {3}} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2+\sqrt {3}}}\right )\right ) \] Input:

Integrate[Cos[x]*Sec[6*x],x]
 

Output:

(-(Sqrt[2]*ArcTanh[Sqrt[2]*Sin[x]]) + Sqrt[2 + Sqrt[3]]*ArcTanh[(2*Sin[x]) 
/Sqrt[2 - Sqrt[3]]] + Sqrt[2 - Sqrt[3]]*ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[3 
]]])/6
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 85, 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, 4856, 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.

\(\displaystyle \int \cos (x) \sec (6 x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (x)}{\cos (6 x)}dx\)

\(\Big \downarrow \) 4856

\(\displaystyle \int \frac {1}{-32 \sin ^6(x)+48 \sin ^4(x)-18 \sin ^2(x)+1}d\sin (x)\)

\(\Big \downarrow \) 2460

\(\displaystyle \int \left (\frac {1}{3 \left (2 \sin ^2(x)-1\right )}-\frac {4 \left (2 \sin ^2(x)-1\right )}{3 \left (16 \sin ^4(x)-16 \sin ^2(x)+1\right )}\right )d\sin (x)\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\text {arctanh}\left (\sqrt {2} \sin (x)\right )}{3 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}+\frac {\text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}}\)

Input:

Int[Cos[x]*Sec[6*x],x]
 

Output:

-1/3*ArcTanh[Sqrt[2]*Sin[x]]/Sqrt[2] + ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[3] 
]]/(6*Sqrt[2 - Sqrt[3]]) + ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[3]]]/(6*Sqrt[2 
 + Sqrt[3]])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2460

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4856

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFacto 
rs[Sin[c*(a + b*x)], x]}, Simp[d/(b*c)   Subst[Int[SubstFor[1, Sin[c*(a + b 
*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a + b*x 
)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])
 

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {2 \,\operatorname {arctanh}\left (\frac {8 \sin \left (x \right )}{2 \sqrt {6}+2 \sqrt {2}}\right )}{3 \left (2 \sqrt {6}+2 \sqrt {2}\right )}+\frac {2 \,\operatorname {arctanh}\left (\frac {8 \sin \left (x \right )}{2 \sqrt {6}-2 \sqrt {2}}\right )}{3 \left (2 \sqrt {6}-2 \sqrt {2}\right )}-\frac {\operatorname {arctanh}\left (\sqrt {2}\, \sin \left (x \right )\right ) \sqrt {2}}{6}\)	\(80\)
risch	\(2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (331776 \textit {\_Z}^{4}-2304 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-13824 i \textit {\_R}^{3}+96 i \textit {\_R} \right ) {\mathrm e}^{i x}-1\right )\right )+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{12}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{12}\)	\(95\)

Input:

int(cos(x)*sec(6*x),x,method=_RETURNVERBOSE)
 

Output:

2/3/(2*6^(1/2)+2*2^(1/2))*arctanh(8*sin(x)/(2*6^(1/2)+2*2^(1/2)))+2/3/(2*6 
^(1/2)-2*2^(1/2))*arctanh(8*sin(x)/(2*6^(1/2)-2*2^(1/2)))-1/6*arctanh(2^(1 
/2)*sin(x))*2^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (67) = 134\).

Time = 0.10 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.81 \[ \int \cos (x) \sec (6 x) \, dx=-\frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} + 2 \, \sin \left (x\right )\right ) + \frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} - 2 \, \sin \left (x\right )\right ) + \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left ({\left (\sqrt {3} + 2\right )} \sqrt {-\sqrt {3} + 2} + 2 \, \sin \left (x\right )\right ) - \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left ({\left (\sqrt {3} + 2\right )} \sqrt {-\sqrt {3} + 2} - 2 \, \sin \left (x\right )\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\frac {2 \, \cos \left (x\right )^{2} + 2 \, \sqrt {2} \sin \left (x\right ) - 3}{2 \, \cos \left (x\right )^{2} - 1}\right ) \] Input:

integrate(cos(x)*sec(6*x),x, algorithm="fricas")
 

Output:

-1/12*sqrt(sqrt(3) + 2)*log(sqrt(sqrt(3) + 2)*(sqrt(3) - 2) + 2*sin(x)) + 
1/12*sqrt(sqrt(3) + 2)*log(sqrt(sqrt(3) + 2)*(sqrt(3) - 2) - 2*sin(x)) + 1 
/12*sqrt(-sqrt(3) + 2)*log((sqrt(3) + 2)*sqrt(-sqrt(3) + 2) + 2*sin(x)) - 
1/12*sqrt(-sqrt(3) + 2)*log((sqrt(3) + 2)*sqrt(-sqrt(3) + 2) - 2*sin(x)) + 
 1/12*sqrt(2)*log(-(2*cos(x)^2 + 2*sqrt(2)*sin(x) - 3)/(2*cos(x)^2 - 1))
 

Sympy [F]

\[ \int \cos (x) \sec (6 x) \, dx=\int \cos {\left (x \right )} \sec {\left (6 x \right )}\, dx \] Input:

integrate(cos(x)*sec(6*x),x)
 

Output:

Integral(cos(x)*sec(6*x), x)
 

Maxima [F]

\[ \int \cos (x) \sec (6 x) \, dx=\int { \cos \left (x\right ) \sec \left (6 \, x\right ) \,d x } \] Input:

integrate(cos(x)*sec(6*x),x, algorithm="maxima")
 

Output:

-1/24*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 + 2*sqrt(2)*cos(x) + 2*sqrt(2)*s 
in(x) + 2) + 1/24*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 + 2*sqrt(2)*cos(x) - 
 2*sqrt(2)*sin(x) + 2) - 1/24*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 - 2*sqrt 
(2)*cos(x) + 2*sqrt(2)*sin(x) + 2) + 1/24*sqrt(2)*log(2*cos(x)^2 + 2*sin(x 
)^2 - 2*sqrt(2)*cos(x) - 2*sqrt(2)*sin(x) + 2) + integrate(-1/3*((cos(7*x) 
 + cos(5*x) + cos(3*x) + cos(x))*cos(8*x) - (cos(4*x) - 1)*cos(7*x) - (cos 
(4*x) - 1)*cos(5*x) - (cos(3*x) + cos(x))*cos(4*x) + (sin(7*x) + sin(5*x) 
+ sin(3*x) + sin(x))*sin(8*x) - (sin(3*x) + sin(x))*sin(4*x) - sin(7*x)*si 
n(4*x) - sin(5*x)*sin(4*x) + cos(3*x) + cos(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)
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.55 \[ \int \cos (x) \sec (6 x) \, dx=\frac {1}{24} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left ({\left | \frac {1}{4} \, \sqrt {6} + \frac {1}{4} \, \sqrt {2} + \sin \left (x\right ) \right |}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left ({\left | \frac {1}{4} \, \sqrt {6} - \frac {1}{4} \, \sqrt {2} + \sin \left (x\right ) \right |}\right ) - \frac {1}{24} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left ({\left | -\frac {1}{4} \, \sqrt {6} + \frac {1}{4} \, \sqrt {2} + \sin \left (x\right ) \right |}\right ) - \frac {1}{24} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left ({\left | -\frac {1}{4} \, \sqrt {6} - \frac {1}{4} \, \sqrt {2} + \sin \left (x\right ) \right |}\right ) + \frac {1}{12} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}\right ) \] Input:

integrate(cos(x)*sec(6*x),x, algorithm="giac")
 

Output:

1/24*(sqrt(6) - sqrt(2))*log(abs(1/4*sqrt(6) + 1/4*sqrt(2) + sin(x))) + 1/ 
24*(sqrt(6) + sqrt(2))*log(abs(1/4*sqrt(6) - 1/4*sqrt(2) + sin(x))) - 1/24 
*(sqrt(6) + sqrt(2))*log(abs(-1/4*sqrt(6) + 1/4*sqrt(2) + sin(x))) - 1/24* 
(sqrt(6) - sqrt(2))*log(abs(-1/4*sqrt(6) - 1/4*sqrt(2) + sin(x))) + 1/12*s 
qrt(2)*log(abs(-2*sqrt(2) + 4*sin(x))/abs(2*sqrt(2) + 4*sin(x)))
 

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39 \[ \int \cos (x) \sec (6 x) \, dx=\mathrm {atanh}\left (\frac {5\,\sqrt {2}\,\sin \left (x\right )}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}+\frac {1}{1048576}\right )}+\frac {3\,\sqrt {6}\,\sin \left (x\right )}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}+\frac {1}{1048576}\right )}\right )\,\left (\frac {\sqrt {2}}{12}+\frac {\sqrt {6}}{12}\right )-\mathrm {atanh}\left (\frac {5\,\sqrt {2}\,\sin \left (x\right )}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}-\frac {1}{1048576}\right )}-\frac {3\,\sqrt {6}\,\sin \left (x\right )}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}-\frac {1}{1048576}\right )}\right )\,\left (\frac {\sqrt {2}}{12}-\frac {\sqrt {6}}{12}\right )-\frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )}{6} \] Input:

int(cos(x)/cos(6*x),x)
 

Output:

atanh((5*2^(1/2)*sin(x))/(2097152*((2^(1/2)*6^(1/2))/4194304 + 1/1048576)) 
 + (3*6^(1/2)*sin(x))/(2097152*((2^(1/2)*6^(1/2))/4194304 + 1/1048576)))*( 
2^(1/2)/12 + 6^(1/2)/12) - atanh((5*2^(1/2)*sin(x))/(2097152*((2^(1/2)*6^( 
1/2))/4194304 - 1/1048576)) - (3*6^(1/2)*sin(x))/(2097152*((2^(1/2)*6^(1/2 
))/4194304 - 1/1048576)))*(2^(1/2)/12 - 6^(1/2)/12) - (2^(1/2)*atanh(2^(1/ 
2)*sin(x)))/6
 

Reduce [F]

\[ \int \cos (x) \sec (6 x) \, dx=-4 \left (\int \frac {1}{\tan \left (\frac {x}{2}\right )^{2} \tan \left (3 x \right )^{2}-\tan \left (\frac {x}{2}\right )^{2}+\tan \left (3 x \right )^{2}-1}d x \right )+\frac {\mathrm {log}\left (\tan \left (3 x \right )-1\right )}{6}-\frac {\mathrm {log}\left (\tan \left (3 x \right )+1\right )}{6}-\sin \left (x \right )-x \] Input:

int(cos(x)*sec(6*x),x)
 

Output:

( - 24*int(1/(tan(x/2)**2*tan(3*x)**2 - tan(x/2)**2 + tan(3*x)**2 - 1),x) 
+ log(tan(3*x) - 1) - log(tan(3*x) + 1) - 6*sin(x) - 6*x)/6