3.109 \(\int \cos (x) \tan (6 x) \, dx\)

Optimal. Leaf size=89 \[ -\cos (x)+\frac{\tanh ^{-1}\left (\sqrt{2} \cos (x)\right )}{3 \sqrt{2}}+\frac{1}{6} \sqrt{2-\sqrt{3}} \tanh ^{-1}\left (\frac{2 \cos (x)}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{6} \sqrt{2+\sqrt{3}} \tanh ^{-1}\left (\frac{2 \cos (x)}{\sqrt{2+\sqrt{3}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.238226, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {12, 6742, 2073, 207, 1166} \[ -\cos (x)+\frac{\tanh ^{-1}\left (\sqrt{2} \cos (x)\right )}{3 \sqrt{2}}+\frac{1}{6} \sqrt{2-\sqrt{3}} \tanh ^{-1}\left (\frac{2 \cos (x)}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{6} \sqrt{2+\sqrt{3}} \tanh ^{-1}\left (\frac{2 \cos (x)}{\sqrt{2+\sqrt{3}}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 2073

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
0]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin{align*} \int \cos (x) \tan (6 x) \, dx &=-\operatorname{Subst}\left (\int \frac{2 x^2 \left (-3+16 x^2-16 x^4\right )}{1-18 x^2+48 x^4-32 x^6} \, dx,x,\cos (x)\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{x^2 \left (-3+16 x^2-16 x^4\right )}{1-18 x^2+48 x^4-32 x^6} \, dx,x,\cos (x)\right )\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \left (\frac{1}{2}-\frac{1-12 x^2+16 x^4}{2 \left (1-18 x^2+48 x^4-32 x^6\right )}\right ) \, dx,x,\cos (x)\right )\right )\\ &=-\cos (x)+\operatorname{Subst}\left (\int \frac{1-12 x^2+16 x^4}{1-18 x^2+48 x^4-32 x^6} \, dx,x,\cos (x)\right )\\ &=-\cos (x)+\operatorname{Subst}\left (\int \left (-\frac{1}{3 \left (-1+2 x^2\right )}-\frac{2 \left (-1+8 x^2\right )}{3 \left (1-16 x^2+16 x^4\right )}\right ) \, dx,x,\cos (x)\right )\\ &=-\cos (x)-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1+2 x^2} \, dx,x,\cos (x)\right )-\frac{2}{3} \operatorname{Subst}\left (\int \frac{-1+8 x^2}{1-16 x^2+16 x^4} \, dx,x,\cos (x)\right )\\ &=\frac{\tanh ^{-1}\left (\sqrt{2} \cos (x)\right )}{3 \sqrt{2}}-\cos (x)-\frac{1}{3} \left (4 \left (2-\sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-8+4 \sqrt{3}+16 x^2} \, dx,x,\cos (x)\right )-\frac{1}{3} \left (4 \left (2+\sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-8-4 \sqrt{3}+16 x^2} \, dx,x,\cos (x)\right )\\ &=\frac{\tanh ^{-1}\left (\sqrt{2} \cos (x)\right )}{3 \sqrt{2}}+\frac{1}{6} \sqrt{2-\sqrt{3}} \tanh ^{-1}\left (\frac{2 \cos (x)}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{6} \sqrt{2+\sqrt{3}} \tanh ^{-1}\left (\frac{2 \cos (x)}{\sqrt{2+\sqrt{3}}}\right )-\cos (x)\\ \end{align*}

Mathematica [C]  time = 8.92697, size = 679, normalized size = 7.63 \[ -\cos (x)+\left (-\frac{1}{6}-\frac{i}{6}\right ) \sqrt [4]{-1} \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt [4]{-1} \sec \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right )+\left (\frac{1}{6}+\frac{i}{6}\right ) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \sec \left (\frac{x}{2}\right ) \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )\right )-\frac{\left (1+\sqrt{2}\right ) \left (x-\log \left (\sec ^2\left (\frac{x}{2}\right )\right )-2 \sqrt{3} \tanh ^{-1}\left (\frac{\left (2+\sqrt{2}\right ) \tan \left (\frac{x}{2}\right )+2}{\sqrt{6}}\right )+\log \left (-\sec ^2\left (\frac{x}{2}\right ) \left (2 \sin (x)-2 \cos (x)+\sqrt{2}\right )\right )\right )}{12 \left (2+\sqrt{2}\right )}+\frac{x-\log \left (\sec ^2\left (\frac{x}{2}\right )\right )+2 \sqrt{3} \tanh ^{-1}\left (\frac{\left (\sqrt{2}-1\right ) \tan \left (\frac{x}{2}\right )+\sqrt{2}}{\sqrt{3}}\right )+\log \left (\sec ^2\left (\frac{x}{2}\right ) \left (-\sqrt{2} \sin (x)+\sqrt{2} \cos (x)+1\right )\right )}{12 \sqrt{2}}-\frac{\left (\sqrt{2}-\sqrt{3} \sin (x)\right ) \left (\left (\sqrt{6}-2\right ) \sin (x)-\left (\sqrt{6}-2\right ) \cos (x)+\sqrt{6}-3\right ) \left (2 \left (\sqrt{6}-2\right ) \tanh ^{-1}\left (\left (\sqrt{2}-\sqrt{3}\right ) \tan \left (\frac{x}{2}\right )+\sqrt{2}\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{2} \sin (x)+\sqrt{2} \cos (x)+\sqrt{3}\right )\right )\right )\right )}{12 \left (20 \sqrt{6} \sin (x)-50 \sin (x)-5 \sqrt{6} \sin (2 x)+12 \sin (2 x)+\left (20-8 \sqrt{6}\right ) \cos (x)+\left (12-5 \sqrt{6}\right ) \cos (2 x)+15 \sqrt{6}-36\right )}+\frac{\left (\sqrt{6} \sin (x)+2\right ) \left (\left (2+\sqrt{6}\right ) \sin (x)-\left (2+\sqrt{6}\right ) \cos (x)+\sqrt{6}+3\right ) \left (\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 (2 \sin (x)-2 \cos (x)+\sqrt{6}\right )\right )\right )-2 \left (\sqrt{2}+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\left (2+\sqrt{6}\right ) \tan \left (\frac{x}{2}\right )+2}{\sqrt{2}}\right )\right )}{12 \left (-20 \sqrt{6} \sin (x)-50 \sin (x)+5 \sqrt{6} \sin (2 x)+12 \sin (2 x)+4 \left (5+2 \sqrt{6}\right ) \cos (x)+\left (12+5 \sqrt{6}\right ) \cos (2 x)-15 \sqrt{6}-36\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-1/6 - I/6)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*Sec[x/2]*(Cos[x/2] + Sin[x/2])] + (1/6 + I/6)*(-1)^(3/4)
*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[x/2]*(Cos[x/2] - Sin[x/2])] - Cos[x] - ((1 + Sqrt[2])*(x - 2*Sqrt[3]*ArcTa
nh[(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]))
]))/(12*(2 + Sqrt[2])) + (x + 2*Sqrt[3]*ArcTanh[(Sqrt[2] + (-1 + Sqrt[2])*Tan[x/2])/Sqrt[3]] - Log[Sec[x/2]^2]
+ Log[Sec[x/2]^2*(1 + Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x])])/(12*Sqrt[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] + Sqr
t[2]*Cos[x] - Sqrt[2]*Sin[x]))]))*(Sqrt[2] - Sqrt[3]*Sin[x])*(-3 + Sqrt[6] - (-2 + Sqrt[6])*Cos[x] + (-2 + Sqr
t[6])*Sin[x]))/(12*(-36 + 15*Sqrt[6] + (20 - 8*Sqrt[6])*Cos[x] + (12 - 5*Sqrt[6])*Cos[2*x] - 50*Sin[x] + 20*Sq
rt[6]*Sin[x] + 12*Sin[2*x] - 5*Sqrt[6]*Sin[2*x])) + ((-2*(Sqrt[2] + Sqrt[3])*ArcTanh[(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]))/(12*(-36 - 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]
))

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Maple [A]  time = 0.049, size = 104, normalized size = 1.2 \begin{align*} -\cos \left ( x \right ) +{\frac{ \left ( -6+4\,\sqrt{3} \right ) \sqrt{3}}{18\,\sqrt{6}-18\,\sqrt{2}}{\it Artanh} \left ( 8\,{\frac{\cos \left ( x \right ) }{2\,\sqrt{6}-2\,\sqrt{2}}} \right ) }+{\frac{ \left ( 6+4\,\sqrt{3} \right ) \sqrt{3}}{18\,\sqrt{6}+18\,\sqrt{2}}{\it Artanh} \left ( 8\,{\frac{\cos \left ( x \right ) }{2\,\sqrt{6}+2\,\sqrt{2}}} \right ) }+{\frac{{\it Artanh} \left ( \cos \left ( x \right ) \sqrt{2} \right ) \sqrt{2}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)+2/9*(-3+2*3^(1/2))*3^(1/2)/(2*6^(1/2)-2*2^(1/2))*arctanh(8*cos(x)/(2*6^(1/2)-2*2^(1/2)))+2/9*(3+2*3^(1
/2))*3^(1/2)/(2*6^(1/2)+2*2^(1/2))*arctanh(8*cos(x)/(2*6^(1/2)+2*2^(1/2)))+1/6*arctanh(cos(x)*2^(1/2))*2^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{24} \, \sqrt{2} \log \left (2 \, \sqrt{2} \sin \left (2 \, x\right ) \sin \left (x\right ) + 2 \,{\left (\sqrt{2} \cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 2 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt{2} \cos \left (x\right ) + 1\right ) - \frac{1}{24} \, \sqrt{2} \log \left (-2 \, \sqrt{2} \sin \left (2 \, x\right ) \sin \left (x\right ) - 2 \,{\left (\sqrt{2} \cos \left (x\right ) - 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 2 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt{2} \cos \left (x\right ) + 1\right ) - \cos \left (x\right ) - \int \frac{{\left (2 \, \sin \left (7 \, x\right ) + \sin \left (5 \, x\right ) - \sin \left (3 \, x\right ) - 2 \, \sin \left (x\right )\right )} \cos \left (8 \, x\right ) +{\left (\sin \left (3 \, x\right ) + 2 \, \sin \left (x\right )\right )} \cos \left (4 \, x\right ) -{\left (2 \, \cos \left (7 \, x\right ) + \cos \left (5 \, x\right ) - \cos \left (3 \, x\right ) - 2 \, \cos \left (x\right )\right )} \sin \left (8 \, x\right ) - 2 \,{\left (\cos \left (4 \, x\right ) - 1\right )} \sin \left (7 \, x\right ) -{\left (\cos \left (4 \, x\right ) - 1\right )} \sin \left (5 \, x\right ) -{\left (\cos \left (3 \, x\right ) + 2 \, \cos \left (x\right )\right )} \sin \left (4 \, x\right ) + 2 \, \cos \left (7 \, x\right ) \sin \left (4 \, x\right ) + \cos \left (5 \, x\right ) \sin \left (4 \, x\right ) - \sin \left (3 \, x\right ) - 2 \, \sin \left (x\right )}{3 \,{\left (2 \,{\left (\cos \left (4 \, x\right ) - 1\right )} \cos \left (8 \, x\right ) - \cos \left (8 \, x\right )^{2} - \cos \left (4 \, x\right )^{2} - \sin \left (8 \, x\right )^{2} + 2 \, \sin \left (8 \, x\right ) \sin \left (4 \, x\right ) - \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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) + c
os(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) + cos(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)

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Fricas [B]  time = 2.75038, size = 435, normalized size = 4.89 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*co
s(x)) + 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(2)*log(-(2*cos(x)^2 + 2*sqrt(2)*cos(x) + 1)/(2*cos(x)^2 - 1)) - cos(x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (x\right ) \tan \left (6 \, x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(x)*tan(6*x), x)