\(\int \sec (6 x) \sin (x) \, dx\) [89]

Optimal result

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

[Out]

-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))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4442, 2082, 213, 1180} \[ \int \sec (6 x) \sin (x) \, dx=-\frac {\text {arctanh}\left (\sqrt {2} \cos (x)\right )}{3 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {2 \cos (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}+\frac {\text {arctanh}\left (\frac {2 \cos (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}} \]

[In]

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

[Out]

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

Rule 213

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

Rule 1180

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]

Rule 2082

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

Rule 4442

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

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{-1+18 x^2-48 x^4+32 x^6} \, dx,x,\cos (x)\right ) \\ & = -\text {Subst}\left (\int \left (-\frac {1}{3 \left (-1+2 x^2\right )}+\frac {4 \left (-1+2 x^2\right )}{3 \left (1-16 x^2+16 x^4\right )}\right ) \, dx,x,\cos (x)\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\cos (x)\right )-\frac {4}{3} \text {Subst}\left (\int \frac {-1+2 x^2}{1-16 x^2+16 x^4} \, dx,x,\cos (x)\right ) \\ & = -\frac {\text {arctanh}\left (\sqrt {2} \cos (x)\right )}{3 \sqrt {2}}-\frac {4}{3} \text {Subst}\left (\int \frac {1}{-8-4 \sqrt {3}+16 x^2} \, dx,x,\cos (x)\right )-\frac {4}{3} \text {Subst}\left (\int \frac {1}{-8+4 \sqrt {3}+16 x^2} \, dx,x,\cos (x)\right ) \\ & = -\frac {\text {arctanh}\left (\sqrt {2} \cos (x)\right )}{3 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {2 \cos (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}+\frac {\text {arctanh}\left (\frac {2 \cos (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.33 (sec) , antiderivative size = 627, normalized size of antiderivative = 7.38 \[ \int \sec (6 x) \sin (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 )+\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 (\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 (1+\sqrt {6} \sin (x)\right ) \left (3+\sqrt {6}-\left (2+\sqrt {6}\right ) \cos (x)+\left (2+\sqrt {6}\right ) \sin (x)\right )}{\left (12+5 \sqrt {6}\right ) \cos (2 x)+2 \cos (x) \left (5+2 \sqrt {6}+5 \sqrt {6} \sin (x)\right )-2 \left (12+5 \sqrt {6}+4 \left (5+2 \sqrt {6}\right ) \sin (x)-6 \sin (2 x)\right )}+\frac {\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}-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 )}{\left (-12+5 \sqrt {6}\right ) \cos (2 x)+2 \cos (x) \left (-5+2 \sqrt {6}+5 \sqrt {6} \sin (x)\right )-2 \left (-12+5 \sqrt {6}+4 \left (-5+2 \sqrt {6}\right ) \sin (x)+6 \sin (2 x)\right )}\right ) \]

[In]

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

[Out]

((-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]]
+ (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 + Sq
rt[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])]) + (2*(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]))]))*(1 + Sqrt[6]*Sin[x])*(3 + Sqrt[6] - (2 + Sqrt[6])*Cos[x] + (2 +
Sqrt[6])*Sin[x]))/((12 + 5*Sqrt[6])*Cos[2*x] + 2*Cos[x]*(5 + 2*Sqrt[6] + 5*Sqrt[6]*Sin[x]) - 2*(12 + 5*Sqrt[6
] + 4*(5 + 2*Sqrt[6])*Sin[x] - 6*Sin[2*x])) + ((-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] + Sqrt[2]*Cos[x] - Sqrt[2]*Sin[
x]))]))*(Sqrt[2] - 2*Sqrt[3]*Sin[x])*(-3 + Sqrt[6] - (-2 + Sqrt[6])*Cos[x] + (-2 + Sqrt[6])*Sin[x]))/((-12 + 5
*Sqrt[6])*Cos[2*x] + 2*Cos[x]*(-5 + 2*Sqrt[6] + 5*Sqrt[6]*Sin[x]) - 2*(-12 + 5*Sqrt[6] + 4*(-5 + 2*Sqrt[6])*Si
n[x] + 6*Sin[2*x])))/24

Maple [A] (verified)

Time = 1.83 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {2 \,\operatorname {arctanh}\left (\frac {8 \cos \left (x \right )}{2 \sqrt {6}+2 \sqrt {2}}\right )}{3 \left (2 \sqrt {6}+2 \sqrt {2}\right )}+\frac {2 \,\operatorname {arctanh}\left (\frac {8 \cos \left (x \right )}{2 \sqrt {6}-2 \sqrt {2}}\right )}{3 \left (2 \sqrt {6}-2 \sqrt {2}\right )}-\frac {\operatorname {arctanh}\left (\cos \left (x \right ) \sqrt {2}\right ) \sqrt {2}}{6}\)	\(80\)
risch	\(\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-\sqrt {2}\, {\mathrm e}^{i x}+1\right )}{12}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+\sqrt {2}\, {\mathrm e}^{i x}+1\right )}{12}-i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (20736 \textit {\_Z}^{4}+576 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-1728 i \textit {\_R}^{3}-48 i \textit {\_R} \right ) {\mathrm e}^{i x}+1\right )\right )\)	\(93\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (67) = 134\).

Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.80 \[ \int \sec (6 x) \sin (x) \, dx=-\frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} + 2 \, \cos \left (x\right )\right ) + \frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} - 2 \, \cos \left (x\right )\right ) + \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left ({\left (\sqrt {3} + 2\right )} \sqrt {-\sqrt {3} + 2} + 2 \, \cos \left (x\right )\right ) - \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left ({\left (\sqrt {3} + 2\right )} \sqrt {-\sqrt {3} + 2} - 2 \, \cos \left (x\right )\right ) + \frac {1}{12} \, \sqrt {2} \log \left (\frac {2 \, \cos \left (x\right )^{2} - 2 \, \sqrt {2} \cos \left (x\right ) + 1}{2 \, \cos \left (x\right )^{2} - 1}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(sin(x)*sec(6*x), x)

Maxima [F]

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

[In]

integrate(sec(6*x)*sin(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) - integrate(1/3*((si
n(7*x) - sin(5*x) + sin(3*x) - sin(x))*cos(8*x) - (sin(3*x) - sin(x))*cos(4*x) - (cos(7*x) - cos(5*x) + cos(3*
x) - cos(x))*sin(8*x) - (cos(4*x) - 1)*sin(7*x) + (cos(4*x) - 1)*sin(5*x) + (cos(3*x) - cos(x))*sin(4*x) + cos
(7*x)*sin(4*x) - cos(5*x)*sin(4*x) + sin(3*x) - 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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (67) = 134\).

Time = 0.28 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.14 \[ \int \sec (6 x) \sin (x) \, dx=-\frac {1}{12} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} - \frac {2 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 6 \right |}}{{\left | 4 \, \sqrt {2} - \frac {2 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 6 \right |}}\right ) - \frac {2.39014968180000 \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 0.0173323801210000\right )}{\frac {268 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + 60.0540532247402} + \frac {5.82951931426000 \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 0.588790706481000\right )}{\frac {268 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + 121.584934401100} + \frac {16.8155413244667 \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1.69839637242000\right )}{\frac {268 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + 559.622604171000} - \frac {7956.25491093333 \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 57.6954805410000\right )}{\frac {268 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 168981.261592000} \]

[In]

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

[Out]

-1/12*sqrt(2)*log(abs(-4*sqrt(2) - 2*(cos(x) - 1)/(cos(x) + 1) - 6)/abs(4*sqrt(2) - 2*(cos(x) - 1)/(cos(x) + 1
) - 6)) - 2.39014968180000*log(-(cos(x) - 1)/(cos(x) + 1) - 0.0173323801210000)/(268*(cos(x) - 1)/(cos(x) + 1)
+ 60.0540532247402) + 5.82951931426000*log(-(cos(x) - 1)/(cos(x) + 1) - 0.588790706481000)/(268*(cos(x) - 1)/
(cos(x) + 1) + 121.584934401100) + 16.8155413244667*log(-(cos(x) - 1)/(cos(x) + 1) - 1.69839637242000)/(268*(c
os(x) - 1)/(cos(x) + 1) + 559.622604171000) - 7956.25491093333*log(-(cos(x) - 1)/(cos(x) + 1) - 57.69548054100
00)/(268*(cos(x) - 1)/(cos(x) + 1) - 168981.261592000)

Mupad [B] (verification not implemented)

Time = 27.36 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39 \[ \int \sec (6 x) \sin (x) \, dx=\mathrm {atanh}\left (\frac {5\,\sqrt {2}\,\cos \left (x\right )}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}+\frac {1}{1048576}\right )}+\frac {3\,\sqrt {6}\,\cos \left (x\right )}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}+\frac {1}{1048576}\right )}\right )\,\left (\frac {\sqrt {2}}{12}+\frac {\sqrt {6}}{12}\right )-\mathrm {atanh}\left (\frac {5\,\sqrt {2}\,\cos \left (x\right )}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}-\frac {1}{1048576}\right )}-\frac {3\,\sqrt {6}\,\cos \left (x\right )}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}-\frac {1}{1048576}\right )}\right )\,\left (\frac {\sqrt {2}}{12}-\frac {\sqrt {6}}{12}\right )-\frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\cos \left (x\right )\right )}{6} \]

[In]

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

[Out]

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