3.118 \(\int \cos (x) \sec (4 x) \, dx\)

Optimal. Leaf size=71 \[ \frac{\tanh ^{-1}\left (\frac{2 \sin (x)}{\sqrt{2-\sqrt{2}}}\right )}{2 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tanh ^{-1}\left (\frac{2 \sin (x)}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2 \left (2+\sqrt{2}\right )}} \]

[Out]

ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) - ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[2]]]/(2*Sqr
t[2*(2 + Sqrt[2])])

________________________________________________________________________________________

Rubi [A]  time = 0.0457058, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4356, 1093, 207} \[ \frac{\tanh ^{-1}\left (\frac{2 \sin (x)}{\sqrt{2-\sqrt{2}}}\right )}{2 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tanh ^{-1}\left (\frac{2 \sin (x)}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2 \left (2+\sqrt{2}\right )}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Sec[4*x],x]

[Out]

ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) - ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[2]]]/(2*Sqr
t[2*(2 + Sqrt[2])])

Rule 4356

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[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])

Rule 1093

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

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])

Rubi steps

\begin{align*} \int \cos (x) \sec (4 x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{1-8 x^2+8 x^4} \, dx,x,\sin (x)\right )\\ &=\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-4-2 \sqrt{2}+8 x^2} \, dx,x,\sin (x)\right )-\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-4+2 \sqrt{2}+8 x^2} \, dx,x,\sin (x)\right )\\ &=\frac{\tanh ^{-1}\left (\frac{2 \sin (x)}{\sqrt{2-\sqrt{2}}}\right )}{2 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tanh ^{-1}\left (\frac{2 \sin (x)}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2 \left (2+\sqrt{2}\right )}}\\ \end{align*}

Mathematica [A]  time = 0.106529, size = 67, normalized size = 0.94 \[ \frac{1}{4} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left (\frac{2 \sin (x)}{\sqrt{2-\sqrt{2}}}\right )-\frac{\tanh ^{-1}\left (\frac{2 \sin (x)}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2 \left (2+\sqrt{2}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Sec[4*x],x]

[Out]

(Sqrt[2 + Sqrt[2]]*ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[2]]])/4 - ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*
(2 + Sqrt[2])])

________________________________________________________________________________________

Maple [A]  time = 0.083, size = 54, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}}{4\,\sqrt{2-\sqrt{2}}}{\it Artanh} \left ( 2\,{\frac{\sin \left ( x \right ) }{\sqrt{2-\sqrt{2}}}} \right ) }-{\frac{\sqrt{2}}{4\,\sqrt{2+\sqrt{2}}}{\it Artanh} \left ( 2\,{\frac{\sin \left ( x \right ) }{\sqrt{2+\sqrt{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*sec(4*x),x)

[Out]

1/4*2^(1/2)/(2-2^(1/2))^(1/2)*arctanh(2*sin(x)/(2-2^(1/2))^(1/2))-1/4*2^(1/2)/(2+2^(1/2))^(1/2)*arctanh(2*sin(
x)/(2+2^(1/2))^(1/2))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (x\right ) \sec \left (4 \, x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sec(4*x),x, algorithm="maxima")

[Out]

integrate(cos(x)*sec(4*x), x)

________________________________________________________________________________________

Fricas [B]  time = 2.60697, size = 393, normalized size = 5.54 \begin{align*} \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \log \left (\sqrt{\sqrt{2} + 2}{\left (\sqrt{2} - 1\right )} + 2 \, \sin \left (x\right )\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \log \left (\sqrt{\sqrt{2} + 2}{\left (\sqrt{2} - 1\right )} - 2 \, \sin \left (x\right )\right ) - \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \log \left ({\left (\sqrt{2} + 1\right )} \sqrt{-\sqrt{2} + 2} + 2 \, \sin \left (x\right )\right ) + \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \log \left ({\left (\sqrt{2} + 1\right )} \sqrt{-\sqrt{2} + 2} - 2 \, \sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sec(4*x),x, algorithm="fricas")

[Out]

1/8*sqrt(sqrt(2) + 2)*log(sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sin(x)) - 1/8*sqrt(sqrt(2) + 2)*log(sqrt(sqrt(2)
+ 2)*(sqrt(2) - 1) - 2*sin(x)) - 1/8*sqrt(-sqrt(2) + 2)*log((sqrt(2) + 1)*sqrt(-sqrt(2) + 2) + 2*sin(x)) + 1/
8*sqrt(-sqrt(2) + 2)*log((sqrt(2) + 1)*sqrt(-sqrt(2) + 2) - 2*sin(x))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos{\left (x \right )} \sec{\left (4 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sec(4*x),x)

[Out]

Integral(cos(x)*sec(4*x), x)

________________________________________________________________________________________

Giac [B]  time = 1.30961, size = 134, normalized size = 1.89 \begin{align*} -\frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \log \left ({\left | \frac{1}{2} \, \sqrt{\sqrt{2} + 2} + \sin \left (x\right ) \right |}\right ) + \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \log \left ({\left | -\frac{1}{2} \, \sqrt{\sqrt{2} + 2} + \sin \left (x\right ) \right |}\right ) + \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \log \left ({\left | \sqrt{-\frac{1}{4} \, \sqrt{2} + \frac{1}{2}} + \sin \left (x\right ) \right |}\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \log \left ({\left | -\sqrt{-\frac{1}{4} \, \sqrt{2} + \frac{1}{2}} + \sin \left (x\right ) \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sec(4*x),x, algorithm="giac")

[Out]

-1/8*sqrt(-sqrt(2) + 2)*log(abs(1/2*sqrt(sqrt(2) + 2) + sin(x))) + 1/8*sqrt(-sqrt(2) + 2)*log(abs(-1/2*sqrt(sq
rt(2) + 2) + sin(x))) + 1/8*sqrt(sqrt(2) + 2)*log(abs(sqrt(-1/4*sqrt(2) + 1/2) + sin(x))) - 1/8*sqrt(sqrt(2) +
2)*log(abs(-sqrt(-1/4*sqrt(2) + 1/2) + sin(x)))