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3.87 \(\int \sec (4 x) \sin (x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 71, normalized size of antiderivative = 1.00, 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 = {4357, 1093, 207} \[ \frac {\tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTanh[(2*Cos[x])/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) + ArcTanh[(2*Cos[x])/Sqrt[2 + Sqrt[2]]]/(2*Sq
rt[2*(2 + Sqrt[2])])

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 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 4357

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

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Mathematica [C]  time = 57.93, size = 4845, normalized size = 68.24 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*(-1)^(3/8)*(1 + Sqrt[2])*x - (2*(-1)^(1/4)*(-2 - (1 - I)*(-1)^(5/8) + (-1)^(5/8)*Sqrt[2])*ArcTan[(-Cos[x]
 + (1 + Sqrt[2])*Sin[x])/(2*(-1)^(3/8) + Cos[x] - Sqrt[2]*Cos[x] + Sin[x])])/((-1 + I) + 2*(-1)^(3/8) + Sqrt[2
]) - (2*(1 - I)^(3/2)*2^(1/4)*((-3 - I) + 2*(-1)^(5/8) + (2 + I)*Sqrt[2] - (2 + 2*I)*(-1)^(3/8)*Sqrt[2] + 2*(-
1)^(5/8)*Sqrt[2])*ArcTan[((1 + I) + I*Sqrt[2] + ((-1 + I) + 2*(-1)^(3/8) + Sqrt[2])*Tan[x/2])/(Sqrt[1 - I]*2^(
3/4))])/((-1 + I) + 2*(-1)^(3/8) + Sqrt[2]) + 2*(-1)^(3/8)*Log[Sec[x/2]^2] + ((-1)^(3/4)*(-2 - (1 - I)*(-1)^(5
/8) + (-1)^(5/8)*Sqrt[2])*Log[-(Sec[x/2]^4*(-2 + (1 - I)*Sqrt[2] + 2*(-1)^(3/8)*(-1 + Sqrt[2])*Cos[x] + Sqrt[2
]*Cos[2*x] - 2*(-1)^(3/8)*Sin[x] + Sqrt[2]*Sin[2*x]))])/((-1 + I) + 2*(-1)^(3/8) + Sqrt[2]))*((-1/2 - I/2)/(((
-1 + I) + Sqrt[1 - I]*Sqrt[1 + I])*(-((-1 - I)^(3/2)*(1 - I)^(1/4)*(1 + I)^(1/4)) - (1 + I)*Cos[x] + I*Sqrt[1
- I]*Sqrt[1 + I]*Cos[x] + (1 - I)*Sin[x] + Sqrt[1 - I]*Sqrt[1 + I]*Sin[x])) - Sin[x]/(Sqrt[-1 - I]*(1 - I)^(1/
4)*(1 + I)^(1/4)*((-1 + I) + Sqrt[1 - I]*Sqrt[1 + I])*(-((-1 - I)^(3/2)*(1 - I)^(1/4)*(1 + I)^(1/4)) - (1 + I)
*Cos[x] + I*Sqrt[1 - I]*Sqrt[1 + I]*Cos[x] + (1 - I)*Sin[x] + Sqrt[1 - I]*Sqrt[1 + I]*Sin[x])) - ((I/2)*Sqrt[-
1 - I]*(1 - I)^(1/4)*(1 + I)^(1/4)*Sin[x])/(((-1 + I) + Sqrt[1 - I]*Sqrt[1 + I])*(-((-1 - I)^(3/2)*(1 - I)^(1/
4)*(1 + I)^(1/4)) - (1 + I)*Cos[x] + I*Sqrt[1 - I]*Sqrt[1 + I]*Cos[x] + (1 - I)*Sin[x] + Sqrt[1 - I]*Sqrt[1 +
I]*Sin[x]))))/(-2*(-1)^(3/8)*(1 + Sqrt[2]) - (2*(-1)^(1/4)*(-2 - (1 - I)*(-1)^(5/8) + (-1)^(5/8)*Sqrt[2])*(((1
 + Sqrt[2])*Cos[x] + Sin[x])/(2*(-1)^(3/8) + Cos[x] - Sqrt[2]*Cos[x] + Sin[x]) - ((Cos[x] - Sin[x] + Sqrt[2]*S
in[x])*(-Cos[x] + (1 + Sqrt[2])*Sin[x]))/(2*(-1)^(3/8) + Cos[x] - Sqrt[2]*Cos[x] + Sin[x])^2))/(((-1 + I) + 2*
(-1)^(3/8) + Sqrt[2])*(1 + (-Cos[x] + (1 + Sqrt[2])*Sin[x])^2/(2*(-1)^(3/8) + Cos[x] - Sqrt[2]*Cos[x] + Sin[x]
)^2)) + 2*(-1)^(3/8)*Tan[x/2] - ((-1)^(3/4)*(-2 - (1 - I)*(-1)^(5/8) + (-1)^(5/8)*Sqrt[2])*Cos[x/2]^4*(-(Sec[x
/2]^4*(-2*(-1)^(3/8)*Cos[x] + 2*Sqrt[2]*Cos[2*x] - 2*(-1)^(3/8)*(-1 + Sqrt[2])*Sin[x] - 2*Sqrt[2]*Sin[2*x])) -
 2*Sec[x/2]^4*(-2 + (1 - I)*Sqrt[2] + 2*(-1)^(3/8)*(-1 + Sqrt[2])*Cos[x] + Sqrt[2]*Cos[2*x] - 2*(-1)^(3/8)*Sin
[x] + Sqrt[2]*Sin[2*x])*Tan[x/2]))/(((-1 + I) + 2*(-1)^(3/8) + Sqrt[2])*(-2 + (1 - I)*Sqrt[2] + 2*(-1)^(3/8)*(
-1 + Sqrt[2])*Cos[x] + Sqrt[2]*Cos[2*x] - 2*(-1)^(3/8)*Sin[x] + Sqrt[2]*Sin[2*x])) - ((1 - I)*((-3 - I) + 2*(-
1)^(5/8) + (2 + I)*Sqrt[2] - (2 + 2*I)*(-1)^(3/8)*Sqrt[2] + 2*(-1)^(5/8)*Sqrt[2])*Sec[x/2]^2)/(Sqrt[2]*(1 + ((
1/4 + I/4)*((1 + I) + I*Sqrt[2] + ((-1 + I) + 2*(-1)^(3/8) + Sqrt[2])*Tan[x/2])^2)/Sqrt[2]))) + ((-4*Sqrt[-1 -
 I]*(-1 + Sqrt[2])*ArcTanh[((-I)*((1 + I) + Sqrt[2]) + ((1 + I) + 2*(-1)^(5/8) - Sqrt[2])*Tan[x/2])/(Sqrt[-1 -
 I]*2^(3/4))] + (-1)^(1/8)*2^(1/4)*(2*ArcTan[(Cos[x] + (1 + Sqrt[2])*Sin[x])/(2*(-1)^(5/8) + (-1 + Sqrt[2])*Co
s[x] + Sin[x])] - I*(2*(1 + Sqrt[2])*x + 2*Log[Sec[x/2]^2] - Log[Sec[x/2]^4*(2 - (1 + I)*Sqrt[2] + 2*(-1)^(5/8
)*(-1 + Sqrt[2])*Cos[x] - Sqrt[2]*Cos[2*x] + 2*(-1)^(5/8)*Sin[x] + Sqrt[2]*Sin[2*x])])))*(2 + I*Sqrt[-1 + I]*2
^(1/4)*((1 + I) + Sqrt[2])*Sin[x]))/(2^(1/4)*(4*Sqrt[-1 + I]*2^(1/4)*((-1 - I) + Sqrt[2]) - 8*(-1 + Sqrt[2])*C
os[x] - 8*Sin[x])*((2*(-1)^(1/8)*(-2 - (1 + I)*Sqrt[2] + (-1)^(1/8)*((1 + I) + I*Sqrt[2])*Cos[x] + (2*I)*(1 +
Sqrt[2])*Cos[2*x] + (-1)^(1/8)*Sin[x] - (-1)^(5/8)*Sin[x] + 3*(-1)^(1/8)*Sqrt[2]*Sin[x] - (2*I)*Sin[2*x]))/(2
- (1 + I)*Sqrt[2] + 2*(-1)^(5/8)*(-1 + Sqrt[2])*Cos[x] - Sqrt[2]*Cos[2*x] + 2*(-1)^(5/8)*Sin[x] + Sqrt[2]*Sin[
2*x]) - (((1 + I) + 2*(-1)^(5/8) - Sqrt[2])*(-1 + Sqrt[2])*Sec[x/2]^2)/(1 + ((1/4 - I/4)*(I*((1 + I) + Sqrt[2]
) + ((-1 - I) - 2*(-1)^(5/8) + Sqrt[2])*Tan[x/2])^2)/Sqrt[2]))) + ((-2*(-1)^(3/8)*Sqrt[2]*(1 + (-1)^(1/4))*x +
 (2*(-2*I + 2*(-1)^(3/4) + 2*(-1)^(1/8)*Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (-1)^(7/8)*Sqrt[2])*ArcTan[Cos[x]/(-((-
1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] + (1 + (-1)^(1/4))*Sin[x])])/(-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2]) - ((
4 + 4*I)*(-1)^(5/8)*((3 - 3*I) - (2 - 2*I)*Sqrt[2] + (-1)^(1/8)*Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (1 - I)*(-1)^(5
/8)*Sqrt[2] + (1 + I)*(-1)^(7/8)*Sqrt[2])*ArcTanh[(1/2 + I/2)*(-1)^(5/8)*(-1 - (-1)^(1/4) + (-I + (-1)^(3/4) +
 (-1)^(1/8)*Sqrt[2])*Tan[x/2])])/(-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2]) - 2*(-1)^(7/8)*Sqrt[2]*(-1 + (-1)^(1/4
))*Log[Sec[x/2]^2] - ((-1 + (-1)^(1/4))*(2 - (-1)^(3/8)*Sqrt[2] + (-1)^(5/8)*Sqrt[2])*Log[(1/4 + I/4)*Sec[x/2]
^4*((2 - 2*I) + 6*Sqrt[2] - (4 - 4*I)*(-1)^(7/8)*Sqrt[2]*Cos[x] - 2*((1 + I) + Sqrt[2])*Cos[2*x] - (4 - 4*I)*(
-1)^(1/8)*Sqrt[2]*Sin[x] - (4 - 4*I)*(-1)^(3/8)*Sqrt[2]*Sin[x] - (2 - 2*I)*Sin[2*x] + (2*I)*Sqrt[2]*Sin[2*x])]
)/(-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2]))*(I/(Sqrt[1 - I]*((-1 + I) + Sqrt[1 - I]*Sqrt[1 + I])^2*(Sqrt[-1 - I]
*(1 - I)^(3/4)*(1 + I)^(1/4) + Sqrt[1 - I]*Cos[x] - Sqrt[1 + I]*Cos[x] + I*Sqrt[1 - I]*Sin[x] + I*Sqrt[1 + I]*
Sin[x])) + 1/(Sqrt[1 + I]*((-1 + I) + Sqrt[1 - I]*Sqrt[1 + I])^2*(Sqrt[-1 - I]*(1 - I)^(3/4)*(1 + I)^(1/4) + S
qrt[1 - I]*Cos[x] - Sqrt[1 + I]*Cos[x] + I*Sqrt[1 - I]*Sin[x] + I*Sqrt[1 + I]*Sin[x])) - (2*Sin[x])/(Sqrt[-1 -
 I]*(1 - I)^(1/4)*(1 + I)^(3/4)*((-1 + I) + Sqrt[1 - I]*Sqrt[1 + I])^2*(Sqrt[-1 - I]*(1 - I)^(3/4)*(1 + I)^(1/
4) + Sqrt[1 - I]*Cos[x] - Sqrt[1 + I]*Cos[x] + I*Sqrt[1 - I]*Sin[x] + I*Sqrt[1 + I]*Sin[x]))))/(-2*(-1)^(3/8)*
Sqrt[2]*(1 + (-1)^(1/4)) + (2*(-2*I + 2*(-1)^(3/4) + 2*(-1)^(1/8)*Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (-1)^(7/8)*Sq
rt[2])*(-((Cos[x]*((1 + (-1)^(1/4))*Cos[x] - (-1)^(3/4)*Sin[x]))/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] +
(1 + (-1)^(1/4))*Sin[x])^2) - Sin[x]/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] + (1 + (-1)^(1/4))*Sin[x])))/(
(-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2])*(1 + Cos[x]^2/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] + (1 + (-1)^(1
/4))*Sin[x])^2)) - 2*(-1)^(7/8)*Sqrt[2]*(-1 + (-1)^(1/4))*Tan[x/2] - ((2 - 2*I)*(-1 + (-1)^(1/4))*(2 - (-1)^(3
/8)*Sqrt[2] + (-1)^(5/8)*Sqrt[2])*Cos[x/2]^4*((1/4 + I/4)*Sec[x/2]^4*((-4 + 4*I)*(-1)^(1/8)*Sqrt[2]*Cos[x] - (
4 - 4*I)*(-1)^(3/8)*Sqrt[2]*Cos[x] - (4 - 4*I)*Cos[2*x] + (4*I)*Sqrt[2]*Cos[2*x] + (4 - 4*I)*(-1)^(7/8)*Sqrt[2
]*Sin[x] + 4*((1 + I) + Sqrt[2])*Sin[2*x]) + (1/2 + I/2)*Sec[x/2]^4*((2 - 2*I) + 6*Sqrt[2] - (4 - 4*I)*(-1)^(7
/8)*Sqrt[2]*Cos[x] - 2*((1 + I) + Sqrt[2])*Cos[2*x] - (4 - 4*I)*(-1)^(1/8)*Sqrt[2]*Sin[x] - (4 - 4*I)*(-1)^(3/
8)*Sqrt[2]*Sin[x] - (2 - 2*I)*Sin[2*x] + (2*I)*Sqrt[2]*Sin[2*x])*Tan[x/2]))/((-I + (-1)^(3/4) + (-1)^(1/8)*Sqr
t[2])*((2 - 2*I) + 6*Sqrt[2] - (4 - 4*I)*(-1)^(7/8)*Sqrt[2]*Cos[x] - 2*((1 + I) + Sqrt[2])*Cos[2*x] - (4 - 4*I
)*(-1)^(1/8)*Sqrt[2]*Sin[x] - (4 - 4*I)*(-1)^(3/8)*Sqrt[2]*Sin[x] - (2 - 2*I)*Sin[2*x] + (2*I)*Sqrt[2]*Sin[2*x
])) + (2*(-1)^(3/4)*((3 - 3*I) - (2 - 2*I)*Sqrt[2] + (-1)^(1/8)*Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (1 - I)*(-1)^(5
/8)*Sqrt[2] + (1 + I)*(-1)^(7/8)*Sqrt[2])*Sec[x/2]^2)/(1 + ((-1)^(3/4)*(-1 - (-1)^(1/4) + (-I + (-1)^(3/4) + (
-1)^(1/8)*Sqrt[2])*Tan[x/2])^2)/2)) + ((2*((-1)^(1/8) + (-1)^(3/8))*x - (2*(-1)^(7/8)*(2 - Sqrt[2] - (-1)^(3/8
)*Sqrt[2] + (-1)^(5/8)*Sqrt[2])*ArcTan[Cos[x]/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] - (1 + (-1)^(1/4))*Si
n[x])])/(-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2]) - ((4 + 4*I)*(-1)^(5/8)*(3*I + (-1)^(1/8) - (-1)^(3/8) - (1 + I
)*(-1)^(5/8) - (2*I)*Sqrt[2] + (1 + I)*(-1)^(5/8)*Sqrt[2])*ArcTanh[(1/2 + I/2)*(-1)^(5/8)*(1 + (-1)^(1/4) + (-
I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2])*Tan[x/2])])/(-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2]) + 2*(-1)^(3/8)*(-I + (
-1)^(1/4))*Log[Sec[x/2]^2] + ((-1)^(3/8)*(2 - Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (-1)^(5/8)*Sqrt[2])*Log[(1/4 + I/
4)*Sec[x/2]^4*((2 - 2*I) + 6*Sqrt[2] - (4 - 4*I)*(-1)^(7/8)*Sqrt[2]*Cos[x] - 2*((1 + I) + Sqrt[2])*Cos[2*x] +
(4 - 4*I)*(-1)^(1/8)*((1 + I) + Sqrt[2])*Sin[x] + (2 - 2*I)*Sin[2*x] - (2*I)*Sqrt[2]*Sin[2*x])])/(-I + (-1)^(3
/4) + (-1)^(1/8)*Sqrt[2]))*(1/(Sqrt[1 - I]*((-1 - I) + Sqrt[1 - I]*Sqrt[1 + I])^2*(-(Sqrt[-1 + I]*(1 - I)^(1/4
)*(1 + I)^(3/4)) + Sqrt[1 - I]*Cos[x] - Sqrt[1 + I]*Cos[x] - I*Sqrt[1 - I]*Sin[x] - I*Sqrt[1 + I]*Sin[x])) - I
/(Sqrt[1 + I]*((-1 - I) + Sqrt[1 - I]*Sqrt[1 + I])^2*(-(Sqrt[-1 + I]*(1 - I)^(1/4)*(1 + I)^(3/4)) + Sqrt[1 - I
]*Cos[x] - Sqrt[1 + I]*Cos[x] - I*Sqrt[1 - I]*Sin[x] - I*Sqrt[1 + I]*Sin[x])) + (2*Sin[x])/(Sqrt[-1 + I]*(1 -
I)^(3/4)*(1 + I)^(1/4)*((-1 - I) + Sqrt[1 - I]*Sqrt[1 + I])^2*(-(Sqrt[-1 + I]*(1 - I)^(1/4)*(1 + I)^(3/4)) + S
qrt[1 - I]*Cos[x] - Sqrt[1 + I]*Cos[x] - I*Sqrt[1 - I]*Sin[x] - I*Sqrt[1 + I]*Sin[x]))))/(2*((-1)^(1/8) + (-1)
^(3/8)) - (2*(-1)^(7/8)*(2 - Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (-1)^(5/8)*Sqrt[2])*(-((Cos[x]*(-((1 + (-1)^(1/4))
*Cos[x]) - (-1)^(3/4)*Sin[x]))/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] - (1 + (-1)^(1/4))*Sin[x])^2) - Sin[
x]/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] - (1 + (-1)^(1/4))*Sin[x])))/((-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt
[2])*(1 + Cos[x]^2/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] - (1 + (-1)^(1/4))*Sin[x])^2)) + 2*(-1)^(3/8)*(-
I + (-1)^(1/4))*Tan[x/2] + ((2 - 2*I)*(-1)^(3/8)*(2 - Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (-1)^(5/8)*Sqrt[2])*Cos[x
/2]^4*((1/4 + I/4)*Sec[x/2]^4*((4 - 4*I)*(-1)^(1/8)*((1 + I) + Sqrt[2])*Cos[x] + (4 - 4*I)*Cos[2*x] - (4*I)*Sq
rt[2]*Cos[2*x] + (4 - 4*I)*(-1)^(7/8)*Sqrt[2]*Sin[x] + 4*((1 + I) + Sqrt[2])*Sin[2*x]) + (1/2 + I/2)*Sec[x/2]^
4*((2 - 2*I) + 6*Sqrt[2] - (4 - 4*I)*(-1)^(7/8)*Sqrt[2]*Cos[x] - 2*((1 + I) + Sqrt[2])*Cos[2*x] + (4 - 4*I)*(-
1)^(1/8)*((1 + I) + Sqrt[2])*Sin[x] + (2 - 2*I)*Sin[2*x] - (2*I)*Sqrt[2]*Sin[2*x])*Tan[x/2]))/((-I + (-1)^(3/4
) + (-1)^(1/8)*Sqrt[2])*((2 - 2*I) + 6*Sqrt[2] - (4 - 4*I)*(-1)^(7/8)*Sqrt[2]*Cos[x] - 2*((1 + I) + Sqrt[2])*C
os[2*x] + (4 - 4*I)*(-1)^(1/8)*((1 + I) + Sqrt[2])*Sin[x] + (2 - 2*I)*Sin[2*x] - (2*I)*Sqrt[2]*Sin[2*x])) + (2
*(-1)^(3/4)*(3*I + (-1)^(1/8) - (-1)^(3/8) - (1 + I)*(-1)^(5/8) - (2*I)*Sqrt[2] + (1 + I)*(-1)^(5/8)*Sqrt[2])*
Sec[x/2]^2)/(1 + ((-1)^(3/4)*(1 + (-1)^(1/4) + (-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2])*Tan[x/2])^2)/2))

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fricas [B]  time = 0.68, size = 121, normalized size = 1.70 \[ -\frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} {\left (\sqrt {2} - 1\right )} + 2 \, \cos \relax (x)\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} {\left (\sqrt {2} - 1\right )} - 2 \, \cos \relax (x)\right ) + \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left ({\left (\sqrt {2} + 1\right )} \sqrt {-\sqrt {2} + 2} + 2 \, \cos \relax (x)\right ) - \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left ({\left (\sqrt {2} + 1\right )} \sqrt {-\sqrt {2} + 2} - 2 \, \cos \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.14, size = 133, normalized size = 1.87 \[ -\frac {2.16139547686000 \, \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 0.0395661298966000\right )}{\frac {140 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} + 28.1312524456150} - \frac {4.18450863968000 \, \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 0.446462692172000\right )}{\frac {140 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} + 44.3876588494000} - \frac {20.9929814212000 \, \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 2.23982880884000\right )}{\frac {140 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} + 404.466590643000} - \frac {1380.66111446200 \, \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 25.2741423691000\right )}{\frac {140 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} - 10892.9855019000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2.16139547686000*log(-(cos(x) - 1)/(cos(x) + 1) - 0.0395661298966000)/(140*(cos(x) - 1)/(cos(x) + 1) + 28.131
2524456150) - 4.18450863968000*log(-(cos(x) - 1)/(cos(x) + 1) - 0.446462692172000)/(140*(cos(x) - 1)/(cos(x) +
 1) + 44.3876588494000) - 20.9929814212000*log(-(cos(x) - 1)/(cos(x) + 1) - 2.23982880884000)/(140*(cos(x) - 1
)/(cos(x) + 1) + 404.466590643000) - 1380.66111446200*log(-(cos(x) - 1)/(cos(x) + 1) - 25.2741423691000)/(140*
(cos(x) - 1)/(cos(x) + 1) - 10892.9855019000)

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maple [A]  time = 0.21, size = 54, normalized size = 0.76 \[ -\frac {\sqrt {2}\, \arctanh \left (\frac {2 \cos \relax (x )}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2-\sqrt {2}}}+\frac {\sqrt {2}\, \arctanh \left (\frac {2 \cos \relax (x )}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2+\sqrt {2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec \left (4 \, x\right ) \sin \relax (x)\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sec(4*x)*sin(x), x)

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mupad [B]  time = 0.09, size = 112, normalized size = 1.58 \[ \frac {\mathrm {atanh}\left (\frac {\cos \relax (x)\,\sqrt {2-\sqrt {2}}}{64\,\left (\frac {\sqrt {2}}{128}-\frac {1}{64}\right )}-\frac {\sqrt {2}\,\cos \relax (x)\,\sqrt {2-\sqrt {2}}}{64\,\left (\frac {\sqrt {2}}{128}-\frac {1}{64}\right )}\right )\,\sqrt {2-\sqrt {2}}}{4}-\frac {\mathrm {atanh}\left (\frac {\cos \relax (x)\,\sqrt {\sqrt {2}+2}}{64\,\left (\frac {\sqrt {2}}{128}+\frac {1}{64}\right )}+\frac {\sqrt {2}\,\cos \relax (x)\,\sqrt {\sqrt {2}+2}}{64\,\left (\frac {\sqrt {2}}{128}+\frac {1}{64}\right )}\right )\,\sqrt {\sqrt {2}+2}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/cos(4*x),x)

[Out]

(atanh((cos(x)*(2 - 2^(1/2))^(1/2))/(64*(2^(1/2)/128 - 1/64)) - (2^(1/2)*cos(x)*(2 - 2^(1/2))^(1/2))/(64*(2^(1
/2)/128 - 1/64)))*(2 - 2^(1/2))^(1/2))/4 - (atanh((cos(x)*(2^(1/2) + 2)^(1/2))/(64*(2^(1/2)/128 + 1/64)) + (2^
(1/2)*cos(x)*(2^(1/2) + 2)^(1/2))/(64*(2^(1/2)/128 + 1/64)))*(2^(1/2) + 2)^(1/2))/4

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin {\relax (x )} \sec {\left (4 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(x)*sec(4*x), x)

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