3.4.75 \(\int \frac {\sin (2 x)}{\cos ^4(x)+\sin ^4(x)} \, dx\) [375]

Optimal. Leaf size=7 \[ -\tan ^{-1}(\cos (2 x)) \]

[Out]

-arctan(cos(2*x))

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {12, 1121, 631, 210} \begin {gather*} -\text {ArcTan}(\cos (2 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[2*x]/(Cos[x]^4 + Sin[x]^4),x]

[Out]

-ArcTan[Cos[2*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 210

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

Rule 631

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

Rule 1121

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rubi steps

\begin {align*} \int \frac {\sin (2 x)}{\cos ^4(x)+\sin ^4(x)} \, dx &=\text {Subst}\left (\int \frac {2 x}{1-2 x^2+2 x^4} \, dx,x,\sin (x)\right )\\ &=2 \text {Subst}\left (\int \frac {x}{1-2 x^2+2 x^4} \, dx,x,\sin (x)\right )\\ &=\text {Subst}\left (\int \frac {1}{1-2 x+2 x^2} \, dx,x,\sin ^2(x)\right )\\ &=\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-2 \sin ^2(x)\right )\\ &=-\tan ^{-1}\left (1-2 \sin ^2(x)\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 7, normalized size = 1.00 \begin {gather*} -\tan ^{-1}(\cos (2 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[2*x]/(Cos[x]^4 + Sin[x]^4),x]

[Out]

-ArcTan[Cos[2*x]]

________________________________________________________________________________________

Maple [A]
time = 0.10, size = 12, normalized size = 1.71

method	result	size
derivativedivides	\(-\arctan \left (2 \left (\cos ^{2}\left (x \right )\right )-1\right )\)	\(12\)
default	\(-\arctan \left (2 \left (\cos ^{2}\left (x \right )\right )-1\right )\)	\(12\)
risch	\(-\frac {i \ln \left ({\mathrm e}^{4 i x}+2 i {\mathrm e}^{2 i x}+1\right )}{2}+\frac {i \ln \left ({\mathrm e}^{4 i x}-2 i {\mathrm e}^{2 i x}+1\right )}{2}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(2*x)/(sin(x)^4+cos(x)^4),x,method=_RETURNVERBOSE)

[Out]

-arctan(2*cos(x)^2-1)

________________________________________________________________________________________

Maxima [A]
time = 2.11, size = 9, normalized size = 1.29 \begin {gather*} \arctan \left (2 \, \sin \left (x\right )^{2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/(cos(x)^4+sin(x)^4),x, algorithm="maxima")

[Out]

arctan(2*sin(x)^2 - 1)

________________________________________________________________________________________

Fricas [A]
time = 1.25, size = 11, normalized size = 1.57 \begin {gather*} -\arctan \left (2 \, \cos \left (x\right )^{2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/(cos(x)^4+sin(x)^4),x, algorithm="fricas")

[Out]

-arctan(2*cos(x)^2 - 1)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/(cos(x)**4+sin(x)**4),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.63, size = 11, normalized size = 1.57 \begin {gather*} -\arctan \left (2 \, \cos \left (x\right )^{2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/(cos(x)^4+sin(x)^4),x, algorithm="giac")

[Out]

-arctan(2*cos(x)^2 - 1)

________________________________________________________________________________________

Mupad [B]
time = 0.27, size = 5, normalized size = 0.71 \begin {gather*} \mathrm {atan}\left ({\mathrm {tan}\left (x\right )}^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(2*x)/(cos(x)^4 + sin(x)^4),x)

[Out]

atan(tan(x)^2)

________________________________________________________________________________________