3.144 \(\int \frac{\sin ^2(x)}{1+\sin ^2(x)} \, dx\)

Optimal. Leaf size=36 \[ -\frac{x}{\sqrt{2}}+x-\frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right )}{\sqrt{2}} \]

[Out]

x - x/Sqrt[2] - ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]/Sqrt[2]

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Rubi [A]  time = 0.0377339, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3171, 3181, 203} \[ -\frac{x}{\sqrt{2}}+x-\frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^2/(1 + Sin[x]^2),x]

[Out]

x - x/Sqrt[2] - ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]/Sqrt[2]

Rule 3171

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(B*x
)/b, x] + Dist[(A*b - a*B)/b, Int[1/(a + b*Sin[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f, A, B}, x]

Rule 3181

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

Rule 203

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

Rubi steps

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

Mathematica [A]  time = 0.0288295, size = 18, normalized size = 0.5 \[ x-\frac{\tan ^{-1}\left (\sqrt{2} \tan (x)\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^2/(1 + Sin[x]^2),x]

[Out]

x - ArcTan[Sqrt[2]*Tan[x]]/Sqrt[2]

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Maple [A]  time = 0.019, size = 15, normalized size = 0.4 \begin{align*} -{\frac{\sqrt{2}\arctan \left ( \tan \left ( x \right ) \sqrt{2} \right ) }{2}}+x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^2/(1+sin(x)^2),x)

[Out]

-1/2*2^(1/2)*arctan(tan(x)*2^(1/2))+x

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Maxima [A]  time = 1.44985, size = 19, normalized size = 0.53 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\sqrt{2} \tan \left (x\right )\right ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/(1+sin(x)^2),x, algorithm="maxima")

[Out]

-1/2*sqrt(2)*arctan(sqrt(2)*tan(x)) + x

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Fricas [A]  time = 1.21058, size = 105, normalized size = 2.92 \begin{align*} \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{3 \, \sqrt{2} \cos \left (x\right )^{2} - 2 \, \sqrt{2}}{4 \, \cos \left (x\right ) \sin \left (x\right )}\right ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/(1+sin(x)^2),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*arctan(1/4*(3*sqrt(2)*cos(x)^2 - 2*sqrt(2))/(cos(x)*sin(x))) + x

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Sympy [B]  time = 64.5499, size = 581, normalized size = 16.14 \begin{align*} \frac{7 \sqrt{2} x \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} + 10 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}} + \frac{10 x \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} + 10 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}} - \frac{3 \sqrt{2 \sqrt{2} + 3} \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{x}{2} \right )}}{\sqrt{3 - 2 \sqrt{2}}} \right )} + \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} + 10 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}} - \frac{2 \sqrt{2} \sqrt{2 \sqrt{2} + 3} \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{x}{2} \right )}}{\sqrt{3 - 2 \sqrt{2}}} \right )} + \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} + 10 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}} - \frac{17 \sqrt{3 - 2 \sqrt{2}} \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{x}{2} \right )}}{\sqrt{2 \sqrt{2} + 3}} \right )} + \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} + 10 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}} - \frac{12 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{x}{2} \right )}}{\sqrt{2 \sqrt{2} + 3}} \right )} + \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} + 10 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**2/(1+sin(x)**2),x)

[Out]

7*sqrt(2)*x*sqrt(3 - 2*sqrt(2))*sqrt(2*sqrt(2) + 3)/(7*sqrt(2)*sqrt(3 - 2*sqrt(2))*sqrt(2*sqrt(2) + 3) + 10*sq
rt(3 - 2*sqrt(2))*sqrt(2*sqrt(2) + 3)) + 10*x*sqrt(3 - 2*sqrt(2))*sqrt(2*sqrt(2) + 3)/(7*sqrt(2)*sqrt(3 - 2*sq
rt(2))*sqrt(2*sqrt(2) + 3) + 10*sqrt(3 - 2*sqrt(2))*sqrt(2*sqrt(2) + 3)) - 3*sqrt(2*sqrt(2) + 3)*(atan(tan(x/2
)/sqrt(3 - 2*sqrt(2))) + pi*floor((x/2 - pi/2)/pi))/(7*sqrt(2)*sqrt(3 - 2*sqrt(2))*sqrt(2*sqrt(2) + 3) + 10*sq
rt(3 - 2*sqrt(2))*sqrt(2*sqrt(2) + 3)) - 2*sqrt(2)*sqrt(2*sqrt(2) + 3)*(atan(tan(x/2)/sqrt(3 - 2*sqrt(2))) + p
i*floor((x/2 - pi/2)/pi))/(7*sqrt(2)*sqrt(3 - 2*sqrt(2))*sqrt(2*sqrt(2) + 3) + 10*sqrt(3 - 2*sqrt(2))*sqrt(2*s
qrt(2) + 3)) - 17*sqrt(3 - 2*sqrt(2))*(atan(tan(x/2)/sqrt(2*sqrt(2) + 3)) + pi*floor((x/2 - pi/2)/pi))/(7*sqrt
(2)*sqrt(3 - 2*sqrt(2))*sqrt(2*sqrt(2) + 3) + 10*sqrt(3 - 2*sqrt(2))*sqrt(2*sqrt(2) + 3)) - 12*sqrt(2)*sqrt(3
- 2*sqrt(2))*(atan(tan(x/2)/sqrt(2*sqrt(2) + 3)) + pi*floor((x/2 - pi/2)/pi))/(7*sqrt(2)*sqrt(3 - 2*sqrt(2))*s
qrt(2*sqrt(2) + 3) + 10*sqrt(3 - 2*sqrt(2))*sqrt(2*sqrt(2) + 3))

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Giac [A]  time = 1.09124, size = 65, normalized size = 1.81 \begin{align*} -\frac{1}{2} \, \sqrt{2}{\left (x + \arctan \left (-\frac{\sqrt{2} \sin \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )}{\sqrt{2} \cos \left (2 \, x\right ) + \sqrt{2} - 2 \, \cos \left (2 \, x\right ) + 2}\right )\right )} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/(1+sin(x)^2),x, algorithm="giac")

[Out]

-1/2*sqrt(2)*(x + arctan(-(sqrt(2)*sin(2*x) - 2*sin(2*x))/(sqrt(2)*cos(2*x) + sqrt(2) - 2*cos(2*x) + 2))) + x