3.207 \(\int \frac{a+b \sin ^2(x)}{c+c \cos ^2(x)} \, dx\)

Optimal. Leaf size=57 \[ \frac{x (a+2 b)}{\sqrt{2} c}-\frac{(a+2 b) \tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right )}{\sqrt{2} c}-\frac{b x}{c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.134189, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {12, 1166, 203} \[ \frac{x (a+2 b)}{\sqrt{2} c}-\frac{(a+2 b) \tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right )}{\sqrt{2} c}-\frac{b x}{c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[x]^2)/(c + c*Cos[x]^2),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1166

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 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{a+b \sin ^2(x)}{c+c \cos ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{a+(a+b) x^2}{c \left (2+3 x^2+x^4\right )} \, dx,x,\tan (x)\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{a+(a+b) x^2}{2+3 x^2+x^4} \, dx,x,\tan (x)\right )}{c}\\ &=-\frac{b \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (x)\right )}{c}+\frac{(a+2 b) \operatorname{Subst}\left (\int \frac{1}{2+x^2} \, dx,x,\tan (x)\right )}{c}\\ &=-\frac{b x}{c}+\frac{(a+2 b) x}{\sqrt{2} c}-\frac{(a+2 b) \tan ^{-1}\left (\frac{\cos (x) \sin (x)}{1+\sqrt{2}+\cos ^2(x)}\right )}{\sqrt{2} c}\\ \end{align*}

Mathematica [A]  time = 0.0885643, size = 34, normalized size = 0.6 \[ -\frac{(-a-2 b) \tan ^{-1}\left (\frac{\tan (x)}{\sqrt{2}}\right )}{\sqrt{2} c}-\frac{b x}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[x]^2)/(c + c*Cos[x]^2),x]

[Out]

-((b*x)/c) - ((-a - 2*b)*ArcTan[Tan[x]/Sqrt[2]])/(Sqrt[2]*c)

________________________________________________________________________________________

Maple [A]  time = 0.032, size = 44, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}a}{2\,c}\arctan \left ({\frac{\tan \left ( x \right ) \sqrt{2}}{2}} \right ) }+{\frac{\sqrt{2}b}{c}\arctan \left ({\frac{\tan \left ( x \right ) \sqrt{2}}{2}} \right ) }-{\frac{b\arctan \left ( \tan \left ( x \right ) \right ) }{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(x)^2)/(c+c*cos(x)^2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.49859, size = 39, normalized size = 0.68 \begin{align*} \frac{\sqrt{2}{\left (a + 2 \, b\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} \tan \left (x\right )\right )}{2 \, c} - \frac{b x}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.56949, size = 128, normalized size = 2.25 \begin{align*} -\frac{\sqrt{2}{\left (a + 2 \, b\right )} \arctan \left (\frac{3 \, \sqrt{2} \cos \left (x\right )^{2} - \sqrt{2}}{4 \, \cos \left (x\right ) \sin \left (x\right )}\right ) + 4 \, b x}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 27.9418, size = 143, normalized size = 2.51 \begin{align*} \frac{\sqrt{2} a \left (\operatorname{atan}{\left (\sqrt{2} \tan{\left (\frac{x}{2} \right )} - 1 \right )} + \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{2 c} + \frac{\sqrt{2} a \left (\operatorname{atan}{\left (\sqrt{2} \tan{\left (\frac{x}{2} \right )} + 1 \right )} + \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{2 c} - \frac{b x}{c} + \frac{\sqrt{2} b \left (\operatorname{atan}{\left (\sqrt{2} \tan{\left (\frac{x}{2} \right )} - 1 \right )} + \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{c} + \frac{\sqrt{2} b \left (\operatorname{atan}{\left (\sqrt{2} \tan{\left (\frac{x}{2} \right )} + 1 \right )} + \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(x)**2)/(c+c*cos(x)**2),x)

[Out]

sqrt(2)*a*(atan(sqrt(2)*tan(x/2) - 1) + pi*floor((x/2 - pi/2)/pi))/(2*c) + sqrt(2)*a*(atan(sqrt(2)*tan(x/2) +
1) + pi*floor((x/2 - pi/2)/pi))/(2*c) - b*x/c + sqrt(2)*b*(atan(sqrt(2)*tan(x/2) - 1) + pi*floor((x/2 - pi/2)/
pi))/c + sqrt(2)*b*(atan(sqrt(2)*tan(x/2) + 1) + pi*floor((x/2 - pi/2)/pi))/c

________________________________________________________________________________________

Giac [A]  time = 1.09982, size = 84, normalized size = 1.47 \begin{align*} \frac{\sqrt{2}{\left (a + 2 \, b\right )}{\left (x + \arctan \left (-\frac{\sqrt{2} \sin \left (2 \, x\right ) - \sin \left (2 \, x\right )}{\sqrt{2} \cos \left (2 \, x\right ) + \sqrt{2} - \cos \left (2 \, x\right ) + 1}\right )\right )}}{2 \, c} - \frac{b x}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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