3.5 \(\int \frac{1}{a+b \sin (x)+c \sin ^2(x)} \, dx\)

Optimal. Leaf size=221 \[ \frac{2 \sqrt{2} c \tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \left (b-\sqrt{b^2-4 a c}\right )+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{2 \sqrt{2} c \tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \left (\sqrt{b^2-4 a c}+b\right )+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}} \]

[Out]

(2*Sqrt[2]*c*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*
a*c]])])/(Sqrt[b^2 - 4*a*c]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) - (2*Sqrt[2]*c*ArcTan[(2*c + (b + S
qrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(Sqrt[b^2 - 4*a*c]*Sqrt[
b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 0.397046, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3248, 2660, 618, 204} \[ \frac{2 \sqrt{2} c \tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \left (b-\sqrt{b^2-4 a c}\right )+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{2 \sqrt{2} c \tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \left (\sqrt{b^2-4 a c}+b\right )+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[x] + c*Sin[x]^2)^(-1),x]

[Out]

(2*Sqrt[2]*c*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*
a*c]])])/(Sqrt[b^2 - 4*a*c]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) - (2*Sqrt[2]*c*ArcTan[(2*c + (b + S
qrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(Sqrt[b^2 - 4*a*c]*Sqrt[
b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])

Rule 3248

Int[((a_.) + (b_.)*sin[(d_.) + (e_.)*(x_)]^(n_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]^(n2_.))^(-1), x_Symbol] :> Mo
dule[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[1/(b - q + 2*c*Sin[d + e*x]^n), x], x] - Dist[(2*c)/q, Int[1/
(b + q + 2*c*Sin[d + e*x]^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

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

Rule 204

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

Rubi steps

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

Mathematica [C]  time = 0.540572, size = 233, normalized size = 1.05 \[ -\frac{2 i c \left (\frac{\tan ^{-1}\left (\frac{2 c+\tan \left (\frac{x}{2}\right ) \left (b-i \sqrt{4 a c-b^2}\right )}{\sqrt{2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right )}{\sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}-\frac{\tan ^{-1}\left (\frac{2 c+\tan \left (\frac{x}{2}\right ) \left (b+i \sqrt{4 a c-b^2}\right )}{\sqrt{2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right )}{\sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right )}{\sqrt{2 a c-\frac{b^2}{2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[x] + c*Sin[x]^2)^(-1),x]

[Out]

((-2*I)*c*(ArcTan[(2*c + (b - I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2
+ 4*a*c]])]/Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]] - ArcTan[(2*c + (b + I*Sqrt[-b^2 + 4*a*c])*Tan[x/
2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])]/Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c
]]))/Sqrt[-b^2/2 + 2*a*c]

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Maple [B]  time = 0.141, size = 610, normalized size = 2.8 \begin{align*} -2\,{\frac{b\sqrt{-4\,ca+{b}^{2}}}{ \left ( 4\,ca-{b}^{2} \right ) \sqrt{4\,ca-2\,{b}^{2}+2\,b\sqrt{-4\,ca+{b}^{2}}+4\,{a}^{2}}}\arctan \left ({\frac{-2\,a\tan \left ( x/2 \right ) +\sqrt{-4\,ca+{b}^{2}}-b}{\sqrt{4\,ca-2\,{b}^{2}+2\,b\sqrt{-4\,ca+{b}^{2}}+4\,{a}^{2}}}} \right ) }-8\,{\frac{ca}{ \left ( 4\,ca-{b}^{2} \right ) \sqrt{4\,ca-2\,{b}^{2}+2\,b\sqrt{-4\,ca+{b}^{2}}+4\,{a}^{2}}}\arctan \left ({\frac{-2\,a\tan \left ( x/2 \right ) +\sqrt{-4\,ca+{b}^{2}}-b}{\sqrt{4\,ca-2\,{b}^{2}+2\,b\sqrt{-4\,ca+{b}^{2}}+4\,{a}^{2}}}} \right ) }+2\,{\frac{{b}^{2}}{ \left ( 4\,ca-{b}^{2} \right ) \sqrt{4\,ca-2\,{b}^{2}+2\,b\sqrt{-4\,ca+{b}^{2}}+4\,{a}^{2}}}\arctan \left ({\frac{-2\,a\tan \left ( x/2 \right ) +\sqrt{-4\,ca+{b}^{2}}-b}{\sqrt{4\,ca-2\,{b}^{2}+2\,b\sqrt{-4\,ca+{b}^{2}}+4\,{a}^{2}}}} \right ) }-2\,{\frac{b\sqrt{-4\,ca+{b}^{2}}}{ \left ( 4\,ca-{b}^{2} \right ) \sqrt{4\,ca-2\,{b}^{2}-2\,b\sqrt{-4\,ca+{b}^{2}}+4\,{a}^{2}}}\arctan \left ({\frac{2\,a\tan \left ( x/2 \right ) +b+\sqrt{-4\,ca+{b}^{2}}}{\sqrt{4\,ca-2\,{b}^{2}-2\,b\sqrt{-4\,ca+{b}^{2}}+4\,{a}^{2}}}} \right ) }+8\,{\frac{ca}{ \left ( 4\,ca-{b}^{2} \right ) \sqrt{4\,ca-2\,{b}^{2}-2\,b\sqrt{-4\,ca+{b}^{2}}+4\,{a}^{2}}}\arctan \left ({\frac{2\,a\tan \left ( x/2 \right ) +b+\sqrt{-4\,ca+{b}^{2}}}{\sqrt{4\,ca-2\,{b}^{2}-2\,b\sqrt{-4\,ca+{b}^{2}}+4\,{a}^{2}}}} \right ) }-2\,{\frac{{b}^{2}}{ \left ( 4\,ca-{b}^{2} \right ) \sqrt{4\,ca-2\,{b}^{2}-2\,b\sqrt{-4\,ca+{b}^{2}}+4\,{a}^{2}}}\arctan \left ({\frac{2\,a\tan \left ( x/2 \right ) +b+\sqrt{-4\,ca+{b}^{2}}}{\sqrt{4\,ca-2\,{b}^{2}-2\,b\sqrt{-4\,ca+{b}^{2}}+4\,{a}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sin(x)+c*sin(x)^2),x)

[Out]

-2/(4*a*c-b^2)/(4*c*a-2*b^2+2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2)*arctan((-2*a*tan(1/2*x)+(-4*a*c+b^2)^(1/2)-b)/
(4*c*a-2*b^2+2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2))*b*(-4*a*c+b^2)^(1/2)-8*a/(4*a*c-b^2)/(4*c*a-2*b^2+2*b*(-4*a*
c+b^2)^(1/2)+4*a^2)^(1/2)*arctan((-2*a*tan(1/2*x)+(-4*a*c+b^2)^(1/2)-b)/(4*c*a-2*b^2+2*b*(-4*a*c+b^2)^(1/2)+4*
a^2)^(1/2))*c+2/(4*a*c-b^2)/(4*c*a-2*b^2+2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2)*arctan((-2*a*tan(1/2*x)+(-4*a*c+b
^2)^(1/2)-b)/(4*c*a-2*b^2+2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2))*b^2-2/(4*a*c-b^2)/(4*c*a-2*b^2-2*b*(-4*a*c+b^2)
^(1/2)+4*a^2)^(1/2)*arctan((2*a*tan(1/2*x)+b+(-4*a*c+b^2)^(1/2))/(4*c*a-2*b^2-2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1
/2))*b*(-4*a*c+b^2)^(1/2)+8*a/(4*a*c-b^2)/(4*c*a-2*b^2-2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2)*arctan((2*a*tan(1/2
*x)+b+(-4*a*c+b^2)^(1/2))/(4*c*a-2*b^2-2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2))*c-2/(4*a*c-b^2)/(4*c*a-2*b^2-2*b*(
-4*a*c+b^2)^(1/2)+4*a^2)^(1/2)*arctan((2*a*tan(1/2*x)+b+(-4*a*c+b^2)^(1/2))/(4*c*a-2*b^2-2*b*(-4*a*c+b^2)^(1/2
)+4*a^2)^(1/2))*b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{c \sin \left (x\right )^{2} + b \sin \left (x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(x)+c*sin(x)^2),x, algorithm="maxima")

[Out]

integrate(1/(c*sin(x)^2 + b*sin(x) + a), x)

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Fricas [B]  time = 4.95249, size = 7071, normalized size = 32. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(x)+c*sin(x)^2),x, algorithm="fricas")

[Out]

-1/4*sqrt(2)*sqrt(-(b^2 - 2*a*c - 2*c^2 + (a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c
)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*
a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^
3 - 3*a*b^2)*c))*log(2*b^2*c*sin(x) + 4*b*c^2 + 2*(4*a*c^4 + (8*a^2 - b^2)*c^3 + 2*(2*a^3 - 3*a*b^2)*c^2 - (a^
2*b^2 - b^4)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 -
2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c))*sin(x) - sqrt(2)*((a^2*b^4 - b^6 + 8*a*c^
5 + 2*(12*a^2 - b^2)*c^4 + 6*(4*a^3 - 3*a*b^2)*c^3 + (8*a^4 - 22*a^2*b^2 + 3*b^4)*c^2 - 2*(3*a^3*b^2 - 4*a*b^4
)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 -
11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c))*cos(x) - (b^4 - 4*a*b^2*c)*cos(x))*sqrt(-(b^2 - 2*a*
c - 2*c^2 + (a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^
4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 -
3*a^3*b^2 + 2*a*b^4)*c)))/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c))) + 1/4*sqrt(2
)*sqrt(-(b^2 - 2*a*c - 2*c^2 - (a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/
(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b
^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2
)*c))*log(2*b^2*c*sin(x) + 4*b*c^2 - 2*(4*a*c^4 + (8*a^2 - b^2)*c^3 + 2*(2*a^3 - 3*a*b^2)*c^2 - (a^2*b^2 - b^4
)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 -
11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c))*sin(x) - sqrt(2)*((a^2*b^4 - b^6 + 8*a*c^5 + 2*(12*a
^2 - b^2)*c^4 + 6*(4*a^3 - 3*a*b^2)*c^3 + (8*a^4 - 22*a^2*b^2 + 3*b^4)*c^2 - 2*(3*a^3*b^2 - 4*a*b^4)*c)*sqrt(b
^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2
+ b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c))*cos(x) + (b^4 - 4*a*b^2*c)*cos(x))*sqrt(-(b^2 - 2*a*c - 2*c^2 -
(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4
*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 +
2*a*b^4)*c)))/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c))) - 1/4*sqrt(2)*sqrt(-(b^
2 - 2*a*c - 2*c^2 - (a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2 -
2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4
*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c))*log(-
2*b^2*c*sin(x) - 4*b*c^2 + 2*(4*a*c^4 + (8*a^2 - b^2)*c^3 + 2*(2*a^3 - 3*a*b^2)*c^2 - (a^2*b^2 - b^4)*c)*sqrt(
b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2
+ b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c))*sin(x) - sqrt(2)*((a^2*b^4 - b^6 + 8*a*c^5 + 2*(12*a^2 - b^2)*
c^4 + 6*(4*a^3 - 3*a*b^2)*c^3 + (8*a^4 - 22*a^2*b^2 + 3*b^4)*c^2 - 2*(3*a^3*b^2 - 4*a*b^4)*c)*sqrt(b^2/(a^4*b^
2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2
- 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c))*cos(x) + (b^4 - 4*a*b^2*c)*cos(x))*sqrt(-(b^2 - 2*a*c - 2*c^2 - (a^2*b^2
- b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (
16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*
c)))/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c))) + 1/4*sqrt(2)*sqrt(-(b^2 - 2*a*c
- 2*c^2 + (a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4
+ b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*
a^3*b^2 + 2*a*b^4)*c)))/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c))*log(-2*b^2*c*si
n(x) - 4*b*c^2 - 2*(4*a*c^4 + (8*a^2 - b^2)*c^3 + 2*(2*a^3 - 3*a*b^2)*c^2 - (a^2*b^2 - b^4)*c)*sqrt(b^2/(a^4*b
^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^
2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c))*sin(x) - sqrt(2)*((a^2*b^4 - b^6 + 8*a*c^5 + 2*(12*a^2 - b^2)*c^4 + 6*(4
*a^3 - 3*a*b^2)*c^3 + (8*a^4 - 22*a^2*b^2 + 3*b^4)*c^2 - 2*(3*a^3*b^2 - 4*a*b^4)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*
b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5
- 3*a^3*b^2 + 2*a*b^4)*c))*cos(x) - (b^4 - 4*a*b^2*c)*cos(x))*sqrt(-(b^2 - 2*a*c - 2*c^2 + (a^2*b^2 - b^4 - 4*
a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b
^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))/(a^2*
b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(x)+c*sin(x)**2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{c \sin \left (x\right )^{2} + b \sin \left (x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(x)+c*sin(x)^2),x, algorithm="giac")

[Out]

integrate(1/(c*sin(x)^2 + b*sin(x) + a), x)