3.1.3 \(\int \frac {\sin ^2(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx\) [3]

Optimal. Leaf size=253 \[ \frac {x}{c}-\frac {\sqrt {2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)-b \sqrt {b^2-4 a c}}}\right )}{c \sqrt {b^2-2 c (a+c)-b \sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)+b \sqrt {b^2-4 a c}}}\right )}{c \sqrt {b^2-2 c (a+c)+b \sqrt {b^2-4 a c}}} \]

[Out]

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

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Rubi [A]
time = 0.70, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3337, 3373, 2739, 632, 210} \begin {gather*} -\frac {\sqrt {2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {ArcTan}\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 )}{c \sqrt {-b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}-\frac {\sqrt {2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \text {ArcTan}\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 )}{c \sqrt {b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}+\frac {x}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 632

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 2739

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 3337

Int[sin[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*sin[(d_.) + (e_.)*(x_)]^(n_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]
^(n2_.))^(p_), x_Symbol] :> Int[ExpandTrig[sin[d + e*x]^m*(a + b*sin[d + e*x]^n + c*sin[d + e*x]^(2*n))^p, x],
x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[m, n, p]

Rule 3373

Int[((A_) + (B_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + (b_.)*sin[(d_.) + (e_.)*(x_)] + (c_.)*sin[(d_.) + (e_.)*(x
_)]^2), x_Symbol] :> Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[B + (b*B - 2*A*c)/q, Int[1/(b + q + 2*c*Sin[d + e*x
]), x], x] + Dist[B - (b*B - 2*A*c)/q, Int[1/(b - q + 2*c*Sin[d + e*x]), x], x]] /; FreeQ[{a, b, c, d, e, A, B
}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.36, size = 310, normalized size = 1.23 \begin {gather*} \frac {x-\frac {\left (i b^2-2 i a c+b \sqrt {-b^2+4 a c}\right ) \tan ^{-1}\left (\frac {2 c+\left (b-i \sqrt {-b^2+4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)-i b \sqrt {-b^2+4 a c}}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {b^2-2 c (a+c)-i b \sqrt {-b^2+4 a c}}}-\frac {\left (-i b^2+2 i a c+b \sqrt {-b^2+4 a c}\right ) \tan ^{-1}\left (\frac {2 c+\left (b+i \sqrt {-b^2+4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)+i b \sqrt {-b^2+4 a c}}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {b^2-2 c (a+c)+i b \sqrt {-b^2+4 a c}}}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x - ((I*b^2 - (2*I)*a*c + b*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b - I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*S
qrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-
b^2 + 4*a*c]]) - (((-I)*b^2 + (2*I)*a*c + 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[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a +
c) + I*b*Sqrt[-b^2 + 4*a*c]]))/c

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Maple [A]
time = 1.12, size = 252, normalized size = 1.00

method	result	size
default	\(\frac {2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{c}+\frac {2 a \left (-\frac {2 \left (b \sqrt {-4 a c +b^{2}}-4 a c +b^{2}\right ) \arctan \left (\frac {-2 a \tan \left (\frac {x}{2}\right )+\sqrt {-4 a c +b^{2}}-b}{\sqrt {4 a c -2 b^{2}+2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}\right )}{\left (8 a c -2 b^{2}\right ) \sqrt {4 a c -2 b^{2}+2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}+\frac {2 \left (-b \sqrt {-4 a c +b^{2}}-4 a c +b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+b +\sqrt {-4 a c +b^{2}}}{\sqrt {4 a c -2 b^{2}-2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}\right )}{\left (8 a c -2 b^{2}\right ) \sqrt {4 a c -2 b^{2}-2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}\right )}{c}\)	\(252\)
risch	\(\frac {x}{c}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (16 a^{4} c^{4}-8 a^{3} b^{2} c^{3}+32 a^{3} c^{5}+a^{2} b^{4} c^{2}-32 a^{2} b^{2} c^{4}+16 a^{2} c^{6}+10 a \,b^{4} c^{3}-8 a \,b^{2} c^{5}-b^{6} c^{2}+b^{4} c^{4}\right ) \textit {\_Z}^{4}+\left (128 a^{4} c^{2}-96 a^{3} b^{2} c +128 a^{3} c^{3}+16 a^{2} b^{4}-288 a^{2} b^{2} c^{2}+128 a \,b^{4} c -16 b^{6}\right ) \textit {\_Z}^{2}+256 a^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+\left (-\frac {2 c^{2} a^{4} b^{2}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}-\frac {18 c^{3} a^{3} b^{2}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {3 c^{2} a^{2} b^{4}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}-\frac {22 c^{4} a^{2} b^{2}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {8 c^{3} a \,b^{4}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}-\frac {6 c^{5} b^{2} a}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {8 c^{3} a^{5}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {24 c^{4} a^{4}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {24 c^{5} a^{3}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {8 c^{6} a^{2}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}-\frac {c^{2} b^{6}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {c^{4} b^{4}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}\right ) \textit {\_R}^{3}+\left (\frac {16 i c^{2} a^{5}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {32 i c^{3} a^{4}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {16 i c^{4} a^{3}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}-\frac {4 i c \,a^{4} b^{2}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}-\frac {24 i c^{2} a^{3} b^{2}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {4 i c \,a^{2} b^{4}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}-\frac {4 i c^{3} b^{2} a^{2}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}\right ) \textit {\_R}^{2}+\left (-\frac {128 c \,a^{3} b^{2}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}-\frac {144 c^{2} b^{2} a^{2}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {96 c \,b^{4} a}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}-\frac {16 b^{6}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {32 c \,a^{5}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}-\frac {16 a^{4} b^{2}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {64 c^{2} a^{4}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {32 c^{3} a^{3}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {32 a^{2} b^{4}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}\right ) \textit {\_R} +\frac {64 i a^{5}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}+\frac {64 i c \,a^{4}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}-\frac {64 i b^{2} a^{3}}{64 a^{4} b +128 c \,a^{3} b -64 b^{3} a^{2}}\right )\right )}{4}\)	\(1212\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^2/(a+b*sin(x)+c*sin(x)^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c*integrate(2*(2*b^2*cos(3*x)^2 + 2*b^2*cos(x)^2 + 2*b^2*sin(3*x)^2 + 2*b^2*sin(x)^2 + 4*(2*a^2 + a*c)*cos(2
*x)^2 + 2*(4*a*b + b*c)*cos(x)*sin(2*x) + 4*(2*a^2 + a*c)*sin(2*x)^2 + b*c*sin(x) - (2*a*c*cos(2*x) + b*c*sin(
3*x) - b*c*sin(x))*cos(4*x) - 2*(2*b^2*cos(x) + (4*a*b + b*c)*sin(2*x))*cos(3*x) - 2*(a*c + (4*a*b + b*c)*sin(
x))*cos(2*x) + (b*c*cos(3*x) - b*c*cos(x) - 2*a*c*sin(2*x))*sin(4*x) - (4*b^2*sin(x) + b*c - 2*(4*a*b + b*c)*c
os(2*x))*sin(3*x))/(c^3*cos(4*x)^2 + 4*b^2*c*cos(3*x)^2 + 4*b^2*c*cos(x)^2 + c^3*sin(4*x)^2 + 4*b^2*c*sin(3*x)
^2 + 4*b^2*c*sin(x)^2 + 4*b*c^2*sin(x) + c^3 + 4*(4*a^2*c + 4*a*c^2 + c^3)*cos(2*x)^2 + 8*(2*a*b*c + b*c^2)*co
s(x)*sin(2*x) + 4*(4*a^2*c + 4*a*c^2 + c^3)*sin(2*x)^2 - 2*(2*b*c^2*sin(3*x) - 2*b*c^2*sin(x) - c^3 + 2*(2*a*c
^2 + c^3)*cos(2*x))*cos(4*x) - 8*(b^2*c*cos(x) + (2*a*b*c + b*c^2)*sin(2*x))*cos(3*x) - 4*(2*a*c^2 + c^3 + 2*(
2*a*b*c + b*c^2)*sin(x))*cos(2*x) + 4*(b*c^2*cos(3*x) - b*c^2*cos(x) - (2*a*c^2 + c^3)*sin(2*x))*sin(4*x) - 4*
(2*b^2*c*sin(x) + b*c^2 - 2*(2*a*b*c + b*c^2)*cos(2*x))*sin(3*x)), x) - x)/c

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4985 vs. \(2 (219) = 438\).
time = 1.13, size = 4985, normalized size = 19.70 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(sqrt(2)*c*sqrt((a^2*b^2 - b^4 - 2*a^2*c^2 - 2*(a^3 - 2*a*b^2)*c + (4*a*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3
- 3*a*b^2)*c^3 - (a^2*b^2 - b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*
c)/(4*a*c^9 + (16*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*
b^2 + 2*a*b^4)*c^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4)))/(4*a*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3
- (a^2*b^2 - b^4)*c^2))*log(8*a^3*b*c^2 + 2*(4*a^3*c^5 + (8*a^4 - a^2*b^2)*c^4 + 2*(2*a^5 - 3*a^3*b^2)*c^3 -
(a^4*b^2 - a^2*b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 +
(16*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)
*c^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4))*sin(x) + 4*(a^4*b - a^2*b^3)*c - sqrt(2)*((8*a^2*c^7 + 6*(4*a^3 - a*b
^2)*c^6 + (24*a^4 - 22*a^2*b^2 + b^4)*c^5 + 2*(4*a^5 - 9*a^3*b^2 + 4*a*b^4)*c^4 - (2*a^4*b^2 - 3*a^2*b^4 + b^6
)*c^3)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a^2 - b^2)*c^8
+ 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 -
2*a^2*b^4 + b^6)*c^4))*cos(x) - (8*a^2*b^2*c^3 + 2*(2*a^3*b^2 - 3*a*b^4)*c^2 - (a^2*b^4 - b^6)*c)*cos(x))*sqrt
((a^2*b^2 - b^4 - 2*a^2*c^2 - 2*(a^3 - 2*a*b^2)*c + (4*a*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3 - (
a^2*b^2 - b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a
^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5
- (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4)))/(4*a*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3 - (a^2*b^2 - b^4)*
c^2)) + 2*(a^4*b^2 - a^2*b^4 + 2*a^3*b^2*c)*sin(x)) - sqrt(2)*c*sqrt((a^2*b^2 - b^4 - 2*a^2*c^2 - 2*(a^3 - 2*a
*b^2)*c - (4*a*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3 - (a^2*b^2 - b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2
*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2
*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4)))/(4*a*
c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3 - (a^2*b^2 - b^4)*c^2))*log(8*a^3*b*c^2 - 2*(4*a^3*c^5 + (8*
a^4 - a^2*b^2)*c^4 + 2*(2*a^5 - 3*a^3*b^2)*c^3 - (a^4*b^2 - a^2*b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4
*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a
^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4))*sin(x) + 4*(a^4*b -
a^2*b^3)*c - sqrt(2)*((8*a^2*c^7 + 6*(4*a^3 - a*b^2)*c^6 + (24*a^4 - 22*a^2*b^2 + b^4)*c^5 + 2*(4*a^5 - 9*a^3*
b^2 + 4*a*b^4)*c^4 - (2*a^4*b^2 - 3*a^2*b^4 + b^6)*c^3)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(
a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6
+ 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4))*cos(x) + (8*a^2*b^2*c^3 + 2*(2*a^3*b^2
- 3*a*b^4)*c^2 - (a^2*b^4 - b^6)*c)*cos(x))*sqrt((a^2*b^2 - b^4 - 2*a^2*c^2 - 2*(a^3 - 2*a*b^2)*c - (4*a*c^5
+ (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3 - (a^2*b^2 - b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*
b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^
2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4)))/(4*a*c^5 + (8*a^2 - b^2)
*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3 - (a^2*b^2 - b^4)*c^2)) + 2*(a^4*b^2 - a^2*b^4 + 2*a^3*b^2*c)*sin(x)) + sqrt(2)
*c*sqrt((a^2*b^2 - b^4 - 2*a^2*c^2 - 2*(a^3 - 2*a*b^2)*c - (4*a*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*
c^3 - (a^2*b^2 - b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9
+ (16*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^
4)*c^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4)))/(4*a*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3 - (a^2*b^2
- b^4)*c^2))*log(-8*a^3*b*c^2 + 2*(4*a^3*c^5 + (8*a^4 - a^2*b^2)*c^4 + 2*(2*a^5 - 3*a^3*b^2)*c^3 - (a^4*b^2 -
a^2*b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a^2 - b
^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4
*b^2 - 2*a^2*b^4 + b^6)*c^4))*sin(x) - 4*(a^4*b - a^2*b^3)*c - sqrt(2)*((8*a^2*c^7 + 6*(4*a^3 - a*b^2)*c^6 + (
24*a^4 - 22*a^2*b^2 + b^4)*c^5 + 2*(4*a^5 - 9*a^3*b^2 + 4*a*b^4)*c^4 - (2*a^4*b^2 - 3*a^2*b^4 + b^6)*c^3)*sqrt
(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a^2 - b^2)*c^8 + 12*(2*a^
3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 - 2*a^2*b^4 +
b^6)*c^4))*cos(x) + (8*a^2*b^2*c^3 + 2*(2*a^3*b^2 - 3*a*b^4)*c^2 - (a^2*b^4 - b^6)*c)*cos(x))*sqrt((a^2*b^2 -
b^4 - 2*a^2*c^2 - 2*(a^3 - 2*a*b^2)*c - (4*a*c...

________________________________________________________________________________________

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

[Out]

Timed out

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

[Out]

Timed out

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Mupad [B]
time = 27.75, size = 2500, normalized size = 9.88 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*atan((147456*a^5*tan(x/2))/(16384*a*b^4 + 393216*a^4*c + 147456*a^5 - 229376*a^3*b^2 + 262144*a^3*c^2 - 131
072*a^2*b^2*c + (32768*a^2*b^4)/c - (32768*a^4*b^2)/c) + (393216*a^4*tan(x/2))/(262144*a^3*c + 393216*a^4 - 13
1072*a^2*b^2 + (147456*a^5)/c + (16384*a*b^4)/c - (229376*a^3*b^2)/c + (32768*a^2*b^4)/c^2 - (32768*a^4*b^2)/c
^2) + (16384*a*b^4*tan(x/2))/(16384*a*b^4 + 393216*a^4*c + 147456*a^5 - 229376*a^3*b^2 + 262144*a^3*c^2 - 1310
72*a^2*b^2*c + (32768*a^2*b^4)/c - (32768*a^4*b^2)/c) + (262144*a^3*c*tan(x/2))/(262144*a^3*c + 393216*a^4 - 1
31072*a^2*b^2 + (147456*a^5)/c + (16384*a*b^4)/c - (229376*a^3*b^2)/c + (32768*a^2*b^4)/c^2 - (32768*a^4*b^2)/
c^2) - (229376*a^3*b^2*tan(x/2))/(16384*a*b^4 + 393216*a^4*c + 147456*a^5 - 229376*a^3*b^2 + 262144*a^3*c^2 -
131072*a^2*b^2*c + (32768*a^2*b^4)/c - (32768*a^4*b^2)/c) - (131072*a^2*b^2*tan(x/2))/(262144*a^3*c + 393216*a
^4 - 131072*a^2*b^2 + (147456*a^5)/c + (16384*a*b^4)/c - (229376*a^3*b^2)/c + (32768*a^2*b^4)/c^2 - (32768*a^4
*b^2)/c^2) + (32768*a^2*b^4*tan(x/2))/(147456*a^5*c + 32768*a^2*b^4 - 32768*a^4*b^2 + 262144*a^3*c^3 + 393216*
a^4*c^2 - 229376*a^3*b^2*c - 131072*a^2*b^2*c^2 + 16384*a*b^4*c) - (32768*a^4*b^2*tan(x/2))/(147456*a^5*c + 32
768*a^2*b^4 - 32768*a^4*b^2 + 262144*a^3*c^3 + 393216*a^4*c^2 - 229376*a^3*b^2*c - 131072*a^2*b^2*c^2 + 16384*
a*b^4*c)))/c - atan((((b^6 - a^2*b^4 - 8*a^3*c^3 - 8*a^4*c^2 - b^3*(-(4*a*c - b^2)^3)^(1/2) + a^2*b*(-(4*a*c -
b^2)^3)^(1/2) + 6*a^3*b^2*c + 18*a^2*b^2*c^2 - 8*a*b^4*c + 2*a*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^6 +
32*a^3*c^5 + 16*a^4*c^4 + b^4*c^4 - b^6*c^2 - 8*a*b^2*c^5 + 10*a*b^4*c^3 - 32*a^2*b^2*c^4 + a^2*b^4*c^2 - 8*a
^3*b^2*c^3)))^(1/2)*(tan(x/2)*(65536*a*b^4 + 131072*a^4*c + 24576*a^5 - 65536*a^3*b^2 + 131072*a^3*c^2 - 26214
4*a^2*b^2*c) + ((b^6 - a^2*b^4 - 8*a^3*c^3 - 8*a^4*c^2 - b^3*(-(4*a*c - b^2)^3)^(1/2) + a^2*b*(-(4*a*c - b^2)^
3)^(1/2) + 6*a^3*b^2*c + 18*a^2*b^2*c^2 - 8*a*b^4*c + 2*a*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^6 + 32*a^
3*c^5 + 16*a^4*c^4 + b^4*c^4 - b^6*c^2 - 8*a*b^2*c^5 + 10*a*b^4*c^3 - 32*a^2*b^2*c^4 + a^2*b^4*c^2 - 8*a^3*b^2
*c^3)))^(1/2)*(tan(x/2)*(32768*a*b^5 - 32768*a^3*b^3 - 65536*a*b^3*c^2 + 262144*a^2*b*c^3 - 196608*a^2*b^3*c +
196608*a^3*b*c^2 + 131072*a^4*b*c) + 24576*a^5*c + 8192*a^2*b^4 - 8192*a^4*b^2 - 131072*a^3*c^3 - 131072*a^4*
c^2 + ((b^6 - a^2*b^4 - 8*a^3*c^3 - 8*a^4*c^2 - b^3*(-(4*a*c - b^2)^3)^(1/2) + a^2*b*(-(4*a*c - b^2)^3)^(1/2)
+ 6*a^3*b^2*c + 18*a^2*b^2*c^2 - 8*a*b^4*c + 2*a*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^6 + 32*a^3*c^5 + 1
6*a^4*c^4 + b^4*c^4 - b^6*c^2 - 8*a*b^2*c^5 + 10*a*b^4*c^3 - 32*a^2*b^2*c^4 + a^2*b^4*c^2 - 8*a^3*b^2*c^3)))^(
1/2)*(tan(x/2)*(16384*a^3*b^4 - 16384*a*b^6 + 524288*a^2*c^5 + 1179648*a^3*c^4 + 786432*a^4*c^3 + 147456*a^5*c
^2 - 131072*a*b^2*c^4 + 196608*a*b^4*c^2 + 131072*a^2*b^4*c - 98304*a^4*b^2*c - 1048576*a^2*b^2*c^3 - 491520*a
^3*b^2*c^2) + ((b^6 - a^2*b^4 - 8*a^3*c^3 - 8*a^4*c^2 - b^3*(-(4*a*c - b^2)^3)^(1/2) + a^2*b*(-(4*a*c - b^2)^3
)^(1/2) + 6*a^3*b^2*c + 18*a^2*b^2*c^2 - 8*a*b^4*c + 2*a*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^6 + 32*a^3
*c^5 + 16*a^4*c^4 + b^4*c^4 - b^6*c^2 - 8*a*b^2*c^5 + 10*a*b^4*c^3 - 32*a^2*b^2*c^4 + a^2*b^4*c^2 - 8*a^3*b^2*
c^3)))^(1/2)*(((b^6 - a^2*b^4 - 8*a^3*c^3 - 8*a^4*c^2 - b^3*(-(4*a*c - b^2)^3)^(1/2) + a^2*b*(-(4*a*c - b^2)^3
)^(1/2) + 6*a^3*b^2*c + 18*a^2*b^2*c^2 - 8*a*b^4*c + 2*a*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^6 + 32*a^3
*c^5 + 16*a^4*c^4 + b^4*c^4 - b^6*c^2 - 8*a*b^2*c^5 + 10*a*b^4*c^3 - 32*a^2*b^2*c^4 + a^2*b^4*c^2 - 8*a^3*b^2*
c^3)))^(1/2)*(tan(x/2)*(524288*a^2*c^7 + 1179648*a^3*c^6 + 851968*a^4*c^5 + 196608*a^5*c^4 - 131072*a*b^2*c^6
+ 139264*a*b^4*c^4 - 16384*a*b^6*c^2 - 851968*a^2*b^2*c^5 + 147456*a^2*b^4*c^3 - 540672*a^3*b^2*c^4 + 16384*a^
3*b^4*c^2 - 114688*a^4*b^2*c^3) - 32768*a*b^3*c^5 + 24576*a*b^5*c^3 + 131072*a^2*b*c^6 + 163840*a^3*b*c^5 + 98
304*a^4*b*c^4 - 139264*a^2*b^3*c^4 - 24576*a^3*b^3*c^3) + tan(x/2)*(32768*a*b^5*c^2 - 65536*a*b^3*c^4 + 262144
*a^2*b*c^5 + 262144*a^3*b*c^4 + 131072*a^4*b*c^3 - 196608*a^2*b^3*c^3 - 32768*a^3*b^3*c^2) + 98304*a^4*c^4 + 9
8304*a^5*c^3 - 24576*a*b^4*c^3 + 98304*a^2*b^2*c^4 + 24576*a^2*b^4*c^2 - 122880*a^3*b^2*c^3 - 24576*a^4*b^2*c^
2) - 32768*a*b^3*c^3 + 131072*a^2*b*c^4 + 65536*a^3*b*c^3 - 24576*a^3*b^3*c + 73728*a^4*b*c^2 - 106496*a^2*b^3
*c^2 + 24576*a*b^5*c) - 8192*a^3*b^2*c + 163840*a^2*b^2*c^2 - 32768*a*b^4*c) - 24576*a^4*b + 32768*a^2*b^3 - 9
8304*a^3*b*c)*1i + ((b^6 - a^2*b^4 - 8*a^3*c^3 - 8*a^4*c^2 - b^3*(-(4*a*c - b^2)^3)^(1/2) + a^2*b*(-(4*a*c - b
^2)^3)^(1/2) + 6*a^3*b^2*c + 18*a^2*b^2*c^2 - 8*a*b^4*c + 2*a*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^6 + 3
2*a^3*c^5 + 16*a^4*c^4 + b^4*c^4 - b^6*c^2 - 8*a*b^2*c^5 + 10*a*b^4*c^3 - 32*a^2*b^2*c^4 + a^2*b^4*c^2 - 8*a^3
*b^2*c^3)))^(1/2)*(tan(x/2)*(65536*a*b^4 + 131072*a^4*c + 24576*a^5 - 65536*a^3*b^2 + 131072*a^3*c^2 - 262144*
a^2*b^2*c) + ((b^6 - a^2*b^4 - 8*a^3*c^3 - 8*a^4*c^2 - b^3*(-(4*a*c - b^2)^3)^(1/2) + a^2*b*(-(4*a*c - b^2)^3)
^(1/2) + 6*a^3*b^2*c + 18*a^2*b^2*c^2 - 8*a*b^4...

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