3.1.1 \(\int \frac {\sin ^4(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx\) [1]

Optimal. Leaf size=323 \[ \frac {x}{2 c}+\frac {\left (b^2-a c\right ) x}{c^3}-\frac {\sqrt {2} \left (b^3-2 a b c-\frac {b^4-4 a b^2 c+2 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)-b \sqrt {b^2-4 a c}}}\right )}{c^3 \sqrt {b^2-2 c (a+c)-b \sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (b^3-2 a b c+\frac {b^4-4 a b^2 c+2 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)+b \sqrt {b^2-4 a c}}}\right )}{c^3 \sqrt {b^2-2 c (a+c)+b \sqrt {b^2-4 a c}}}+\frac {b \cos (x)}{c^2}-\frac {\cos (x) \sin (x)}{2 c} \]

[Out]

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

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Rubi [A]
time = 2.07, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3337, 2718, 2715, 8, 3373, 2739, 632, 210} \begin {gather*} -\frac {\sqrt {2} \left (-\frac {2 a^2 c^2-4 a b^2 c+b^4}{\sqrt {b^2-4 a c}}-2 a b c+b^3\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^3 \sqrt {-b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}-\frac {\sqrt {2} \left (\frac {2 a^2 c^2-4 a b^2 c+b^4}{\sqrt {b^2-4 a c}}-2 a b c+b^3\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^3 \sqrt {b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}+\frac {x \left (b^2-a c\right )}{c^3}+\frac {b \cos (x)}{c^2}+\frac {x}{2 c}-\frac {\sin (x) \cos (x)}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]^4/(a + b*Sin[x] + c*Sin[x]^2),x]

[Out]

x/(2*c) + ((b^2 - a*c)*x)/c^3 - (Sqrt[2]*(b^3 - 2*a*b*c - (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^2 - 4*a*c])*Arc
Tan[(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^3*Sq
rt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(b^3 - 2*a*b*c + (b^4 - 4*a*b^2*c + 2*a^2*c^2)/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^3*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) + (b*Cos[x])/c^2 - (Cos[x]*Sin[x])/(2*c)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 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 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.85, size = 410, normalized size = 1.27 \begin {gather*} \frac {4 b^2 x+2 c (-2 a+c) x-\frac {4 \left (i b^4-4 i a b^2 c+2 i a^2 c^2+b^3 \sqrt {-b^2+4 a c}-2 a b c \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 {4 \left (-i b^4+4 i a b^2 c-2 i a^2 c^2+b^3 \sqrt {-b^2+4 a c}-2 a b c \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}}}+4 b c \cos (x)-c^2 \sin (2 x)}{4 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^4/(a + b*Sin[x] + c*Sin[x]^2),x]

[Out]

(4*b^2*x + 2*c*(-2*a + c)*x - (4*(I*b^4 - (4*I)*a*b^2*c + (2*I)*a^2*c^2 + b^3*Sqrt[-b^2 + 4*a*c] - 2*a*b*c*Sqr
t[-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]]) - (4*((-I)*b^4 +
(4*I)*a*b^2*c - (2*I)*a^2*c^2 + b^3*Sqrt[-b^2 + 4*a*c] - 2*a*b*c*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]]) + 4*b*c*Cos[x] - c^2*Sin[2*x])/(4*c^3)

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Maple [A]
time = 1.24, size = 372, normalized size = 1.15

method	result	size
default	\(-\frac {2 \left (\frac {-\frac {c^{2} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2}-b c \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\frac {c^{2} \tan \left (\frac {x}{2}\right )}{2}-b c}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {\left (2 a c -2 b^{2}-c^{2}\right ) \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{2}\right )}{c^{3}}+\frac {2 a \left (-\frac {2 \left (-3 \sqrt {-4 a c +b^{2}}\, a b c +\sqrt {-4 a c +b^{2}}\, b^{3}+4 a^{2} c^{2}-5 a \,b^{2} c +b^{4}\right ) \arctan \left (\frac {-2 a \tan \left (\frac {x}{2}\right )+\sqrt {-4 a c +b^{2}}-b}{\sqrt {4 a c -2 b^{2}+2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}\right )}{\left (8 a c -2 b^{2}\right ) \sqrt {4 a c -2 b^{2}+2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}+\frac {2 \left (3 \sqrt {-4 a c +b^{2}}\, a b c -\sqrt {-4 a c +b^{2}}\, b^{3}+4 a^{2} c^{2}-5 a \,b^{2} c +b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+b +\sqrt {-4 a c +b^{2}}}{\sqrt {4 a c -2 b^{2}-2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}\right )}{\left (8 a c -2 b^{2}\right ) \sqrt {4 a c -2 b^{2}-2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}\right )}{c^{3}}\)	\(372\)
risch	\(\text {Expression too large to display}\)	\(3919\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^4/(a+b*sin(x)+c*sin(x)^2),x,method=_RETURNVERBOSE)

[Out]

-2/c^3*((-1/2*c^2*tan(1/2*x)^3-b*c*tan(1/2*x)^2+1/2*c^2*tan(1/2*x)-b*c)/(tan(1/2*x)^2+1)^2+1/2*(2*a*c-2*b^2-c^
2)*arctan(tan(1/2*x)))+2/c^3*a*(-2*(-3*(-4*a*c+b^2)^(1/2)*a*b*c+(-4*a*c+b^2)^(1/2)*b^3+4*a^2*c^2-5*a*b^2*c+b^4
)/(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*(3*(-4*a*c+b^2)^(1/2)*a*b*c-(-4*a*c+b^2)^(1/2)*b^3+4*a^2*
c^2-5*a*b^2*c+b^4)/(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)^4/(a+b*sin(x)+c*sin(x)^2),x, algorithm="maxima")

[Out]

1/4*(4*c^3*integrate(-2*(2*(b^4 - 2*a*b^2*c)*cos(3*x)^2 + 4*(2*a^2*b^2 - a^2*c^2 - (2*a^3 - a*b^2)*c)*cos(2*x)
^2 + 2*(b^4 - 2*a*b^2*c)*cos(x)^2 + 2*(b^4 - 2*a*b^2*c)*sin(3*x)^2 + 2*(4*a*b^3 - 2*a*b*c^2 - (6*a^2*b - b^3)*
c)*cos(x)*sin(2*x) + 4*(2*a^2*b^2 - a^2*c^2 - (2*a^3 - a*b^2)*c)*sin(2*x)^2 + 2*(b^4 - 2*a*b^2*c)*sin(x)^2 - (
2*(a*b^2*c - a^2*c^2)*cos(2*x) + (b^3*c - 2*a*b*c^2)*sin(3*x) - (b^3*c - 2*a*b*c^2)*sin(x))*cos(4*x) - 2*(2*(b
^4 - 2*a*b^2*c)*cos(x) + (4*a*b^3 - 2*a*b*c^2 - (6*a^2*b - b^3)*c)*sin(2*x))*cos(3*x) - 2*(a*b^2*c - a^2*c^2 +
(4*a*b^3 - 2*a*b*c^2 - (6*a^2*b - b^3)*c)*sin(x))*cos(2*x) + ((b^3*c - 2*a*b*c^2)*cos(3*x) - (b^3*c - 2*a*b*c
^2)*cos(x) - 2*(a*b^2*c - a^2*c^2)*sin(2*x))*sin(4*x) - (b^3*c - 2*a*b*c^2 - 2*(4*a*b^3 - 2*a*b*c^2 - (6*a^2*b
- b^3)*c)*cos(2*x) + 4*(b^4 - 2*a*b^2*c)*sin(x))*sin(3*x) + (b^3*c - 2*a*b*c^2)*sin(x))/(c^5*cos(4*x)^2 + 4*b
^2*c^3*cos(3*x)^2 + 4*b^2*c^3*cos(x)^2 + c^5*sin(4*x)^2 + 4*b^2*c^3*sin(3*x)^2 + 4*b^2*c^3*sin(x)^2 + 4*b*c^4*
sin(x) + c^5 + 4*(4*a^2*c^3 + 4*a*c^4 + c^5)*cos(2*x)^2 + 8*(2*a*b*c^3 + b*c^4)*cos(x)*sin(2*x) + 4*(4*a^2*c^3
+ 4*a*c^4 + c^5)*sin(2*x)^2 - 2*(2*b*c^4*sin(3*x) - 2*b*c^4*sin(x) - c^5 + 2*(2*a*c^4 + c^5)*cos(2*x))*cos(4*
x) - 8*(b^2*c^3*cos(x) + (2*a*b*c^3 + b*c^4)*sin(2*x))*cos(3*x) - 4*(2*a*c^4 + c^5 + 2*(2*a*b*c^3 + b*c^4)*sin
(x))*cos(2*x) + 4*(b*c^4*cos(3*x) - b*c^4*cos(x) - (2*a*c^4 + c^5)*sin(2*x))*sin(4*x) - 4*(2*b^2*c^3*sin(x) +
b*c^4 - 2*(2*a*b*c^3 + b*c^4)*cos(2*x))*sin(3*x)), x) + 4*b*c*cos(x) - c^2*sin(2*x) + 2*(2*b^2 - 2*a*c + c^2)*
x)/c^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 8169 vs. \(2 (285) = 570\).
time = 5.10, size = 8169, normalized size = 25.29 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(2)*c^3*sqrt((a^2*b^6 - b^8 - 2*a^4*c^4 - 2*(a^5 - 8*a^3*b^2)*c^3 + (9*a^4*b^2 - 20*a^2*b^4)*c^2 - 2
*(3*a^3*b^4 - 4*a*b^6)*c + (4*a*c^9 + (8*a^2 - b^2)*c^8 + 2*(2*a^3 - 3*a*b^2)*c^7 - (a^2*b^2 - b^4)*c^6)*sqrt(
-(a^4*b^10 - 2*a^2*b^12 + b^14 + 16*a^6*b^2*c^6 + 8*(3*a^7*b^2 - 10*a^5*b^4)*c^5 + (9*a^8*b^2 - 92*a^6*b^4 + 1
48*a^4*b^6)*c^4 - 4*(6*a^7*b^4 - 31*a^5*b^6 + 32*a^3*b^8)*c^3 + 2*(11*a^6*b^6 - 37*a^4*b^8 + 28*a^2*b^10)*c^2
- 4*(2*a^5*b^8 - 5*a^3*b^10 + 3*a*b^12)*c)/(4*a*c^17 + (16*a^2 - b^2)*c^16 + 12*(2*a^3 - a*b^2)*c^15 + 2*(8*a^
4 - 11*a^2*b^2 + b^4)*c^14 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^13 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^12)))/(4*a*c^9
+ (8*a^2 - b^2)*c^8 + 2*(2*a^3 - 3*a*b^2)*c^7 - (a^2*b^2 - b^4)*c^6))*log(16*a^7*b*c^4 + 4*(3*a^8*b - 10*a^6*
b^3)*c^3 - 8*(2*a^7*b^3 - 3*a^5*b^5)*c^2 + 2*(4*a^5*c^9 + (8*a^6 - a^4*b^2)*c^8 + 2*(2*a^7 - 3*a^5*b^2)*c^7 -
(a^6*b^2 - a^4*b^4)*c^6)*sqrt(-(a^4*b^10 - 2*a^2*b^12 + b^14 + 16*a^6*b^2*c^6 + 8*(3*a^7*b^2 - 10*a^5*b^4)*c^5
+ (9*a^8*b^2 - 92*a^6*b^4 + 148*a^4*b^6)*c^4 - 4*(6*a^7*b^4 - 31*a^5*b^6 + 32*a^3*b^8)*c^3 + 2*(11*a^6*b^6 -
37*a^4*b^8 + 28*a^2*b^10)*c^2 - 4*(2*a^5*b^8 - 5*a^3*b^10 + 3*a*b^12)*c)/(4*a*c^17 + (16*a^2 - b^2)*c^16 + 12*
(2*a^3 - a*b^2)*c^15 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^14 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^13 - (a^4*b^2 - 2*a
^2*b^4 + b^6)*c^12))*sin(x) + 4*(a^6*b^5 - a^4*b^7)*c - sqrt(2)*((8*a^3*c^12 + 6*(4*a^4 - 3*a^2*b^2)*c^11 + 2*
(12*a^5 - 25*a^3*b^2 + 4*a*b^4)*c^10 + (8*a^6 - 38*a^4*b^2 + 35*a^2*b^4 - b^6)*c^9 - 2*(3*a^5*b^2 - 8*a^3*b^4
+ 5*a*b^6)*c^8 + (a^4*b^4 - 2*a^2*b^6 + b^8)*c^7)*sqrt(-(a^4*b^10 - 2*a^2*b^12 + b^14 + 16*a^6*b^2*c^6 + 8*(3*
a^7*b^2 - 10*a^5*b^4)*c^5 + (9*a^8*b^2 - 92*a^6*b^4 + 148*a^4*b^6)*c^4 - 4*(6*a^7*b^4 - 31*a^5*b^6 + 32*a^3*b^
8)*c^3 + 2*(11*a^6*b^6 - 37*a^4*b^8 + 28*a^2*b^10)*c^2 - 4*(2*a^5*b^8 - 5*a^3*b^10 + 3*a*b^12)*c)/(4*a*c^17 +
(16*a^2 - b^2)*c^16 + 12*(2*a^3 - a*b^2)*c^15 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^14 + 4*(a^5 - 3*a^3*b^2 + 2*a*b
^4)*c^13 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^12))*cos(x) - (32*a^5*b^2*c^6 + 8*(5*a^6*b^2 - 13*a^4*b^4)*c^5 + 2*(6
*a^7*b^2 - 47*a^5*b^4 + 56*a^3*b^6)*c^4 - (19*a^6*b^4 - 69*a^4*b^6 + 54*a^2*b^8)*c^3 + 4*(2*a^5*b^6 - 5*a^3*b^
8 + 3*a*b^10)*c^2 - (a^4*b^8 - 2*a^2*b^10 + b^12)*c)*cos(x))*sqrt((a^2*b^6 - b^8 - 2*a^4*c^4 - 2*(a^5 - 8*a^3*
b^2)*c^3 + (9*a^4*b^2 - 20*a^2*b^4)*c^2 - 2*(3*a^3*b^4 - 4*a*b^6)*c + (4*a*c^9 + (8*a^2 - b^2)*c^8 + 2*(2*a^3
- 3*a*b^2)*c^7 - (a^2*b^2 - b^4)*c^6)*sqrt(-(a^4*b^10 - 2*a^2*b^12 + b^14 + 16*a^6*b^2*c^6 + 8*(3*a^7*b^2 - 10
*a^5*b^4)*c^5 + (9*a^8*b^2 - 92*a^6*b^4 + 148*a^4*b^6)*c^4 - 4*(6*a^7*b^4 - 31*a^5*b^6 + 32*a^3*b^8)*c^3 + 2*(
11*a^6*b^6 - 37*a^4*b^8 + 28*a^2*b^10)*c^2 - 4*(2*a^5*b^8 - 5*a^3*b^10 + 3*a*b^12)*c)/(4*a*c^17 + (16*a^2 - b^
2)*c^16 + 12*(2*a^3 - a*b^2)*c^15 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^14 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^13 - (
a^4*b^2 - 2*a^2*b^4 + b^6)*c^12)))/(4*a*c^9 + (8*a^2 - b^2)*c^8 + 2*(2*a^3 - 3*a*b^2)*c^7 - (a^2*b^2 - b^4)*c^
6)) + 2*(a^6*b^6 - a^4*b^8 + 4*a^7*b^2*c^3 + (3*a^8*b^2 - 10*a^6*b^4)*c^2 - 2*(2*a^7*b^4 - 3*a^5*b^6)*c)*sin(x
)) - sqrt(2)*c^3*sqrt((a^2*b^6 - b^8 - 2*a^4*c^4 - 2*(a^5 - 8*a^3*b^2)*c^3 + (9*a^4*b^2 - 20*a^2*b^4)*c^2 - 2*
(3*a^3*b^4 - 4*a*b^6)*c - (4*a*c^9 + (8*a^2 - b^2)*c^8 + 2*(2*a^3 - 3*a*b^2)*c^7 - (a^2*b^2 - b^4)*c^6)*sqrt(-
(a^4*b^10 - 2*a^2*b^12 + b^14 + 16*a^6*b^2*c^6 + 8*(3*a^7*b^2 - 10*a^5*b^4)*c^5 + (9*a^8*b^2 - 92*a^6*b^4 + 14
8*a^4*b^6)*c^4 - 4*(6*a^7*b^4 - 31*a^5*b^6 + 32*a^3*b^8)*c^3 + 2*(11*a^6*b^6 - 37*a^4*b^8 + 28*a^2*b^10)*c^2 -
4*(2*a^5*b^8 - 5*a^3*b^10 + 3*a*b^12)*c)/(4*a*c^17 + (16*a^2 - b^2)*c^16 + 12*(2*a^3 - a*b^2)*c^15 + 2*(8*a^4
- 11*a^2*b^2 + b^4)*c^14 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^13 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^12)))/(4*a*c^9
+ (8*a^2 - b^2)*c^8 + 2*(2*a^3 - 3*a*b^2)*c^7 - (a^2*b^2 - b^4)*c^6))*log(16*a^7*b*c^4 + 4*(3*a^8*b - 10*a^6*b
^3)*c^3 - 8*(2*a^7*b^3 - 3*a^5*b^5)*c^2 - 2*(4*a^5*c^9 + (8*a^6 - a^4*b^2)*c^8 + 2*(2*a^7 - 3*a^5*b^2)*c^7 - (
a^6*b^2 - a^4*b^4)*c^6)*sqrt(-(a^4*b^10 - 2*a^2*b^12 + b^14 + 16*a^6*b^2*c^6 + 8*(3*a^7*b^2 - 10*a^5*b^4)*c^5
+ (9*a^8*b^2 - 92*a^6*b^4 + 148*a^4*b^6)*c^4 - 4*(6*a^7*b^4 - 31*a^5*b^6 + 32*a^3*b^8)*c^3 + 2*(11*a^6*b^6 - 3
7*a^4*b^8 + 28*a^2*b^10)*c^2 - 4*(2*a^5*b^8 - 5*a^3*b^10 + 3*a*b^12)*c)/(4*a*c^17 + (16*a^2 - b^2)*c^16 + 12*(
2*a^3 - a*b^2)*c^15 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^14 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^13 - (a^4*b^2 - 2*a^
2*b^4 + b^6)*c^12))*sin(x) + 4*(a^6*b^5 - a^4*b^7)*c - sqrt(2)*((8*a^3*c^12 + 6*(4*a^4 - 3*a^2*b^2)*c^11 + 2*(
12*a^5 - 25*a^3*b^2 + 4*a*b^4)*c^10 + (8*a^6 - 38*a^4*b^2 + 35*a^2*b^4 - b^6)*c^9 - 2*(3*a^5*b^2 - 8*a^3*b^4 +
5*a*b^6)*c^8 + (a^4*b^4 - 2*a^2*b^6 + b^8)*c^7)*sqrt(-(a^4*b^10 - 2*a^2*b^12 + b^14 + 16*a^6*b^2*c^6 + 8*(3*a
^7*b^2 - 10*a^5*b^4)*c^5 + (9*a^8*b^2 - 92*a^6*b^4 + 148*a^4*b^6)*c^4 - 4*(6*a^7*b^4 - 31*a^5*b^6 + 32*a^3*b^8
)*c^3 + 2*(11*a^6*b^6 - 37*a^4*b^8 + 28*a^2*b^10)*c^2 - 4*(2*a^5*b^8 - 5*a^3*b^10 + 3*a*b^12)*c)/(4*a*c^17 + (
16*a^2 - b^2)*c^16 + 12*(2*a^3 - a*b^2)*c^15 + ...

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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)**4/(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)^4/(a+b*sin(x)+c*sin(x)^2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 26.41, size = 2500, normalized size = 7.74 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^4/(a + c*sin(x)^2 + b*sin(x)),x)

[Out]

((2*b)/c^2 - tan(x/2)/c + tan(x/2)^3/c + (2*b*tan(x/2)^2)/c^2)/(2*tan(x/2)^2 + tan(x/2)^4 + 1) - atan(((((2048
*(44*a^5*c^9 - 16*a^4*c^10 - 4*a^6*c^8 - 64*a^7*c^7 + 12*a^8*c^6 + 4*a*b^6*c^7 + 15*a*b^8*c^5 + 14*a*b^10*c^3
- 28*a^2*b^4*c^8 - 119*a^2*b^6*c^6 - 128*a^2*b^8*c^4 - 8*a^2*b^10*c^2 + 52*a^3*b^2*c^9 + 290*a^3*b^4*c^7 + 397
*a^3*b^6*c^5 + 62*a^3*b^8*c^3 - 227*a^4*b^2*c^8 - 491*a^4*b^4*c^6 - 148*a^4*b^6*c^4 + 8*a^4*b^8*c^2 + 221*a^5*
b^2*c^7 + 102*a^5*b^4*c^5 - 60*a^5*b^6*c^3 + 68*a^6*b^2*c^6 + 136*a^6*b^4*c^4 - 100*a^7*b^2*c^5))/c^8 - (-(a^2
*b^8 - b^10 + 8*a^5*c^5 + 8*a^6*c^4 - b^7*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^6*c + a^2*b^5*(-(4*a*c - b^2)^3)
^(1/2) - 52*a^2*b^6*c^2 + 96*a^3*b^4*c^3 - 66*a^4*b^2*c^4 + 33*a^4*b^4*c^2 - 38*a^5*b^2*c^3 + 12*a*b^8*c + 4*a
^3*b*c^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a^3*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 3*a^4*b*c^2*(-(4*a*c - b^2)^3)^(1/2
) - 10*a^2*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^10 + 32*a^3*c^9
+ 16*a^4*c^8 + b^4*c^8 - b^6*c^6 - 8*a*b^2*c^9 + 10*a*b^4*c^7 - 32*a^2*b^2*c^8 + a^2*b^4*c^6 - 8*a^3*b^2*c^7)
))^(1/2)*((2048*(4*a*b^3*c^11 + 13*a*b^5*c^9 + 4*a*b^7*c^7 - 12*a*b^9*c^5 - 16*a^2*b*c^12 + 44*a^3*b*c^11 + 4*
a^4*b*c^10 + 80*a^5*b*c^9 + 12*a^6*b*c^8 - 63*a^2*b^3*c^10 - 16*a^2*b^5*c^8 + 76*a^2*b^7*c^6 - a^3*b^3*c^9 - 1
04*a^3*b^5*c^7 + 12*a^3*b^7*c^5 - 56*a^4*b^3*c^8 - 60*a^4*b^5*c^6 + 48*a^5*b^3*c^7))/c^8 - (((2048*(12*a*b^5*c
^11 - 16*a*b^3*c^13 + 64*a^2*b*c^14 + 80*a^3*b*c^13 + 48*a^4*b*c^12 - 68*a^2*b^3*c^12 - 12*a^3*b^3*c^11))/c^8
+ (2048*tan(x/2)*(256*a^2*c^15 + 576*a^3*c^14 + 416*a^4*c^13 + 96*a^5*c^12 - 64*a*b^2*c^14 + 68*a*b^4*c^12 - 8
*a*b^6*c^10 - 416*a^2*b^2*c^13 + 72*a^2*b^4*c^11 - 264*a^3*b^2*c^12 + 8*a^3*b^4*c^10 - 56*a^4*b^2*c^11))/c^8)*
(-(a^2*b^8 - b^10 + 8*a^5*c^5 + 8*a^6*c^4 - b^7*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^6*c + a^2*b^5*(-(4*a*c - b
^2)^3)^(1/2) - 52*a^2*b^6*c^2 + 96*a^3*b^4*c^3 - 66*a^4*b^2*c^4 + 33*a^4*b^4*c^2 - 38*a^5*b^2*c^3 + 12*a*b^8*c
+ 4*a^3*b*c^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a^3*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 3*a^4*b*c^2*(-(4*a*c - b^2)^3
)^(1/2) - 10*a^2*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^10 + 32*a
^3*c^9 + 16*a^4*c^8 + b^4*c^8 - b^6*c^6 - 8*a*b^2*c^9 + 10*a*b^4*c^7 - 32*a^2*b^2*c^8 + a^2*b^4*c^6 - 8*a^3*b^
2*c^7)))^(1/2) - (2048*(32*a^3*c^13 + 64*a^4*c^12 - 16*a^5*c^11 - 48*a^6*c^10 + 2*a*b^4*c^11 - 14*a*b^6*c^9 -
16*a^2*b^2*c^12 + 96*a^2*b^4*c^10 + 8*a^2*b^6*c^8 - 176*a^3*b^2*c^11 - 46*a^3*b^4*c^9 + 60*a^4*b^2*c^10 - 8*a^
4*b^4*c^8 + 44*a^5*b^2*c^9))/c^8 + (2048*tan(x/2)*(32*a*b^5*c^10 - 16*a*b^7*c^8 + 256*a^3*b*c^12 + 320*a^4*b*c
^11 + 128*a^5*b*c^10 - 192*a^2*b^3*c^11 + 128*a^2*b^5*c^9 - 336*a^3*b^3*c^10 + 16*a^3*b^5*c^8 - 96*a^4*b^3*c^9
))/c^8)*(-(a^2*b^8 - b^10 + 8*a^5*c^5 + 8*a^6*c^4 - b^7*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^6*c + a^2*b^5*(-(4
*a*c - b^2)^3)^(1/2) - 52*a^2*b^6*c^2 + 96*a^3*b^4*c^3 - 66*a^4*b^2*c^4 + 33*a^4*b^4*c^2 - 38*a^5*b^2*c^3 + 12
*a*b^8*c + 4*a^3*b*c^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a^3*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 3*a^4*b*c^2*(-(4*a*c
- b^2)^3)^(1/2) - 10*a^2*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^1
0 + 32*a^3*c^9 + 16*a^4*c^8 + b^4*c^8 - b^6*c^6 - 8*a*b^2*c^9 + 10*a*b^4*c^7 - 32*a^2*b^2*c^8 + a^2*b^4*c^6 -
8*a^3*b^2*c^7)))^(1/2) + (2048*tan(x/2)*(128*a^3*c^12 - 64*a^2*c^13 + 184*a^4*c^11 - 296*a^5*c^10 - 352*a^6*c^
9 - 72*a^7*c^8 + 16*a*b^2*c^12 + 48*a*b^4*c^10 + a*b^6*c^8 - 92*a*b^8*c^6 + 8*a*b^10*c^4 - 224*a^2*b^2*c^11 +
56*a^2*b^4*c^9 + 732*a^2*b^6*c^7 - 88*a^2*b^8*c^5 - 286*a^3*b^2*c^10 - 1817*a^3*b^4*c^8 + 440*a^3*b^6*c^6 - 8*
a^3*b^8*c^4 + 1502*a^4*b^2*c^9 - 1140*a^4*b^4*c^7 + 72*a^4*b^6*c^5 + 1208*a^5*b^2*c^8 - 220*a^5*b^4*c^6 + 256*
a^6*b^2*c^7))/c^8) + (2048*tan(x/2)*(8*a*b^5*c^8 + 28*a*b^7*c^6 + 16*a*b^9*c^4 - 16*a*b^11*c^2 + 64*a^3*b*c^10
- 176*a^4*b*c^9 - 32*a^5*b*c^8 + 128*a^6*b*c^7 + 112*a^7*b*c^6 - 48*a^2*b^3*c^9 - 192*a^2*b^5*c^7 - 112*a^2*b
^7*c^5 + 160*a^2*b^9*c^3 + 364*a^3*b^3*c^8 + 212*a^3*b^5*c^6 - 592*a^3*b^7*c^4 + 16*a^3*b^9*c^2 - 72*a^4*b^3*c
^7 + 1008*a^4*b^5*c^5 - 128*a^4*b^7*c^3 - 720*a^5*b^3*c^6 + 336*a^5*b^5*c^4 - 352*a^6*b^3*c^5))/c^8)*(-(a^2*b^
8 - b^10 + 8*a^5*c^5 + 8*a^6*c^4 - b^7*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^6*c + a^2*b^5*(-(4*a*c - b^2)^3)^(1
/2) - 52*a^2*b^6*c^2 + 96*a^3*b^4*c^3 - 66*a^4*b^2*c^4 + 33*a^4*b^4*c^2 - 38*a^5*b^2*c^3 + 12*a*b^8*c + 4*a^3*
b*c^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a^3*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 3*a^4*b*c^2*(-(4*a*c - b^2)^3)^(1/2) -
10*a^2*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^10 + 32*a^3*c^9 +
16*a^4*c^8 + b^4*c^8 - b^6*c^6 - 8*a*b^2*c^9 + 10*a*b^4*c^7 - 32*a^2*b^2*c^8 + a^2*b^4*c^6 - 8*a^3*b^2*c^7)))^
(1/2) + (2048*(16*a^2*b^11 - 12*a^4*b^9 - 144*a^3*b^9*c - 28*a^5*b*c^7 + 84*a^5*b^7*c + 97*a^6*b*c^6 - 52*a^7*
b*c^5 - 60*a^8*b*c^4 + 4*a^2*b^7*c^4 + 16*a^2*b^9*c^2 - 28*a^3*b^5*c^5 - 128*a^3*b^7*c^3 + 56*a^4*b^3*c^6 + 33
3*a^4*b^5*c^4 + 452*a^4*b^7*c^2 - 321*a^5*b^3*c...

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