3.1.8 \(\int \frac {\csc ^3(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx\) [8]

Optimal. Leaf size=331 \[ -\frac {\sqrt {2} c \left (b^3-3 a b c+\sqrt {b^2-4 a c} \left (b^2-a c\right )\right ) \tan ^{-1}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)-b \sqrt {b^2-4 a c}}}\right )}{a^3 \sqrt {b^2-4 a c} \sqrt {b^2-2 c (a+c)-b \sqrt {b^2-4 a c}}}+\frac {\sqrt {2} c \left (b^3-3 a b c-\sqrt {b^2-4 a c} \left (b^2-a c\right )\right ) \tan ^{-1}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)+b \sqrt {b^2-4 a c}}}\right )}{a^3 \sqrt {b^2-4 a c} \sqrt {b^2-2 c (a+c)+b \sqrt {b^2-4 a c}}}-\frac {\tanh ^{-1}(\cos (x))}{2 a}-\frac {\left (b^2-a c\right ) \tanh ^{-1}(\cos (x))}{a^3}+\frac {b \cot (x)}{a^2}-\frac {\cot (x) \csc (x)}{2 a} \]

[Out]

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

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Rubi [A]
time = 2.15, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3337, 3855, 3852, 8, 3853, 3373, 2739, 632, 210} \begin {gather*} -\frac {\sqrt {2} c \left (\sqrt {b^2-4 a c} \left (b^2-a c\right )-3 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 )}{a^3 \sqrt {b^2-4 a c} \sqrt {-b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}+\frac {\sqrt {2} c \left (-\sqrt {b^2-4 a c} \left (b^2-a c\right )-3 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 )}{a^3 \sqrt {b^2-4 a c} \sqrt {b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}-\frac {\left (b^2-a c\right ) \tanh ^{-1}(\cos (x))}{a^3}+\frac {b \cot (x)}{a^2}-\frac {\tanh ^{-1}(\cos (x))}{2 a}-\frac {\cot (x) \csc (x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3/(a + b*Sin[x] + c*Sin[x]^2),x]

[Out]

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

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 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]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

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

Rule 3855

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.96, size = 481, normalized size = 1.45 \begin {gather*} \frac {\csc ^2(x) (-2 a-c+c \cos (2 x)-2 b \sin (x)) \left (\frac {8 c \left (-i b^3+3 i a b c+b^2 \sqrt {-b^2+4 a c}-a 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 {8 c \left (i b^3-3 i a b c+b^2 \sqrt {-b^2+4 a c}-a 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 a b \cot \left (\frac {x}{2}\right )+a^2 \csc ^2\left (\frac {x}{2}\right )+4 \left (a^2+2 b^2-2 a c\right ) \log \left (\cos \left (\frac {x}{2}\right )\right )-4 \left (a^2+2 b^2-2 a c\right ) \log \left (\sin \left (\frac {x}{2}\right )\right )-a^2 \sec ^2\left (\frac {x}{2}\right )+4 a b \tan \left (\frac {x}{2}\right )\right )}{16 a^3 \left (c+b \csc (x)+a \csc ^2(x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^3/(a + b*Sin[x] + c*Sin[x]^2),x]

[Out]

(Csc[x]^2*(-2*a - c + c*Cos[2*x] - 2*b*Sin[x])*((8*c*((-I)*b^3 + (3*I)*a*b*c + b^2*Sqrt[-b^2 + 4*a*c] - a*c*Sq
rt[-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*Sqr
t[-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]]) + (8*c*(I*b^3 -
(3*I)*a*b*c + b^2*Sqrt[-b^2 + 4*a*c] - a*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*a*b*Cot[x/2] + a^2*Csc[x/2]^2 + 4*(a^2 + 2*b^2 - 2*a*c)*Log[Cos[x/2]] - 4*(a^
2 + 2*b^2 - 2*a*c)*Log[Sin[x/2]] - a^2*Sec[x/2]^2 + 4*a*b*Tan[x/2]))/(16*a^3*(c + b*Csc[x] + a*Csc[x]^2))

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Maple [A]
time = 5.83, size = 414, normalized size = 1.25

method	result	size
default	\(\frac {\frac {a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-2 b \tan \left (\frac {x}{2}\right )}{4 a^{2}}-\frac {1}{8 a \tan \left (\frac {x}{2}\right )^{2}}+\frac {\left (2 a^{2}-4 a c +4 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tan \left (\frac {x}{2}\right )}+\frac {-\frac {2 \left (-2 \sqrt {-4 a c +b^{2}}\, a^{2} c^{2}+4 \sqrt {-4 a c +b^{2}}\, b^{2} c a -\sqrt {-4 a c +b^{2}}\, b^{4}+8 a^{2} b \,c^{2}-6 b^{3} c a +b^{5}\right ) \arctan \left (\frac {-2 a \tan \left (\frac {x}{2}\right )+\sqrt {-4 a c +b^{2}}-b}{\sqrt {4 a c -2 b^{2}+2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}\right )}{a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}+2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}+\frac {2 \left (2 \sqrt {-4 a c +b^{2}}\, a^{2} c^{2}-4 \sqrt {-4 a c +b^{2}}\, b^{2} c a +\sqrt {-4 a c +b^{2}}\, b^{4}+8 a^{2} b \,c^{2}-6 b^{3} c a +b^{5}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+b +\sqrt {-4 a c +b^{2}}}{\sqrt {4 a c -2 b^{2}-2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}\right )}{a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}-2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}}{a^{2}}\)	\(414\)
risch	\(\text {Expression too large to display}\)	\(4146\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^3/(a+b*sin(x)+c*sin(x)^2),x,method=_RETURNVERBOSE)

[Out]

1/4/a^2*(1/2*a*tan(1/2*x)^2-2*b*tan(1/2*x))-1/8/a/tan(1/2*x)^2+1/4/a^3*(2*a^2-4*a*c+4*b^2)*ln(tan(1/2*x))+1/2*
b/a^2/tan(1/2*x)+2/a^2*(-(-2*(-4*a*c+b^2)^(1/2)*a^2*c^2+4*(-4*a*c+b^2)^(1/2)*b^2*c*a-(-4*a*c+b^2)^(1/2)*b^4+8*
a^2*b*c^2-6*b^3*c*a+b^5)/a/(4*a*c-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*(-4*a*c+b^2)^(1/2)*a^2*c^2-4*(-4*
a*c+b^2)^(1/2)*b^2*c*a+(-4*a*c+b^2)^(1/2)*b^4+8*a^2*b*c^2-6*b^3*c*a+b^5)/a/(4*a*c-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(csc(x)^3/(a+b*sin(x)+c*sin(x)^2),x, algorithm="maxima")

[Out]

-1/4*(8*a^2*cos(2*x)*cos(x) + 8*a^2*sin(3*x)*sin(2*x) - 4*a^2*cos(x) - 4*(a^2*cos(3*x) + a^2*cos(x) - 2*a*b*si
n(2*x))*cos(4*x) + 4*(2*a^2*cos(2*x) - a^2)*cos(3*x) - 4*(a^3*cos(4*x)^2 + 4*a^3*cos(2*x)^2 + a^3*sin(4*x)^2 -
4*a^3*sin(4*x)*sin(2*x) + 4*a^3*sin(2*x)^2 - 4*a^3*cos(2*x) + a^3 - 2*(2*a^3*cos(2*x) - a^3)*cos(4*x))*integr
ate(-2*(2*(b^3*c - a*b*c^2)*cos(3*x)^2 + 4*(2*a*b^3 - 2*a*b*c^2 - (4*a^2*b - b^3)*c)*cos(2*x)^2 + 2*(b^3*c - a
*b*c^2)*cos(x)^2 + 2*(b^3*c - a*b*c^2)*sin(3*x)^2 + 2*(2*b^4 - 2*a*b^2*c - a*c^3 - (2*a^2 - b^2)*c^2)*cos(x)*s
in(2*x) + 4*(2*a*b^3 - 2*a*b*c^2 - (4*a^2*b - b^3)*c)*sin(2*x)^2 + 2*(b^3*c - a*b*c^2)*sin(x)^2 - (2*(b^3*c -
2*a*b*c^2)*cos(2*x) + (b^2*c^2 - a*c^3)*sin(3*x) - (b^2*c^2 - a*c^3)*sin(x))*cos(4*x) - 2*(2*(b^3*c - a*b*c^2)
*cos(x) + (2*b^4 - 2*a*b^2*c - a*c^3 - (2*a^2 - b^2)*c^2)*sin(2*x))*cos(3*x) - 2*(b^3*c - 2*a*b*c^2 + (2*b^4 -
2*a*b^2*c - a*c^3 - (2*a^2 - b^2)*c^2)*sin(x))*cos(2*x) + ((b^2*c^2 - a*c^3)*cos(3*x) - (b^2*c^2 - a*c^3)*cos
(x) - 2*(b^3*c - 2*a*b*c^2)*sin(2*x))*sin(4*x) - (b^2*c^2 - a*c^3 - 2*(2*b^4 - 2*a*b^2*c - a*c^3 - (2*a^2 - b^
2)*c^2)*cos(2*x) + 4*(b^3*c - a*b*c^2)*sin(x))*sin(3*x) + (b^2*c^2 - a*c^3)*sin(x))/(a^3*c^2*cos(4*x)^2 + 4*a^
3*b^2*cos(3*x)^2 + 4*a^3*b^2*cos(x)^2 + a^3*c^2*sin(4*x)^2 + 4*a^3*b^2*sin(3*x)^2 + 4*a^3*b^2*sin(x)^2 + 4*a^3
*b*c*sin(x) + a^3*c^2 + 4*(4*a^5 + 4*a^4*c + a^3*c^2)*cos(2*x)^2 + 8*(2*a^4*b + a^3*b*c)*cos(x)*sin(2*x) + 4*(
4*a^5 + 4*a^4*c + a^3*c^2)*sin(2*x)^2 - 2*(2*a^3*b*c*sin(3*x) - 2*a^3*b*c*sin(x) - a^3*c^2 + 2*(2*a^4*c + a^3*
c^2)*cos(2*x))*cos(4*x) - 8*(a^3*b^2*cos(x) + (2*a^4*b + a^3*b*c)*sin(2*x))*cos(3*x) - 4*(2*a^4*c + a^3*c^2 +
2*(2*a^4*b + a^3*b*c)*sin(x))*cos(2*x) + 4*(a^3*b*c*cos(3*x) - a^3*b*c*cos(x) - (2*a^4*c + a^3*c^2)*sin(2*x))*
sin(4*x) - 4*(2*a^3*b^2*sin(x) + a^3*b*c - 2*(2*a^4*b + a^3*b*c)*cos(2*x))*sin(3*x)), x) + ((a^2 + 2*b^2 - 2*a
*c)*cos(4*x)^2 + 4*(a^2 + 2*b^2 - 2*a*c)*cos(2*x)^2 + (a^2 + 2*b^2 - 2*a*c)*sin(4*x)^2 - 4*(a^2 + 2*b^2 - 2*a*
c)*sin(4*x)*sin(2*x) + 4*(a^2 + 2*b^2 - 2*a*c)*sin(2*x)^2 + a^2 + 2*b^2 - 2*a*c + 2*(a^2 + 2*b^2 - 2*a*c - 2*(
a^2 + 2*b^2 - 2*a*c)*cos(2*x))*cos(4*x) - 4*(a^2 + 2*b^2 - 2*a*c)*cos(2*x))*log(cos(x)^2 + sin(x)^2 + 2*cos(x)
+ 1) - ((a^2 + 2*b^2 - 2*a*c)*cos(4*x)^2 + 4*(a^2 + 2*b^2 - 2*a*c)*cos(2*x)^2 + (a^2 + 2*b^2 - 2*a*c)*sin(4*x
)^2 - 4*(a^2 + 2*b^2 - 2*a*c)*sin(4*x)*sin(2*x) + 4*(a^2 + 2*b^2 - 2*a*c)*sin(2*x)^2 + a^2 + 2*b^2 - 2*a*c + 2
*(a^2 + 2*b^2 - 2*a*c - 2*(a^2 + 2*b^2 - 2*a*c)*cos(2*x))*cos(4*x) - 4*(a^2 + 2*b^2 - 2*a*c)*cos(2*x))*log(cos
(x)^2 + sin(x)^2 - 2*cos(x) + 1) - 4*(2*a*b*cos(2*x) + a^2*sin(3*x) + a^2*sin(x) - 2*a*b)*sin(4*x) + 8*(a^2*si
n(x) - a*b)*sin(2*x))/(a^3*cos(4*x)^2 + 4*a^3*cos(2*x)^2 + a^3*sin(4*x)^2 - 4*a^3*sin(4*x)*sin(2*x) + 4*a^3*si
n(2*x)^2 - 4*a^3*cos(2*x) + a^3 - 2*(2*a^3*cos(2*x) - a^3)*cos(4*x))

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Fricas [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(csc(x)^3/(a+b*sin(x)+c*sin(x)^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc ^{3}{\left (x \right )}}{a + b \sin {\left (x \right )} + c \sin ^{2}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)**3/(a+b*sin(x)+c*sin(x)**2),x)

[Out]

Integral(csc(x)**3/(a + b*sin(x) + c*sin(x)**2), x)

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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(csc(x)^3/(a+b*sin(x)+c*sin(x)^2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 24.32, size = 2500, normalized size = 7.55 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^3*(a + c*sin(x)^2 + b*sin(x))),x)

[Out]

atan(-(((8*a^4*c^6 - b^10 + 8*a^5*c^5 - b^7*(-(4*a*c - b^2)^3)^(1/2) + b^8*c^2 - 10*a*b^6*c^3 + 33*a^2*b^4*c^4
- 52*a^2*b^6*c^2 - 38*a^3*b^2*c^5 + 96*a^3*b^4*c^3 - 66*a^4*b^2*c^4 + b^5*c^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a
*b^8*c - 4*a*b^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^4*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^3*(-(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*(a^8*b^4 - a^
6*b^6 + 16*a^8*c^4 + 32*a^9*c^3 + 16*a^10*c^2 + 10*a^7*b^4*c - 8*a^9*b^2*c + a^6*b^4*c^2 - 8*a^7*b^2*c^3 - 32*
a^8*b^2*c^2)))^(1/2)*(((8*a^4*c^6 - b^10 + 8*a^5*c^5 - b^7*(-(4*a*c - b^2)^3)^(1/2) + b^8*c^2 - 10*a*b^6*c^3 +
33*a^2*b^4*c^4 - 52*a^2*b^6*c^2 - 38*a^3*b^2*c^5 + 96*a^3*b^4*c^3 - 66*a^4*b^2*c^4 + b^5*c^2*(-(4*a*c - b^2)^
3)^(1/2) + 12*a*b^8*c - 4*a*b^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^4*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*
c^3*(-(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*(a^8*b^4 - a^6*b^6 + 16*a^8*c^4 + 32*a^9*c^3 + 16*a^10*c^2 + 10*a^7*b^4*c - 8*a^9*b^2*c + a^6*b^4*c^2 - 8*a^
7*b^2*c^3 - 32*a^8*b^2*c^2)))^(1/2)*(((8*a^4*c^6 - b^10 + 8*a^5*c^5 - b^7*(-(4*a*c - b^2)^3)^(1/2) + b^8*c^2 -
10*a*b^6*c^3 + 33*a^2*b^4*c^4 - 52*a^2*b^6*c^2 - 38*a^3*b^2*c^5 + 96*a^3*b^4*c^3 - 66*a^4*b^2*c^4 + b^5*c^2*(
-(4*a*c - b^2)^3)^(1/2) + 12*a*b^8*c - 4*a*b^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^4*(-(4*a*c - b^2)^3)^(
1/2) + 4*a^3*b*c^3*(-(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*(a^8*b^4 - a^6*b^6 + 16*a^8*c^4 + 32*a^9*c^3 + 16*a^10*c^2 + 10*a^7*b^4*c - 8*a^9*b^2*c + a^6
*b^4*c^2 - 8*a^7*b^2*c^3 - 32*a^8*b^2*c^2)))^(1/2)*((16*(4*a^7*b^5 - 16*a^5*b^7 + 3*a^9*b^3 + 122*a^6*b^5*c +
96*a^7*b*c^4 + 160*a^8*b*c^3 - 17*a^8*b^3*c + 4*a^9*b*c^2 + 16*a^5*b^5*c^2 - 88*a^6*b^3*c^3 - 272*a^7*b^3*c^2
- 12*a^10*b*c))/a^6 + ((16*(8*a^8*b^5 - 6*a^10*b^3 + 32*a^9*b*c^3 - 50*a^9*b^3*c + 72*a^10*b*c^2 - 8*a^8*b^3*c
^2 + 24*a^11*b*c))/a^6 - (16*tan(x/2)*(16*a^12*c - 32*a^7*b^6 + 34*a^9*b^4 - 4*a^11*b^2 + 384*a^9*c^4 + 768*a^
10*c^3 + 400*a^11*c^2 + 288*a^8*b^4*c - 236*a^10*b^2*c + 32*a^7*b^4*c^2 - 224*a^8*b^2*c^3 - 832*a^9*b^2*c^2))/
a^6)*((8*a^4*c^6 - b^10 + 8*a^5*c^5 - b^7*(-(4*a*c - b^2)^3)^(1/2) + b^8*c^2 - 10*a*b^6*c^3 + 33*a^2*b^4*c^4 -
52*a^2*b^6*c^2 - 38*a^3*b^2*c^5 + 96*a^3*b^4*c^3 - 66*a^4*b^2*c^4 + b^5*c^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b
^8*c - 4*a*b^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^4*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^3*(-(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*(a^8*b^4 - a^6*
b^6 + 16*a^8*c^4 + 32*a^9*c^3 + 16*a^10*c^2 + 10*a^7*b^4*c - 8*a^9*b^2*c + a^6*b^4*c^2 - 8*a^7*b^2*c^3 - 32*a^
8*b^2*c^2)))^(1/2) + (16*tan(x/2)*(8*a^11*c - 32*a^4*b^8 + 18*a^6*b^6 + 5*a^8*b^4 - 2*a^10*b^2 - 192*a^7*c^5 -
288*a^8*c^4 - 48*a^9*c^3 + 56*a^10*c^2 + 288*a^5*b^6*c - 118*a^7*b^4*c - 34*a^9*b^2*c + 32*a^4*b^6*c^2 - 224*
a^5*b^4*c^3 + 432*a^6*b^2*c^4 - 864*a^6*b^4*c^2 + 968*a^7*b^2*c^3 + 196*a^8*b^2*c^2))/a^6) + (16*(8*a^2*b^9 +
2*a^4*b^7 - a^6*b^5 - 78*a^3*b^7*c + 104*a^5*b*c^5 - 18*a^5*b^5*c + 114*a^6*b*c^4 - 36*a^7*b*c^3 + 6*a^7*b^3*c
- 8*a^8*b*c^2 - 8*a^2*b^7*c^2 + 64*a^3*b^5*c^3 - 152*a^4*b^3*c^4 + 256*a^4*b^5*c^2 - 318*a^5*b^3*c^3 + 49*a^6
*b^3*c^2))/a^6 + (16*tan(x/2)*(2*a^3*b^8 - 4*a^5*b^6 + 96*a^5*c^6 + 96*a^6*c^5 + 20*a^7*c^4 + 16*a^8*c^3 + 32*
a^2*b^8*c - 24*a^4*b^6*c + 28*a^6*b^4*c - 32*a^2*b^6*c^3 + 224*a^3*b^4*c^4 - 288*a^3*b^6*c^2 - 400*a^4*b^2*c^5
+ 824*a^4*b^4*c^3 - 768*a^5*b^2*c^4 + 92*a^5*b^4*c^2 - 116*a^6*b^2*c^3 - 52*a^7*b^2*c^2))/a^6) + (16*(6*b^9*c
- 8*b^7*c^3 + 48*a*b^5*c^4 - 48*a*b^7*c^2 + 3*a^2*b^7*c + 48*a^3*b*c^6 + 26*a^4*b*c^5 - 21*a^5*b*c^4 - 80*a^2
*b^3*c^5 + 122*a^2*b^5*c^3 - 108*a^3*b^3*c^4 - 21*a^3*b^5*c^2 + 42*a^4*b^3*c^3))/a^6 - (16*tan(x/2)*(2*b^10 +
a^2*b^8 - 48*a^3*c^7 - 24*a^4*c^6 + 12*a^5*c^5 + 2*a^6*c^4 + 16*b^6*c^4 - 16*b^8*c^2 - 80*a*b^4*c^5 + 112*a*b^
6*c^3 - 8*a^3*b^6*c + 96*a^2*b^2*c^6 - 232*a^2*b^4*c^4 + 48*a^2*b^6*c^2 + 152*a^3*b^2*c^5 - 24*a^3*b^4*c^3 - 3
6*a^4*b^2*c^4 + 20*a^4*b^4*c^2 - 16*a^5*b^2*c^3 - 18*a*b^8*c))/a^6)*1i - ((8*a^4*c^6 - b^10 + 8*a^5*c^5 - b^7*
(-(4*a*c - b^2)^3)^(1/2) + b^8*c^2 - 10*a*b^6*c^3 + 33*a^2*b^4*c^4 - 52*a^2*b^6*c^2 - 38*a^3*b^2*c^5 + 96*a^3*
b^4*c^3 - 66*a^4*b^2*c^4 + b^5*c^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^8*c - 4*a*b^3*c^3*(-(4*a*c - b^2)^3)^(1/2
) + 3*a^2*b*c^4*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^3*(-(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*(a^8*b^4 - a^6*b^6 + 16*a^8*c^4 + 32*a^9*c^3 + 16*a^10*c^
2 + 10*a^7*b^4*c - 8*a^9*b^2*c + a^6*b^4*c^2 - 8*a^7*b^2*c^3 - 32*a^8*b^2*c^2)))^(1/2)*(((8*a^4*c^6 - b^10 + 8
*a^5*c^5 - b^7*(-(4*a*c - b^2)^3)^(1/2) + b^8*c^2 - 10*a*b^6*c^3 + 33*a^2*b^4*c^4 - 52*a^2*b^6*c^2 - 38*a^3*b^
2*c^5 + 96*a^3*b^4*c^3 - 66*a^4*b^2*c^4 + b^5*c^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^8*c - 4*a*b^3*c^3*(-(4*a*c
- b^2)^3)^(1/2) + 3*a^2*b*c^4*(-(4*a*c - b^2)^...

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