3.1.6 \(\int \frac {\csc (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx\) [6]

Optimal. Leaf size=244 \[ -\frac {\sqrt {2} c \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)-b \sqrt {b^2-4 a c}}}\right )}{a \sqrt {b^2-2 c (a+c)-b \sqrt {b^2-4 a c}}}-\frac {\sqrt {2} c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)+b \sqrt {b^2-4 a c}}}\right )}{a \sqrt {b^2-2 c (a+c)+b \sqrt {b^2-4 a c}}}-\frac {\tanh ^{-1}(\cos (x))}{a} \]

[Out]

-arctanh(cos(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)*(1+b/(-4*a*c+b^2)^(1/2))/a/(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)*(1-b/(-4*a*c+b^
2)^(1/2))/a/(b^2-2*c*(a+c)+b*(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 0.51, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3337, 3855, 3373, 2739, 632, 210} \begin {gather*} -\frac {\sqrt {2} c \left (\frac {b}{\sqrt {b^2-4 a c}}+1\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 \sqrt {-b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}-\frac {\sqrt {2} c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\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 \sqrt {b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}-\frac {\tanh ^{-1}(\cos (x))}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[2]*c*(1 + b/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]])])/(a*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])) - (Sqrt[2]*c*(1 - b/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]])])/(a*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) - ArcTanh[Cos[x]]/a

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.69, size = 306, normalized size = 1.25 \begin {gather*} -\frac {\frac {c \left (-i 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 {c \left (i 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}}}+\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((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[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]]
) + (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[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])
+ Log[Cos[x/2]] - Log[Sin[x/2]])/a)

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Maple [A]
time = 1.30, size = 284, normalized size = 1.16

method	result	size
default	\(\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a}-\frac {2 \left (2 \sqrt {-4 a c +b^{2}}\, a c -\sqrt {-4 a c +b^{2}}\, b^{2}-4 a b c +b^{3}\right ) \arctan \left (\frac {-2 a \tan \left (\frac {x}{2}\right )+\sqrt {-4 a c +b^{2}}-b}{\sqrt {4 a c -2 b^{2}+2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}\right )}{a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}+2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}+\frac {2 \left (-2 \sqrt {-4 a c +b^{2}}\, a c +\sqrt {-4 a c +b^{2}}\, b^{2}-4 a b c +b^{3}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+b +\sqrt {-4 a c +b^{2}}}{\sqrt {4 a c -2 b^{2}-2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}\right )}{a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}-2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}\)	\(284\)
risch	\(2 i \left (\munderset {\textit {\_R} =\RootOf \left (\left (256 a^{6} c^{2}-128 a^{5} b^{2} c +512 a^{5} c^{3}+16 a^{4} b^{4}-512 a^{4} b^{2} c^{2}+256 a^{4} c^{4}+160 a^{3} b^{4} c -128 a^{3} b^{2} c^{3}-16 a^{2} b^{6}+16 a^{2} b^{4} c^{2}\right ) \textit {\_Z}^{4}+\left (32 a^{3} c^{3}-72 a^{2} b^{2} c^{2}+32 a^{2} c^{4}+32 a \,b^{4} c -24 a \,b^{2} c^{3}-4 b^{6}+4 b^{4} c^{2}\right ) \textit {\_Z}^{2}+c^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+\left (\frac {8 i a^{4} b^{5}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}-\frac {8 i a^{2} b^{7}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {96 i a^{6} b \,c^{2}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}-\frac {56 i a^{5} b^{3} c}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {224 i a^{5} b \,c^{3}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}-\frac {216 i a^{4} b^{3} c^{2}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {160 i a^{4} b \,c^{4}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {72 i a^{3} b^{5} c}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}-\frac {104 i a^{3} b^{3} c^{3}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {32 i c^{5} a^{3} b}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {16 i a^{2} b^{5} c^{2}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}-\frac {8 i b^{3} a^{2} c^{4}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}\right ) \textit {\_R}^{3}+\left (-\frac {16 i a^{5} c^{3}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}-\frac {32 i a^{4} c^{4}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}-\frac {16 i c^{5} a^{3}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {4 i a^{4} b^{2} c^{2}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {24 i a^{3} b^{2} c^{3}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}-\frac {4 i a^{2} b^{4} c^{2}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {4 i b^{2} a^{2} c^{4}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}\right ) \textit {\_R}^{2}+\left (\frac {4 i b^{5} c^{2}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}-\frac {2 i b^{3} c^{4}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {14 i a \,b^{5} c}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}-\frac {20 i a \,b^{3} c^{3}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {6 i c^{5} a b}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {14 i a^{3} b \,c^{3}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}-\frac {28 i a^{2} b^{3} c^{2}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {20 i a^{2} b \,c^{4}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}-\frac {2 i b^{7}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}\right ) \textit {\_R} -\frac {i a^{2} c^{4}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}-\frac {i c^{5} a}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}-\frac {i b^{4} c^{2}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {i b^{2} c^{4}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}+\frac {3 i a \,b^{2} c^{3}}{2 a b \,c^{4}-c^{3} b^{3}+c^{5} b}\right )\right )-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{a}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{a}\)	\(1303\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5296 vs. \(2 (208) = 416\).
time = 73.48, size = 5296, normalized size = 21.70 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(sqrt(2)*a*sqrt(-(b^4 - 4*a*b^2*c + 2*a*c^3 + (2*a^2 - b^2)*c^2 + (a^4*b^2 - a^2*b^4 - 4*a^3*c^3 - (8*a^4
- a^2*b^2)*c^2 - 2*(2*a^5 - 3*a^3*b^2)*c)*sqrt((b^6 - 4*a*b^4*c + 4*a*b^2*c^3 + b^2*c^4 + 2*(2*a^2*b^2 - b^4)*
c^2)/(a^8*b^2 - 2*a^6*b^4 + a^4*b^6 - 4*a^5*c^5 - (16*a^6 - a^4*b^2)*c^4 - 12*(2*a^7 - a^5*b^2)*c^3 - 2*(8*a^8
- 11*a^6*b^2 + a^4*b^4)*c^2 - 4*(a^9 - 3*a^7*b^2 + 2*a^5*b^4)*c)))/(a^4*b^2 - a^2*b^4 - 4*a^3*c^3 - (8*a^4 -
a^2*b^2)*c^2 - 2*(2*a^5 - 3*a^3*b^2)*c))*log(4*b^3*c^3 - 8*a*b*c^4 - 4*b*c^5 + 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((b^6 - 4*a*b^4*c + 4*a*b^2*c^3 + b^2*c^4 + 2
*(2*a^2*b^2 - b^4)*c^2)/(a^8*b^2 - 2*a^6*b^4 + a^4*b^6 - 4*a^5*c^5 - (16*a^6 - a^4*b^2)*c^4 - 12*(2*a^7 - a^5*
b^2)*c^3 - 2*(8*a^8 - 11*a^6*b^2 + a^4*b^4)*c^2 - 4*(a^9 - 3*a^7*b^2 + 2*a^5*b^4)*c))*sin(x) - sqrt(2)*((a^4*b
^5 - a^2*b^7 + 4*a^3*b*c^5 + (20*a^4*b - a^2*b^3)*c^4 + (28*a^5*b - 13*a^3*b^3)*c^3 + (12*a^6*b - 27*a^4*b^3 +
2*a^2*b^5)*c^2 - (7*a^5*b^3 - 9*a^3*b^5)*c)*sqrt((b^6 - 4*a*b^4*c + 4*a*b^2*c^3 + b^2*c^4 + 2*(2*a^2*b^2 - b^
4)*c^2)/(a^8*b^2 - 2*a^6*b^4 + a^4*b^6 - 4*a^5*c^5 - (16*a^6 - a^4*b^2)*c^4 - 12*(2*a^7 - a^5*b^2)*c^3 - 2*(8*
a^8 - 11*a^6*b^2 + a^4*b^4)*c^2 - 4*(a^9 - 3*a^7*b^2 + 2*a^5*b^4)*c))*cos(x) - (b^7 - 7*a*b^5*c - 4*a*b*c^5 -
(12*a^2*b - b^3)*c^4 - (8*a^3*b - 11*a*b^3)*c^3 + 2*(7*a^2*b^3 - b^5)*c^2)*cos(x))*sqrt(-(b^4 - 4*a*b^2*c + 2*
a*c^3 + (2*a^2 - b^2)*c^2 + (a^4*b^2 - a^2*b^4 - 4*a^3*c^3 - (8*a^4 - a^2*b^2)*c^2 - 2*(2*a^5 - 3*a^3*b^2)*c)*
sqrt((b^6 - 4*a*b^4*c + 4*a*b^2*c^3 + b^2*c^4 + 2*(2*a^2*b^2 - b^4)*c^2)/(a^8*b^2 - 2*a^6*b^4 + a^4*b^6 - 4*a^
5*c^5 - (16*a^6 - a^4*b^2)*c^4 - 12*(2*a^7 - a^5*b^2)*c^3 - 2*(8*a^8 - 11*a^6*b^2 + a^4*b^4)*c^2 - 4*(a^9 - 3*
a^7*b^2 + 2*a^5*b^4)*c)))/(a^4*b^2 - a^2*b^4 - 4*a^3*c^3 - (8*a^4 - a^2*b^2)*c^2 - 2*(2*a^5 - 3*a^3*b^2)*c)) +
2*(b^4*c^2 - 2*a*b^2*c^3 - b^2*c^4)*sin(x)) - sqrt(2)*a*sqrt(-(b^4 - 4*a*b^2*c + 2*a*c^3 + (2*a^2 - b^2)*c^2
- (a^4*b^2 - a^2*b^4 - 4*a^3*c^3 - (8*a^4 - a^2*b^2)*c^2 - 2*(2*a^5 - 3*a^3*b^2)*c)*sqrt((b^6 - 4*a*b^4*c + 4*
a*b^2*c^3 + b^2*c^4 + 2*(2*a^2*b^2 - b^4)*c^2)/(a^8*b^2 - 2*a^6*b^4 + a^4*b^6 - 4*a^5*c^5 - (16*a^6 - a^4*b^2)
*c^4 - 12*(2*a^7 - a^5*b^2)*c^3 - 2*(8*a^8 - 11*a^6*b^2 + a^4*b^4)*c^2 - 4*(a^9 - 3*a^7*b^2 + 2*a^5*b^4)*c)))/
(a^4*b^2 - a^2*b^4 - 4*a^3*c^3 - (8*a^4 - a^2*b^2)*c^2 - 2*(2*a^5 - 3*a^3*b^2)*c))*log(4*b^3*c^3 - 8*a*b*c^4 -
4*b*c^5 - 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((b
^6 - 4*a*b^4*c + 4*a*b^2*c^3 + b^2*c^4 + 2*(2*a^2*b^2 - b^4)*c^2)/(a^8*b^2 - 2*a^6*b^4 + a^4*b^6 - 4*a^5*c^5 -
(16*a^6 - a^4*b^2)*c^4 - 12*(2*a^7 - a^5*b^2)*c^3 - 2*(8*a^8 - 11*a^6*b^2 + a^4*b^4)*c^2 - 4*(a^9 - 3*a^7*b^2
+ 2*a^5*b^4)*c))*sin(x) - sqrt(2)*((a^4*b^5 - a^2*b^7 + 4*a^3*b*c^5 + (20*a^4*b - a^2*b^3)*c^4 + (28*a^5*b -
13*a^3*b^3)*c^3 + (12*a^6*b - 27*a^4*b^3 + 2*a^2*b^5)*c^2 - (7*a^5*b^3 - 9*a^3*b^5)*c)*sqrt((b^6 - 4*a*b^4*c +
4*a*b^2*c^3 + b^2*c^4 + 2*(2*a^2*b^2 - b^4)*c^2)/(a^8*b^2 - 2*a^6*b^4 + a^4*b^6 - 4*a^5*c^5 - (16*a^6 - a^4*b
^2)*c^4 - 12*(2*a^7 - a^5*b^2)*c^3 - 2*(8*a^8 - 11*a^6*b^2 + a^4*b^4)*c^2 - 4*(a^9 - 3*a^7*b^2 + 2*a^5*b^4)*c)
)*cos(x) + (b^7 - 7*a*b^5*c - 4*a*b*c^5 - (12*a^2*b - b^3)*c^4 - (8*a^3*b - 11*a*b^3)*c^3 + 2*(7*a^2*b^3 - b^5
)*c^2)*cos(x))*sqrt(-(b^4 - 4*a*b^2*c + 2*a*c^3 + (2*a^2 - b^2)*c^2 - (a^4*b^2 - a^2*b^4 - 4*a^3*c^3 - (8*a^4
- a^2*b^2)*c^2 - 2*(2*a^5 - 3*a^3*b^2)*c)*sqrt((b^6 - 4*a*b^4*c + 4*a*b^2*c^3 + b^2*c^4 + 2*(2*a^2*b^2 - b^4)*
c^2)/(a^8*b^2 - 2*a^6*b^4 + a^4*b^6 - 4*a^5*c^5 - (16*a^6 - a^4*b^2)*c^4 - 12*(2*a^7 - a^5*b^2)*c^3 - 2*(8*a^8
- 11*a^6*b^2 + a^4*b^4)*c^2 - 4*(a^9 - 3*a^7*b^2 + 2*a^5*b^4)*c)))/(a^4*b^2 - a^2*b^4 - 4*a^3*c^3 - (8*a^4 -
a^2*b^2)*c^2 - 2*(2*a^5 - 3*a^3*b^2)*c)) + 2*(b^4*c^2 - 2*a*b^2*c^3 - b^2*c^4)*sin(x)) + sqrt(2)*a*sqrt(-(b^4
- 4*a*b^2*c + 2*a*c^3 + (2*a^2 - b^2)*c^2 - (a^4*b^2 - a^2*b^4 - 4*a^3*c^3 - (8*a^4 - a^2*b^2)*c^2 - 2*(2*a^5
- 3*a^3*b^2)*c)*sqrt((b^6 - 4*a*b^4*c + 4*a*b^2*c^3 + b^2*c^4 + 2*(2*a^2*b^2 - b^4)*c^2)/(a^8*b^2 - 2*a^6*b^4
+ a^4*b^6 - 4*a^5*c^5 - (16*a^6 - a^4*b^2)*c^4 - 12*(2*a^7 - a^5*b^2)*c^3 - 2*(8*a^8 - 11*a^6*b^2 + a^4*b^4)*c
^2 - 4*(a^9 - 3*a^7*b^2 + 2*a^5*b^4)*c)))/(a^4*b^2 - a^2*b^4 - 4*a^3*c^3 - (8*a^4 - a^2*b^2)*c^2 - 2*(2*a^5 -
3*a^3*b^2)*c))*log(-4*b^3*c^3 + 8*a*b*c^4 + 4*b*c^5 + 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((b^6 - 4*a*b^4*c + 4*a*b^2*c^3 + b^2*c^4 + 2*(2*a^2*b^2 - b^4)*c^2)/(
a^8*b^2 - 2*a^6*b^4 + a^4*b^6 - 4*a^5*c^5 - (16*a^6 - a^4*b^2)*c^4 - 12*(2*a^7 - a^5*b^2)*c^3 - 2*(8*a^8 - 11*
a^6*b^2 + a^4*b^4)*c^2 - 4*(a^9 - 3*a^7*b^2 + 2*a^5*b^4)*c))*sin(x) - sqrt(2)*((a^4*b^5 - a^2*b^7 + 4*a^3*b*c^
5 + (20*a^4*b - a^2*b^3)*c^4 + (28*a^5*b - 13*a^3*b^3)*c^3 + (12*a^6*b - 27*a^4*b^3 + 2*a^2*b^5)*c^2 - (7*a^5*
b^3 - 9*a^3*b^5)*c)*sqrt((b^6 - 4*a*b^4*c + 4*a*b^2*c^3 + b^2*c^4 + 2*(2*a^2*b^2 - b^4)*c^2)/(a^8*b^2 - 2*a^6*
b^4 + a^4*b^6 - 4*a^5*c^5 - (16*a^6 - a^4*b^2)*...

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

[Out]

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

[Out]

Timed out

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Mupad [B]
time = 26.52, size = 2500, normalized size = 10.25 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan((((8*a^2*c^4 - b^6 + 8*a^3*c^3 - b^3*(-(4*a*c - b^2)^3)^(1/2) + b^4*c^2 - 6*a*b^2*c^3 + b*c^2*(-(4*a*c -
b^2)^3)^(1/2) - 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*(a^4*b^4 - a^2*b^6 + 16*a^4*
c^4 + 32*a^5*c^3 + 16*a^6*c^2 + 10*a^3*b^4*c - 8*a^5*b^2*c + a^2*b^4*c^2 - 8*a^3*b^2*c^3 - 32*a^4*b^2*c^2)))^(
1/2)*(((8*a^2*c^4 - b^6 + 8*a^3*c^3 - b^3*(-(4*a*c - b^2)^3)^(1/2) + b^4*c^2 - 6*a*b^2*c^3 + b*c^2*(-(4*a*c -
b^2)^3)^(1/2) - 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*(a^4*b^4 - a^2*b^6 + 16*a^4*
c^4 + 32*a^5*c^3 + 16*a^6*c^2 + 10*a^3*b^4*c - 8*a^5*b^2*c + a^2*b^4*c^2 - 8*a^3*b^2*c^3 - 32*a^4*b^2*c^2)))^(
1/2)*(((8*a^2*c^4 - b^6 + 8*a^3*c^3 - b^3*(-(4*a*c - b^2)^3)^(1/2) + b^4*c^2 - 6*a*b^2*c^3 + b*c^2*(-(4*a*c -
b^2)^3)^(1/2) - 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*(a^4*b^4 - a^2*b^6 + 16*a^4*
c^4 + 32*a^5*c^3 + 16*a^6*c^2 + 10*a^3*b^4*c - 8*a^5*b^2*c + a^2*b^4*c^2 - 8*a^3*b^2*c^3 - 32*a^4*b^2*c^2)))^(
1/2)*(((8*a^2*c^4 - b^6 + 8*a^3*c^3 - b^3*(-(4*a*c - b^2)^3)^(1/2) + b^4*c^2 - 6*a*b^2*c^3 + b*c^2*(-(4*a*c -
b^2)^3)^(1/2) - 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*(a^4*b^4 - a^2*b^6 + 16*a^4*
c^4 + 32*a^5*c^3 + 16*a^6*c^2 + 10*a^3*b^4*c - 8*a^5*b^2*c + a^2*b^4*c^2 - 8*a^3*b^2*c^3 - 32*a^4*b^2*c^2)))^(
1/2)*(tan(x/2)*(256*a^6*c - 512*a*b^6 + 544*a^3*b^4 - 64*a^5*b^2 + 6144*a^3*c^4 + 12288*a^4*c^3 + 6400*a^5*c^2
+ 512*a*b^4*c^2 + 4608*a^2*b^4*c - 3776*a^4*b^2*c - 3584*a^2*b^2*c^3 - 13312*a^3*b^2*c^2) - 128*a^2*b^5 + 96*
a^4*b^3 - 512*a^3*b*c^3 + 800*a^3*b^3*c - 1152*a^4*b*c^2 + 128*a^2*b^3*c^2 - 384*a^5*b*c) - tan(x/2)*(256*a^5*
c - 512*b^6 + 416*a^2*b^4 - 64*a^4*b^2 + 3072*a^2*c^4 + 5632*a^3*c^3 + 2816*a^4*c^2 + 512*b^4*c^2 - 2816*a*b^2
*c^3 - 2368*a^3*b^2*c - 8576*a^2*b^2*c^2 + 3840*a*b^4*c) + 256*a*b^5 - 128*a^3*b^3 - 256*a*b^3*c^2 + 1024*a^2*
b*c^3 - 1568*a^2*b^3*c + 2176*a^3*b*c^2 + 512*a^4*b*c) - 128*b^5 + tan(x/2)*(96*a*b^4 - 1536*a*c^4 - 256*b^4*c
- 1024*a^2*c^3 + 448*a^3*c^2 + 256*b^2*c^3 + 1408*a*b^2*c^2 - 512*a^2*b^2*c) + 32*a^2*b^3 + 128*b^3*c^2 - 131
2*a^2*b*c^2 - 640*a*b*c^3 + 864*a*b^3*c - 128*a^3*b*c) + tan(x/2)*(640*a*c^3 + 32*b^4 + 768*c^4 + 64*a^2*c^2 -
256*b^2*c^2 - 128*a*b^2*c) + 128*b*c^3 - 96*b^3*c + 320*a*b*c^2)*1i + ((8*a^2*c^4 - b^6 + 8*a^3*c^3 - b^3*(-(
4*a*c - b^2)^3)^(1/2) + b^4*c^2 - 6*a*b^2*c^3 + b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 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*(a^4*b^4 - a^2*b^6 + 16*a^4*c^4 + 32*a^5*c^3 + 16*a^6*c^2 + 10*a^3*b^4*c
- 8*a^5*b^2*c + a^2*b^4*c^2 - 8*a^3*b^2*c^3 - 32*a^4*b^2*c^2)))^(1/2)*(tan(x/2)*(640*a*c^3 + 32*b^4 + 768*c^4
+ 64*a^2*c^2 - 256*b^2*c^2 - 128*a*b^2*c) - ((8*a^2*c^4 - b^6 + 8*a^3*c^3 - b^3*(-(4*a*c - b^2)^3)^(1/2) + b^4
*c^2 - 6*a*b^2*c^3 + b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 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*(a^4*b^4 - a^2*b^6 + 16*a^4*c^4 + 32*a^5*c^3 + 16*a^6*c^2 + 10*a^3*b^4*c - 8*a^5*b^2*c + a^2*b^4*c^2
- 8*a^3*b^2*c^3 - 32*a^4*b^2*c^2)))^(1/2)*(((8*a^2*c^4 - b^6 + 8*a^3*c^3 - b^3*(-(4*a*c - b^2)^3)^(1/2) + b^4
*c^2 - 6*a*b^2*c^3 + b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 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*(a^4*b^4 - a^2*b^6 + 16*a^4*c^4 + 32*a^5*c^3 + 16*a^6*c^2 + 10*a^3*b^4*c - 8*a^5*b^2*c + a^2*b^4*c^2
- 8*a^3*b^2*c^3 - 32*a^4*b^2*c^2)))^(1/2)*(tan(x/2)*(256*a^5*c - 512*b^6 + 416*a^2*b^4 - 64*a^4*b^2 + 3072*a^
2*c^4 + 5632*a^3*c^3 + 2816*a^4*c^2 + 512*b^4*c^2 - 2816*a*b^2*c^3 - 2368*a^3*b^2*c - 8576*a^2*b^2*c^2 + 3840*
a*b^4*c) + ((8*a^2*c^4 - b^6 + 8*a^3*c^3 - b^3*(-(4*a*c - b^2)^3)^(1/2) + b^4*c^2 - 6*a*b^2*c^3 + b*c^2*(-(4*a
*c - b^2)^3)^(1/2) - 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*(a^4*b^4 - a^2*b^6 + 16
*a^4*c^4 + 32*a^5*c^3 + 16*a^6*c^2 + 10*a^3*b^4*c - 8*a^5*b^2*c + a^2*b^4*c^2 - 8*a^3*b^2*c^3 - 32*a^4*b^2*c^2
)))^(1/2)*(tan(x/2)*(256*a^6*c - 512*a*b^6 + 544*a^3*b^4 - 64*a^5*b^2 + 6144*a^3*c^4 + 12288*a^4*c^3 + 6400*a^
5*c^2 + 512*a*b^4*c^2 + 4608*a^2*b^4*c - 3776*a^4*b^2*c - 3584*a^2*b^2*c^3 - 13312*a^3*b^2*c^2) - 128*a^2*b^5
+ 96*a^4*b^3 - 512*a^3*b*c^3 + 800*a^3*b^3*c - 1152*a^4*b*c^2 + 128*a^2*b^3*c^2 - 384*a^5*b*c) - 256*a*b^5 + 1
28*a^3*b^3 + 256*a*b^3*c^2 - 1024*a^2*b*c^3 + 1568*a^2*b^3*c - 2176*a^3*b*c^2 - 512*a^4*b*c) - 128*b^5 + tan(x
/2)*(96*a*b^4 - 1536*a*c^4 - 256*b^4*c - 1024*a^2*c^3 + 448*a^3*c^2 + 256*b^2*c^3 + 1408*a*b^2*c^2 - 512*a^2*b
^2*c) + 32*a^2*b^3 + 128*b^3*c^2 - 1312*a^2*b*c^2 - 640*a*b*c^3 + 864*a*b^3*c - 128*a^3*b*c) + 128*b*c^3 - 96*
b^3*c + 320*a*b*c^2)*1i)/(((8*a^2*c^4 - b^6 + 8*a^3*c^3 - b^3*(-(4*a*c - b^2)^3)^(1/2) + b^4*c^2 - 6*a*b^2*c^3
+ b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 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*(a^4*b^4
- a^2*b^6 + 16*a^4*c^4 + 32*a^5*c^3 + 16*a^6*c^2 + 10*a^3*b^4*c - 8*a^5*b^2*c + a^2*b^4*c^2 - 8*a^3*b^2*c^3 -
32*a^4*b^2*c^2)))^(1/2)*(tan(x/2)*(640*a*c^3 + 32*b^4 + 768*c^4 + 64*a^2*c^2 - 256*b^2*c^2 - 128*a*b^2*c) - (
(8*a^2*c^4 - b^6 + 8*a^3*c^3 - b^3*(-(4*a*c - b...

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