3.27 \(\int \frac{\csc ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx\)

Optimal. Leaf size=184 \[ -\frac{2 b^2 \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{a^4 \sqrt{a^2+b^2}}-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{a^4}-\frac{\tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{2 a^2 \sqrt{a^2+b^2}}-\frac{2 b}{a^3 (a \cos (x)+b \sin (x))}-\frac{b \cos (x)-a \sin (x)}{2 a^2 (a \cos (x)+b \sin (x))^2}+\frac{3 b \tanh ^{-1}(\cos (x))}{a^4}-\frac{\csc (x)}{a^3} \]

[Out]

(3*b*ArcTanh[Cos[x]])/a^4 - ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]]/(2*a^2*Sqrt[a^2 + b^2]) - (2*b^2*Ar
cTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^4*Sqrt[a^2 + b^2]) - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[x] - a*S
in[x])/Sqrt[a^2 + b^2]])/a^4 - Csc[x]/a^3 - (b*Cos[x] - a*Sin[x])/(2*a^2*(a*Cos[x] + b*Sin[x])^2) - (2*b)/(a^3
*(a*Cos[x] + b*Sin[x]))

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Rubi [A]  time = 0.224382, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {3105, 3076, 3074, 206, 3103, 3770, 3093} \[ -\frac{2 b^2 \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{a^4 \sqrt{a^2+b^2}}-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{a^4}-\frac{\tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{2 a^2 \sqrt{a^2+b^2}}-\frac{2 b}{a^3 (a \cos (x)+b \sin (x))}-\frac{b \cos (x)-a \sin (x)}{2 a^2 (a \cos (x)+b \sin (x))^2}+\frac{3 b \tanh ^{-1}(\cos (x))}{a^4}-\frac{\csc (x)}{a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*ArcTanh[Cos[x]])/a^4 - ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]]/(2*a^2*Sqrt[a^2 + b^2]) - (2*b^2*Ar
cTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^4*Sqrt[a^2 + b^2]) - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[x] - a*S
in[x])/Sqrt[a^2 + b^2]])/a^4 - Csc[x]/a^3 - (b*Cos[x] - a*Sin[x])/(2*a^2*(a*Cos[x] + b*Sin[x])^2) - (2*b)/(a^3
*(a*Cos[x] + b*Sin[x]))

Rule 3105

Int[sin[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbo
l] :> Dist[(a^2 + b^2)/a^2, Int[Sin[c + d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^n, x], x] + (Dist[1/a^2
, Int[Sin[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] - Dist[(2*b)/a^2, Int[Sin[c + d*x]^(m +
1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n
, -1] && LtQ[m, -1]

Rule 3076

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[((b*Cos[c + d*x] -
a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1))/(d*(n + 1)*(a^2 + b^2)), x] + Dist[(n + 2)/((n + 1
)*(a^2 + b^2)), Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b
^2, 0] && LtQ[n, -1] && NeQ[n, -2]

Rule 3074

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

Rule 206

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

Rule 3103

Int[sin[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[Sin[c + d*x]^(m + 1)/(a*d*(m + 1)), x] + (-Dist[b/a^2, Int[Sin[c + d*x]^(m + 1), x], x] + Dist[(a^2 + b^
2)/a^2, Int[Sin[c + d*x]^(m + 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 + b^2, 0] && LtQ[m, -1]

Rule 3770

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

Rule 3093

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_)/sin[(c_.) + (d_.)*(x_)], x_Symbol] :>
-Simp[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/(a*d*(n + 1)), x] + (Dist[1/a^2, Int[(a*Cos[c + d*x] + b*Sin[
c + d*x])^(n + 2)/Sin[c + d*x], x], x] - Dist[b/a^2, Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x], x]) /;
FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rubi steps

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

Mathematica [A]  time = 0.743807, size = 193, normalized size = 1.05 \[ \frac{\csc ^3(x) (a \cos (x)+b \sin (x)) \left (a \left (a^2+b^2\right ) \sin (x)+\frac{6 \left (a^2+2 b^2\right ) (a \cos (x)+b \sin (x))^2 \tanh ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )-b}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}}-5 a b (a \cos (x)+b \sin (x))+6 b \log \left (\cos \left (\frac{x}{2}\right )\right ) (a \cos (x)+b \sin (x))^2-6 b \log \left (\sin \left (\frac{x}{2}\right )\right ) (a \cos (x)+b \sin (x))^2-a \tan \left (\frac{x}{2}\right ) (a \cos (x)+b \sin (x))^2-a \cot \left (\frac{x}{2}\right ) (a \cos (x)+b \sin (x))^2\right )}{2 a^4 (a \cot (x)+b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[x]^3*(a*Cos[x] + b*Sin[x])*(a*(a^2 + b^2)*Sin[x] - 5*a*b*(a*Cos[x] + b*Sin[x]) + (6*(a^2 + 2*b^2)*ArcTanh
[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]]*(a*Cos[x] + b*Sin[x])^2)/Sqrt[a^2 + b^2] - a*Cot[x/2]*(a*Cos[x] + b*Sin[x]
)^2 + 6*b*Log[Cos[x/2]]*(a*Cos[x] + b*Sin[x])^2 - 6*b*Log[Sin[x/2]]*(a*Cos[x] + b*Sin[x])^2 - a*(a*Cos[x] + b*
Sin[x])^2*Tan[x/2]))/(2*a^4*(b + a*Cot[x])^3)

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Maple [A]  time = 0.162, size = 333, normalized size = 1.8 \begin{align*} -{\frac{1}{2\,{a}^{3}}\tan \left ({\frac{x}{2}} \right ) }+{\frac{1}{a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}a-2\,b\tan \left ( x/2 \right ) -a \right ) ^{-2}}+6\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{3}{b}^{2}}{{a}^{3} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}a-2\,b\tan \left ( x/2 \right ) -a \right ) ^{2}}}+5\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}b}{{a}^{2} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}a-2\,b\tan \left ( x/2 \right ) -a \right ) ^{2}}}-10\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}{b}^{3}}{{a}^{4} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}a-2\,b\tan \left ( x/2 \right ) -a \right ) ^{2}}}+{\frac{1}{a}\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}a-2\,b\tan \left ( x/2 \right ) -a \right ) ^{-2}}-14\,{\frac{\tan \left ( x/2 \right ){b}^{2}}{{a}^{3} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}a-2\,b\tan \left ( x/2 \right ) -a \right ) ^{2}}}-5\,{\frac{b}{{a}^{2} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}a-2\,b\tan \left ( x/2 \right ) -a \right ) ^{2}}}+3\,{\frac{1}{\sqrt{{a}^{2}+{b}^{2}}{a}^{2}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tan \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }+6\,{\frac{{b}^{2}}{{a}^{4}\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tan \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }-{\frac{1}{2\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-3\,{\frac{b\ln \left ( \tan \left ( x/2 \right ) \right ) }{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^2/(a*cos(x)+b*sin(x))^3,x)

[Out]

-1/2/a^3*tan(1/2*x)+1/a/(tan(1/2*x)^2*a-2*b*tan(1/2*x)-a)^2*tan(1/2*x)^3+6/a^3/(tan(1/2*x)^2*a-2*b*tan(1/2*x)-
a)^2*tan(1/2*x)^3*b^2+5/a^2/(tan(1/2*x)^2*a-2*b*tan(1/2*x)-a)^2*tan(1/2*x)^2*b-10/a^4/(tan(1/2*x)^2*a-2*b*tan(
1/2*x)-a)^2*tan(1/2*x)^2*b^3+1/a/(tan(1/2*x)^2*a-2*b*tan(1/2*x)-a)^2*tan(1/2*x)-14/a^3/(tan(1/2*x)^2*a-2*b*tan
(1/2*x)-a)^2*tan(1/2*x)*b^2-5/a^2/(tan(1/2*x)^2*a-2*b*tan(1/2*x)-a)^2*b+3/a^2/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a
*tan(1/2*x)-2*b)/(a^2+b^2)^(1/2))+6/a^4/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan(1/2*x)-2*b)/(a^2+b^2)^(1/2))*b^2-
1/2/a^3/tan(1/2*x)-3/a^4*b*ln(tan(1/2*x))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2/(a*cos(x)+b*sin(x))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.721376, size = 1084, normalized size = 5.89 \begin{align*} -\frac{2 \, a^{5} - 10 \, a^{3} b^{2} - 12 \, a b^{4} - 6 \,{\left (a^{5} - a^{3} b^{2} - 2 \, a b^{4}\right )} \cos \left (x\right )^{2} - 18 \,{\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (x\right ) \sin \left (x\right ) - 3 \,{\left (2 \,{\left (a^{3} b + 2 \, a b^{3}\right )} \cos \left (x\right )^{3} - 2 \,{\left (a^{3} b + 2 \, a b^{3}\right )} \cos \left (x\right ) -{\left (a^{2} b^{2} + 2 \, b^{4} +{\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )} \sqrt{a^{2} + b^{2}} \log \left (-\frac{2 \, a b \cos \left (x\right ) \sin \left (x\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt{a^{2} + b^{2}}{\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right ) - 6 \,{\left (2 \,{\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{3} - 2 \,{\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right ) -{\left (a^{2} b^{3} + b^{5} +{\left (a^{4} b - b^{5}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 6 \,{\left (2 \,{\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{3} - 2 \,{\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right ) -{\left (a^{2} b^{3} + b^{5} +{\left (a^{4} b - b^{5}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{4 \,{\left (2 \,{\left (a^{7} b + a^{5} b^{3}\right )} \cos \left (x\right )^{3} - 2 \,{\left (a^{7} b + a^{5} b^{3}\right )} \cos \left (x\right ) -{\left (a^{6} b^{2} + a^{4} b^{4} +{\left (a^{8} - a^{4} b^{4}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2/(a*cos(x)+b*sin(x))^3,x, algorithm="fricas")

[Out]

-1/4*(2*a^5 - 10*a^3*b^2 - 12*a*b^4 - 6*(a^5 - a^3*b^2 - 2*a*b^4)*cos(x)^2 - 18*(a^4*b + a^2*b^3)*cos(x)*sin(x
) - 3*(2*(a^3*b + 2*a*b^3)*cos(x)^3 - 2*(a^3*b + 2*a*b^3)*cos(x) - (a^2*b^2 + 2*b^4 + (a^4 + a^2*b^2 - 2*b^4)*
cos(x)^2)*sin(x))*sqrt(a^2 + b^2)*log(-(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 - 2*a^2 - b^2 + 2*sqrt(a^2
+ b^2)*(b*cos(x) - a*sin(x)))/(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2)) - 6*(2*(a^3*b^2 + a*b^4)*cos
(x)^3 - 2*(a^3*b^2 + a*b^4)*cos(x) - (a^2*b^3 + b^5 + (a^4*b - b^5)*cos(x)^2)*sin(x))*log(1/2*cos(x) + 1/2) +
6*(2*(a^3*b^2 + a*b^4)*cos(x)^3 - 2*(a^3*b^2 + a*b^4)*cos(x) - (a^2*b^3 + b^5 + (a^4*b - b^5)*cos(x)^2)*sin(x)
)*log(-1/2*cos(x) + 1/2))/(2*(a^7*b + a^5*b^3)*cos(x)^3 - 2*(a^7*b + a^5*b^3)*cos(x) - (a^6*b^2 + a^4*b^4 + (a
^8 - a^4*b^4)*cos(x)^2)*sin(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.27457, size = 286, normalized size = 1.55 \begin{align*} -\frac{3 \, b \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{a^{4}} - \frac{\tan \left (\frac{1}{2} \, x\right )}{2 \, a^{3}} - \frac{3 \,{\left (a^{2} + 2 \, b^{2}\right )} \log \left (\frac{{\left | 2 \, a \tan \left (\frac{1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac{1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{2 \, \sqrt{a^{2} + b^{2}} a^{4}} + \frac{6 \, b \tan \left (\frac{1}{2} \, x\right ) - a}{2 \, a^{4} \tan \left (\frac{1}{2} \, x\right )} + \frac{a^{3} \tan \left (\frac{1}{2} \, x\right )^{3} + 6 \, a b^{2} \tan \left (\frac{1}{2} \, x\right )^{3} + 5 \, a^{2} b \tan \left (\frac{1}{2} \, x\right )^{2} - 10 \, b^{3} \tan \left (\frac{1}{2} \, x\right )^{2} + a^{3} \tan \left (\frac{1}{2} \, x\right ) - 14 \, a b^{2} \tan \left (\frac{1}{2} \, x\right ) - 5 \, a^{2} b}{{\left (a \tan \left (\frac{1}{2} \, x\right )^{2} - 2 \, b \tan \left (\frac{1}{2} \, x\right ) - a\right )}^{2} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*b*log(abs(tan(1/2*x)))/a^4 - 1/2*tan(1/2*x)/a^3 - 3/2*(a^2 + 2*b^2)*log(abs(2*a*tan(1/2*x) - 2*b - 2*sqrt(a
^2 + b^2))/abs(2*a*tan(1/2*x) - 2*b + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^4) + 1/2*(6*b*tan(1/2*x) - a)/(a^
4*tan(1/2*x)) + (a^3*tan(1/2*x)^3 + 6*a*b^2*tan(1/2*x)^3 + 5*a^2*b*tan(1/2*x)^2 - 10*b^3*tan(1/2*x)^2 + a^3*ta
n(1/2*x) - 14*a*b^2*tan(1/2*x) - 5*a^2*b)/((a*tan(1/2*x)^2 - 2*b*tan(1/2*x) - a)^2*a^4)