Integrand size = 16, antiderivative size = 118 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {\text {arctanh}(\cos (x))}{2 a^2}-\frac {2 b^2 \text {arctanh}(\cos (x))}{a^4}-\frac {\left (a^2+b^2\right ) \text {arctanh}(\cos (x))}{a^4}+\frac {3 b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^4}+\frac {2 b \csc (x)}{a^3}-\frac {\cot (x) \csc (x)}{2 a^2}+\frac {a^2+b^2}{a^3 (a \cos (x)+b \sin (x))} \]
-1/2*arctanh(cos(x))/a^2-2*b^2*arctanh(cos(x))/a^4-(a^2+b^2)*arctanh(cos(x ))/a^4+2*b*csc(x)/a^3-1/2*cot(x)*csc(x)/a^2+(a^2+b^2)/a^3/(a*cos(x)+b*sin( x))+3*b*arctanh((b*cos(x)-a*sin(x))/(a^2+b^2)^(1/2))*(a^2+b^2)^(1/2)/a^4
Leaf count is larger than twice the leaf count of optimal. \(270\) vs. \(2(118)=236\).
Time = 2.42 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.29 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {-48 b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right ) (b+a \cot (x))+8 a^3 \csc (x)+8 a b^2 \csc (x)-12 a^2 b \log \left (\cos \left (\frac {x}{2}\right )\right )-24 b^3 \log \left (\cos \left (\frac {x}{2}\right )\right )-12 a^3 \cot (x) \log \left (\cos \left (\frac {x}{2}\right )\right )-24 a b^2 \cot (x) \log \left (\cos \left (\frac {x}{2}\right )\right )+12 a^2 b \log \left (\sin \left (\frac {x}{2}\right )\right )+24 b^3 \log \left (\sin \left (\frac {x}{2}\right )\right )+12 a^3 \cot (x) \log \left (\sin \left (\frac {x}{2}\right )\right )+24 a b^2 \cot (x) \log \left (\sin \left (\frac {x}{2}\right )\right )+a^2 b \sec ^2\left (\frac {x}{2}\right )+a^3 \cot (x) \sec ^2\left (\frac {x}{2}\right )-a \csc ^2\left (\frac {x}{2}\right ) \left (-4 a b \cos (x)+a^2 \cot (x)+b (a-4 b \sin (x))\right )+8 a b^2 \tan \left (\frac {x}{2}\right )+8 a^2 b \cot (x) \tan \left (\frac {x}{2}\right )}{8 a^4 (b+a \cot (x))} \]
(-48*b*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]]*(b + a*C ot[x]) + 8*a^3*Csc[x] + 8*a*b^2*Csc[x] - 12*a^2*b*Log[Cos[x/2]] - 24*b^3*L og[Cos[x/2]] - 12*a^3*Cot[x]*Log[Cos[x/2]] - 24*a*b^2*Cot[x]*Log[Cos[x/2]] + 12*a^2*b*Log[Sin[x/2]] + 24*b^3*Log[Sin[x/2]] + 12*a^3*Cot[x]*Log[Sin[x /2]] + 24*a*b^2*Cot[x]*Log[Sin[x/2]] + a^2*b*Sec[x/2]^2 + a^3*Cot[x]*Sec[x /2]^2 - a*Csc[x/2]^2*(-4*a*b*Cos[x] + a^2*Cot[x] + b*(a - 4*b*Sin[x])) + 8 *a*b^2*Tan[x/2] + 8*a^2*b*Cot[x]*Tan[x/2])/(8*a^4*(b + a*Cot[x]))
Time = 0.94 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.32, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {3042, 3584, 3042, 3572, 3042, 3553, 219, 3582, 3042, 3553, 219, 4255, 3042, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (x)^3 (a \cos (x)+b \sin (x))^2}dx\) |
| \(\Big \downarrow \) 3584 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {\csc (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2}-\frac {2 b \int \frac {\csc ^2(x)}{a \cos (x)+b \sin (x)}dx}{a^2}+\frac {\int \csc ^3(x)dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {1}{\sin (x) (a \cos (x)+b \sin (x))^2}dx}{a^2}-\frac {2 b \int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))}dx}{a^2}+\frac {\int \csc (x)^3dx}{a^2}\) |
| \(\Big \downarrow \) 3572 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2}+\frac {\int \csc (x)dx}{a^2}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}-\frac {2 b \int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))}dx}{a^2}+\frac {\int \csc (x)^3dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2}+\frac {\int \csc (x)dx}{a^2}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}-\frac {2 b \int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))}dx}{a^2}+\frac {\int \csc (x)^3dx}{a^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {b \int \frac {1}{a^2+b^2-(b \cos (x)-a \sin (x))^2}d(b \cos (x)-a \sin (x))}{a^2}+\frac {\int \csc (x)dx}{a^2}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}-\frac {2 b \int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))}dx}{a^2}+\frac {\int \csc (x)^3dx}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\int \csc (x)dx}{a^2}+\frac {b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}-\frac {2 b \int \frac {1}{\sin (x)^2 (a \cos (x)+b \sin (x))}dx}{a^2}+\frac {\int \csc (x)^3dx}{a^2}\) |
| \(\Big \downarrow \) 3582 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\int \csc (x)dx}{a^2}+\frac {b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}-\frac {2 b \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2}-\frac {b \int \csc (x)dx}{a^2}-\frac {\csc (x)}{a}\right )}{a^2}+\frac {\int \csc (x)^3dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\int \csc (x)dx}{a^2}+\frac {b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}-\frac {2 b \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2}-\frac {b \int \csc (x)dx}{a^2}-\frac {\csc (x)}{a}\right )}{a^2}+\frac {\int \csc (x)^3dx}{a^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\int \csc (x)dx}{a^2}+\frac {b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}-\frac {2 b \left (-\frac {\left (a^2+b^2\right ) \int \frac {1}{a^2+b^2-(b \cos (x)-a \sin (x))^2}d(b \cos (x)-a \sin (x))}{a^2}-\frac {b \int \csc (x)dx}{a^2}-\frac {\csc (x)}{a}\right )}{a^2}+\frac {\int \csc (x)^3dx}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\int \csc (x)dx}{a^2}+\frac {b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}-\frac {2 b \left (-\frac {b \int \csc (x)dx}{a^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2}-\frac {\csc (x)}{a}\right )}{a^2}+\frac {\int \csc (x)^3dx}{a^2}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\int \csc (x)dx}{a^2}+\frac {b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}-\frac {2 b \left (-\frac {b \int \csc (x)dx}{a^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2}-\frac {\csc (x)}{a}\right )}{a^2}+\frac {\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\int \csc (x)dx}{a^2}+\frac {b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}-\frac {2 b \left (-\frac {b \int \csc (x)dx}{a^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2}-\frac {\csc (x)}{a}\right )}{a^2}+\frac {\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)}{a^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}-\frac {\text {arctanh}(\cos (x))}{a^2}+\frac {1}{a (a \cos (x)+b \sin (x))}\right )}{a^2}-\frac {2 b \left (-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2}+\frac {b \text {arctanh}(\cos (x))}{a^2}-\frac {\csc (x)}{a}\right )}{a^2}+\frac {-\frac {1}{2} \text {arctanh}(\cos (x))-\frac {1}{2} \cot (x) \csc (x)}{a^2}\) |
(-2*b*((b*ArcTanh[Cos[x]])/a^2 - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[x] - a*Si n[x])/Sqrt[a^2 + b^2]])/a^2 - Csc[x]/a))/a^2 + (-1/2*ArcTanh[Cos[x]] - (Co t[x]*Csc[x])/2)/a^2 + ((a^2 + b^2)*(-(ArcTanh[Cos[x]]/a^2) + (b*ArcTanh[(b *Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2*Sqrt[a^2 + b^2]) + 1/(a*(a*Cos[ x] + b*Sin[x]))))/a^2
3.1.21.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x _Symbol] :> Simp[-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]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_)/si n[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-(a*Cos[c + d*x] + b*Sin[c + d*x]) ^(n + 1)/(a*d*(n + 1)), x] + (Simp[1/a^2 Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2)/Sin[c + d*x], x], x] - Simp[b/a^2 Int[(a*Cos[c + d*x] + b*S in[c + d*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0 ] && LtQ[n, -1]
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] + (-Simp[b/a^2 Int[Sin[c + d*x]^(m + 1), x], x] + Simp[(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]
Int[sin[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(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] + (Simp[1/a^2 I nt[Sin[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] - Simp[ 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]
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] + Simp[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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.71 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.34
| method | result | size |
| default | \(-\frac {1}{8 a^{2} \tan \left (\frac {x}{2}\right )^{2}}+\frac {\left (6 a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tan \left (\frac {x}{2}\right )}+\frac {\frac {\tan \left (\frac {x}{2}\right )^{2} a}{2}+4 b \tan \left (\frac {x}{2}\right )}{4 a^{3}}+\frac {\frac {4 \left (\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tan \left (\frac {x}{2}\right )-\frac {\left (a^{2}+b^{2}\right ) a}{2}\right )}{\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a}-6 b \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{4}}\) | \(158\) |
| risch | \(\frac {{\mathrm e}^{i x} \left (3 i a b \,{\mathrm e}^{4 i x}+3 a^{2} {\mathrm e}^{4 i x}+6 b^{2} {\mathrm e}^{4 i x}-2 a^{2} {\mathrm e}^{2 i x}-12 b^{2} {\mathrm e}^{2 i x}-3 i b a +3 a^{2}+6 b^{2}\right )}{\left ({\mathrm e}^{2 i x}-1\right )^{2} \left (-i b \,{\mathrm e}^{2 i x}+a \,{\mathrm e}^{2 i x}+i b +a \right ) a^{3}}-\frac {3 i \sqrt {-a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i x}-\frac {\sqrt {-a^{2}-b^{2}}\, \left (i b +a \right )}{a^{2}+b^{2}}\right )}{a^{4}}+\frac {3 i \sqrt {-a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i x}+\frac {\sqrt {-a^{2}-b^{2}}\, \left (i b +a \right )}{a^{2}+b^{2}}\right )}{a^{4}}+\frac {3 \ln \left ({\mathrm e}^{i x}-1\right )}{2 a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i x}-1\right ) b^{2}}{a^{4}}-\frac {3 \ln \left ({\mathrm e}^{i x}+1\right )}{2 a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i x}+1\right ) b^{2}}{a^{4}}\) | \(297\) |
-1/8/a^2/tan(1/2*x)^2+1/4/a^4*(6*a^2+12*b^2)*ln(tan(1/2*x))+b/a^3/tan(1/2* x)+1/4/a^3*(1/2*tan(1/2*x)^2*a+4*b*tan(1/2*x))+4/a^4*(((-1/2*a^2*b-1/2*b^3 )*tan(1/2*x)-1/2*(a^2+b^2)*a)/(tan(1/2*x)^2*a-2*b*tan(1/2*x)-a)-3/2*b*(a^2 +b^2)^(1/2)*arctanh(1/2*(2*a*tan(1/2*x)-2*b)/(a^2+b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (110) = 220\).
Time = 0.32 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.92 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {6 \, a^{2} b \cos \left (x\right ) \sin \left (x\right ) + 4 \, a^{3} + 12 \, a b^{2} - 6 \, {\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (x\right )^{2} - 6 \, {\left (a b \cos \left (x\right )^{3} - a b \cos \left (x\right ) + {\left (b^{2} \cos \left (x\right )^{2} - b^{2}\right )} \sin \left (x\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right ) + 3 \, {\left ({\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (x\right )^{3} - {\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (x\right ) - {\left (a^{2} b + 2 \, b^{3} - {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 3 \, {\left ({\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (x\right )^{3} - {\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (x\right ) - {\left (a^{2} b + 2 \, b^{3} - {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{4 \, {\left (a^{5} \cos \left (x\right )^{3} - a^{5} \cos \left (x\right ) + {\left (a^{4} b \cos \left (x\right )^{2} - a^{4} b\right )} \sin \left (x\right )\right )}} \]
-1/4*(6*a^2*b*cos(x)*sin(x) + 4*a^3 + 12*a*b^2 - 6*(a^3 + 2*a*b^2)*cos(x)^ 2 - 6*(a*b*cos(x)^3 - a*b*cos(x) + (b^2*cos(x)^2 - b^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*sq rt(a^2 + b^2)*(b*cos(x) - a*sin(x)))/(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*co s(x)^2 + b^2)) + 3*((a^3 + 2*a*b^2)*cos(x)^3 - (a^3 + 2*a*b^2)*cos(x) - (a ^2*b + 2*b^3 - (a^2*b + 2*b^3)*cos(x)^2)*sin(x))*log(1/2*cos(x) + 1/2) - 3 *((a^3 + 2*a*b^2)*cos(x)^3 - (a^3 + 2*a*b^2)*cos(x) - (a^2*b + 2*b^3 - (a^ 2*b + 2*b^3)*cos(x)^2)*sin(x))*log(-1/2*cos(x) + 1/2))/(a^5*cos(x)^3 - a^5 *cos(x) + (a^4*b*cos(x)^2 - a^4*b)*sin(x))
\[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\int \frac {\csc ^{3}{\left (x \right )}}{\left (a \cos {\left (x \right )} + b \sin {\left (x \right )}\right )^{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (110) = 220\).
Time = 0.31 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.05 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {a^{3} - \frac {6 \, a^{2} b \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {{\left (17 \, a^{3} + 32 \, a b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {8 \, {\left (a^{2} b + 2 \, b^{3}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{8 \, {\left (\frac {a^{5} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {2 \, a^{4} b \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {a^{5} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}\right )}} + \frac {\frac {8 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a^{3}} + \frac {3 \, {\left (a^{2} + 2 \, b^{2}\right )} \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, a^{4}} + \frac {3 \, {\left (a^{2} b + b^{3}\right )} \log \left (\frac {b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} \]
-1/8*(a^3 - 6*a^2*b*sin(x)/(cos(x) + 1) - (17*a^3 + 32*a*b^2)*sin(x)^2/(co s(x) + 1)^2 - 8*(a^2*b + 2*b^3)*sin(x)^3/(cos(x) + 1)^3)/(a^5*sin(x)^2/(co s(x) + 1)^2 + 2*a^4*b*sin(x)^3/(cos(x) + 1)^3 - a^5*sin(x)^4/(cos(x) + 1)^ 4) + 1/8*(8*b*sin(x)/(cos(x) + 1) + a*sin(x)^2/(cos(x) + 1)^2)/a^3 + 3/2*( a^2 + 2*b^2)*log(sin(x)/(cos(x) + 1))/a^4 + 3*(a^2*b + b^3)*log((b - a*sin (x)/(cos(x) + 1) + sqrt(a^2 + b^2))/(b - a*sin(x)/(cos(x) + 1) - sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^4)
Time = 0.31 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.82 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {3 \, {\left (a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{4}} + \frac {3 \, {\left (a^{2} b + b^{3}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} + \frac {a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 8 \, a b \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{4}} - \frac {2 \, {\left (a^{2} b \tan \left (\frac {1}{2} \, x\right ) + b^{3} \tan \left (\frac {1}{2} \, x\right ) + a^{3} + a b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, x\right ) - a\right )} a^{4}} - \frac {18 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 36 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, x\right ) + a^{2}}{8 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{2}} \]
3/2*(a^2 + 2*b^2)*log(abs(tan(1/2*x)))/a^4 + 3*(a^2*b + b^3)*log(abs(2*a*t an(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) + 1/8*(a^2*tan(1/2*x)^2 + 8*a*b*tan(1/2*x) )/a^4 - 2*(a^2*b*tan(1/2*x) + b^3*tan(1/2*x) + a^3 + a*b^2)/((a*tan(1/2*x) ^2 - 2*b*tan(1/2*x) - a)*a^4) - 1/8*(18*a^2*tan(1/2*x)^2 + 36*b^2*tan(1/2* x)^2 - 8*a*b*tan(1/2*x) + a^2)/(a^4*tan(1/2*x)^2)
Time = 22.09 (sec) , antiderivative size = 511, normalized size of antiderivative = 4.33 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (\frac {17\,a^2}{2}+16\,b^2\right )-\frac {a^2}{2}+3\,a\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (a^2\,b+2\,b^3\right )}{a}}{-4\,a^4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+4\,a^4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+8\,b\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a^2}+\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\,\left (3\,a^2+6\,b^2\right )}{2\,a^4}+\frac {b\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^3}-\frac {6\,b\,\mathrm {atanh}\left (\frac {54\,b^2\,\sqrt {a^2+b^2}}{18\,a^2\,b+90\,b^3+\frac {72\,b^5}{a^2}+\frac {216\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{a}+\frac {144\,b^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^3}+72\,a\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )}+\frac {72\,b^4\,\sqrt {a^2+b^2}}{18\,a^4\,b+72\,b^5+90\,a^2\,b^3+72\,a^3\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {144\,b^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{a}+216\,a\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}+\frac {144\,b^3\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}}{216\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )+90\,a\,b^3+18\,a^3\,b+\frac {72\,b^5}{a}+72\,a^2\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {144\,b^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2}}+\frac {144\,b^5\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}}{18\,a^5\,b+72\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^4\,b^2+90\,a^3\,b^3+216\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^2\,b^4+72\,a\,b^5+144\,\mathrm {tan}\left (\frac {x}{2}\right )\,b^6}+\frac {18\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}}{18\,a\,b+72\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {90\,b^3}{a}+\frac {72\,b^5}{a^3}+\frac {216\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2}+\frac {144\,b^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^4}}\right )\,\sqrt {a^2+b^2}}{a^4} \]
(tan(x/2)^2*((17*a^2)/2 + 16*b^2) - a^2/2 + 3*a*b*tan(x/2) + (4*tan(x/2)^3 *(a^2*b + 2*b^3))/a)/(4*a^4*tan(x/2)^2 - 4*a^4*tan(x/2)^4 + 8*a^3*b*tan(x/ 2)^3) + tan(x/2)^2/(8*a^2) + (log(tan(x/2))*(3*a^2 + 6*b^2))/(2*a^4) + (b* tan(x/2))/a^3 - (6*b*atanh((54*b^2*(a^2 + b^2)^(1/2))/(18*a^2*b + 90*b^3 + (72*b^5)/a^2 + (216*b^4*tan(x/2))/a + (144*b^6*tan(x/2))/a^3 + 72*a*b^2*t an(x/2)) + (72*b^4*(a^2 + b^2)^(1/2))/(18*a^4*b + 72*b^5 + 90*a^2*b^3 + 72 *a^3*b^2*tan(x/2) + (144*b^6*tan(x/2))/a + 216*a*b^4*tan(x/2)) + (144*b^3* tan(x/2)*(a^2 + b^2)^(1/2))/(216*b^4*tan(x/2) + 90*a*b^3 + 18*a^3*b + (72* b^5)/a + 72*a^2*b^2*tan(x/2) + (144*b^6*tan(x/2))/a^2) + (144*b^5*tan(x/2) *(a^2 + b^2)^(1/2))/(144*b^6*tan(x/2) + 72*a*b^5 + 18*a^5*b + 90*a^3*b^3 + 216*a^2*b^4*tan(x/2) + 72*a^4*b^2*tan(x/2)) + (18*b*tan(x/2)*(a^2 + b^2)^ (1/2))/(18*a*b + 72*b^2*tan(x/2) + (90*b^3)/a + (72*b^5)/a^3 + (216*b^4*ta n(x/2))/a^2 + (144*b^6*tan(x/2))/a^4))*(a^2 + b^2)^(1/2))/a^4