Integrand size = 16, antiderivative size = 117 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {3 b \cot (x)}{a^4}-\frac {\cot ^2(x)}{2 a^3}+\frac {2 \left (a^2+3 b^2\right ) \log (\tan (x))}{a^5}-\frac {2 \left (a^2+3 b^2\right ) \log (a+b \tan (x))}{a^5}+\frac {\left (a^2+b^2\right )^2}{2 a^3 b^2 (a+b \tan (x))^2}-\frac {\left (a^2-3 b^2\right ) \left (a^2+b^2\right )}{a^4 b^2 (a+b \tan (x))} \] Output:
3*b*cot(x)/a^4-1/2*cot(x)^2/a^3+2*(a^2+3*b^2)*ln(tan(x))/a^5-2*(a^2+3*b^2) *ln(a+b*tan(x))/a^5+1/2*(a^2+b^2)^2/a^3/b^2/(a+b*tan(x))^2-(a^2-3*b^2)*(a^ 2+b^2)/a^4/b^2/(a+b*tan(x))
Time = 1.25 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.78 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {6 a^3 b \cot ^3(x)+a^4 \csc ^2(x)-2 a b \cot (x) \left (3 a^2+a^2 \csc ^2(x)-4 \left (a^2+3 b^2\right ) \log (\sin (x))+4 a^2 \log (a \cos (x)+b \sin (x))+12 b^2 \log (a \cos (x)+b \sin (x))\right )+2 b^2 \left (-3 \left (a^2+b^2\right )+2 \left (a^2+3 b^2\right ) \log (\sin (x))-2 \left (a^2+3 b^2\right ) \log (a \cos (x)+b \sin (x))\right )+\cot ^2(x) \left (-a^4 \csc ^2(x)+4 a^2 \left (3 b^2+\left (a^2+3 b^2\right ) \log (\sin (x))-\left (a^2+3 b^2\right ) \log (a \cos (x)+b \sin (x))\right )\right )}{2 a^5 (b+a \cot (x))^2} \] Input:
Integrate[Csc[x]^3/(a*Cos[x] + b*Sin[x])^3,x]
Output:
(6*a^3*b*Cot[x]^3 + a^4*Csc[x]^2 - 2*a*b*Cot[x]*(3*a^2 + a^2*Csc[x]^2 - 4* (a^2 + 3*b^2)*Log[Sin[x]] + 4*a^2*Log[a*Cos[x] + b*Sin[x]] + 12*b^2*Log[a* Cos[x] + b*Sin[x]]) + 2*b^2*(-3*(a^2 + b^2) + 2*(a^2 + 3*b^2)*Log[Sin[x]] - 2*(a^2 + 3*b^2)*Log[a*Cos[x] + b*Sin[x]]) + Cot[x]^2*(-(a^4*Csc[x]^2) + 4*a^2*(3*b^2 + (a^2 + 3*b^2)*Log[Sin[x]] - (a^2 + 3*b^2)*Log[a*Cos[x] + b* Sin[x]])))/(2*a^5*(b + a*Cot[x])^2)
Time = 0.35 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3566, 522, 2009}
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.
| \(\displaystyle \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (x)^3 (a \cos (x)+b \sin (x))^3}dx\) |
| \(\Big \downarrow \) 3566 |
| \(\displaystyle \int \frac {\left (\tan ^2(x)+1\right )^2 \cot ^3(x)}{(a+b \tan (x))^3}d\tan (x)\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \int \left (-\frac {3 b \cot ^2(x)}{a^4}+\frac {\cot ^3(x)}{a^3}-\frac {2 b \left (a^2+3 b^2\right )}{a^5 (a+b \tan (x))}+\frac {2 \left (a^2+3 b^2\right ) \cot (x)}{a^5}+\frac {a^4-2 a^2 b^2-3 b^4}{a^4 b (a+b \tan (x))^2}-\frac {\left (a^2+b^2\right )^2}{a^3 b (a+b \tan (x))^3}\right )d\tan (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 b \cot (x)}{a^4}-\frac {\cot ^2(x)}{2 a^3}+\frac {2 \left (a^2+3 b^2\right ) \log (\tan (x))}{a^5}-\frac {2 \left (a^2+3 b^2\right ) \log (a+b \tan (x))}{a^5}-\frac {\left (a^2-3 b^2\right ) \left (a^2+b^2\right )}{a^4 b^2 (a+b \tan (x))}+\frac {\left (a^2+b^2\right )^2}{2 a^3 b^2 (a+b \tan (x))^2}\) |
Input:
Int[Csc[x]^3/(a*Cos[x] + b*Sin[x])^3,x]
Output:
(3*b*Cot[x])/a^4 - Cot[x]^2/(2*a^3) + (2*(a^2 + 3*b^2)*Log[Tan[x]])/a^5 - (2*(a^2 + 3*b^2)*Log[a + b*Tan[x]])/a^5 + (a^2 + b^2)^2/(2*a^3*b^2*(a + b* Tan[x])^2) - ((a^2 - 3*b^2)*(a^2 + b^2))/(a^4*b^2*(a + b*Tan[x]))
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[sin[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*si n[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[1/d Subst[Int[x^m*((a + b* x)^n/(1 + x^2)^((m + n + 2)/2)), x], x, Tan[c + d*x]], x] /; FreeQ[{a, b, c , d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] && !(GtQ[n, 0 ] && GtQ[m, 1])
Time = 0.83 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.09
| method | result | size |
| default | \(-\frac {-a^{4}-2 a^{2} b^{2}-b^{4}}{2 b^{2} a^{3} \left (a +b \tan \left (x \right )\right )^{2}}-\frac {a^{4}-2 a^{2} b^{2}-3 b^{4}}{a^{4} b^{2} \left (a +b \tan \left (x \right )\right )}-\frac {2 \left (a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (x \right )\right )}{a^{5}}-\frac {1}{2 a^{3} \tan \left (x \right )^{2}}+\frac {\left (2 a^{2}+6 b^{2}\right ) \ln \left (\tan \left (x \right )\right )}{a^{5}}+\frac {3 b}{a^{4} \tan \left (x \right )}\) | \(128\) |
| norman | \(\frac {\frac {b \tan \left (\frac {x}{2}\right )}{a^{2}}+\frac {b \left (-13 a^{2}-24 b^{2}\right ) \tan \left (\frac {x}{2}\right )^{3}}{a^{4}}-\frac {1}{8 a}-\frac {\tan \left (\frac {x}{2}\right )^{8}}{8 a}-\frac {\left (-9 a^{4}+56 a^{2} b^{2}+144 b^{4}\right ) \tan \left (\frac {x}{2}\right )^{4}}{4 a^{5}}-\frac {b \tan \left (\frac {x}{2}\right )^{7}}{a^{2}}-\frac {b \left (-13 a^{2}-24 b^{2}\right ) \tan \left (\frac {x}{2}\right )^{5}}{a^{4}}}{\tan \left (\frac {x}{2}\right )^{2} \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )^{2}}+\frac {2 \left (a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{5}}-\frac {2 \left (a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )}{a^{5}}\) | \(196\) |
| parallelrisch | \(\frac {-8 \left (\left (a^{2}-b^{2}\right ) \cos \left (2 x \right )+2 a b \sin \left (2 x \right )+a^{2}+b^{2}\right ) \left (a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )+8 \left (\left (a^{2}-b^{2}\right ) \cos \left (2 x \right )+2 a b \sin \left (2 x \right )+a^{2}+b^{2}\right ) \left (a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )\right )-\left (\frac {1}{8}-\frac {\cos \left (4 x \right )}{8}+\cos \left (2 x \right )\right ) \sec \left (\frac {x}{2}\right )^{2} a^{4} \csc \left (\frac {x}{2}\right )^{2}+\left (\cos \left (x \right )+7 \cos \left (3 x \right )\right ) b \,a^{3} \sec \left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right )+2 \left (7 a^{2} b^{2}+18 b^{4}\right ) \cos \left (2 x \right )-48 \sin \left (2 x \right ) a \,b^{3}-14 a^{2} b^{2}-36 b^{4}}{4 \left (\left (a^{2}-b^{2}\right ) \cos \left (2 x \right )+2 a b \sin \left (2 x \right )+a^{2}+b^{2}\right ) a^{5}}\) | \(237\) |
| risch | \(\frac {4 i \left (i a^{3} {\mathrm e}^{2 i x}+6 i a \,b^{2}-3 b^{3}+3 a^{2} b -a^{2} b \,{\mathrm e}^{2 i x}-9 i a \,b^{2} {\mathrm e}^{2 i x}+3 i a \,b^{2} {\mathrm e}^{6 i x}+a^{2} b \,{\mathrm e}^{6 i x}-3 a^{2} b \,{\mathrm e}^{4 i x}+i a^{3} {\mathrm e}^{6 i x}+3 b^{3} {\mathrm e}^{6 i x}-9 b^{3} {\mathrm e}^{4 i x}+9 b^{3} {\mathrm e}^{2 i x}\right )}{\left ({\mathrm e}^{2 i x}-1\right )^{2} \left (b \,{\mathrm e}^{2 i x}+i a \,{\mathrm e}^{2 i x}-b +i a \right )^{2} a^{4}}+\frac {2 \ln \left ({\mathrm e}^{2 i x}-1\right )}{a^{3}}+\frac {6 \ln \left ({\mathrm e}^{2 i x}-1\right ) b^{2}}{a^{5}}-\frac {2 \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{3}}-\frac {6 \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right ) b^{2}}{a^{5}}\) | \(262\) |
Input:
int(csc(x)^3/(a*cos(x)+b*sin(x))^3,x,method=_RETURNVERBOSE)
Output:
-1/2*(-a^4-2*a^2*b^2-b^4)/b^2/a^3/(a+b*tan(x))^2-(a^4-2*a^2*b^2-3*b^4)/a^4 /b^2/(a+b*tan(x))-2*(a^2+3*b^2)*ln(a+b*tan(x))/a^5-1/2/a^3/tan(x)^2+(2*a^2 +6*b^2)/a^5*ln(tan(x))+3/a^4*b/tan(x)
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (113) = 226\).
Time = 0.10 (sec) , antiderivative size = 385, normalized size of antiderivative = 3.29 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {24 \, a^{2} b^{2} \cos \left (x\right )^{4} - a^{4} + 6 \, a^{2} b^{2} + 2 \, {\left (a^{4} - 15 \, a^{2} b^{2}\right )} \cos \left (x\right )^{2} - 2 \, {\left ({\left (a^{4} + 2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (x\right )^{4} - a^{2} b^{2} - 3 \, b^{4} - {\left (a^{4} + a^{2} b^{2} - 6 \, b^{4}\right )} \cos \left (x\right )^{2} + 2 \, {\left ({\left (a^{3} b + 3 \, a b^{3}\right )} \cos \left (x\right )^{3} - {\left (a^{3} b + 3 \, a b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) + 2 \, {\left ({\left (a^{4} + 2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (x\right )^{4} - a^{2} b^{2} - 3 \, b^{4} - {\left (a^{4} + a^{2} b^{2} - 6 \, b^{4}\right )} \cos \left (x\right )^{2} + 2 \, {\left ({\left (a^{3} b + 3 \, a b^{3}\right )} \cos \left (x\right )^{3} - {\left (a^{3} b + 3 \, a b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )\right )} \log \left (-\frac {1}{4} \, \cos \left (x\right )^{2} + \frac {1}{4}\right ) - 4 \, {\left (3 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (x\right )^{3} - {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{2 \, {\left (a^{5} b^{2} - {\left (a^{7} - a^{5} b^{2}\right )} \cos \left (x\right )^{4} + {\left (a^{7} - 2 \, a^{5} b^{2}\right )} \cos \left (x\right )^{2} - 2 \, {\left (a^{6} b \cos \left (x\right )^{3} - a^{6} b \cos \left (x\right )\right )} \sin \left (x\right )\right )}} \] Input:
integrate(csc(x)^3/(a*cos(x)+b*sin(x))^3,x, algorithm="fricas")
Output:
-1/2*(24*a^2*b^2*cos(x)^4 - a^4 + 6*a^2*b^2 + 2*(a^4 - 15*a^2*b^2)*cos(x)^ 2 - 2*((a^4 + 2*a^2*b^2 - 3*b^4)*cos(x)^4 - a^2*b^2 - 3*b^4 - (a^4 + a^2*b ^2 - 6*b^4)*cos(x)^2 + 2*((a^3*b + 3*a*b^3)*cos(x)^3 - (a^3*b + 3*a*b^3)*c os(x))*sin(x))*log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) + 2*( (a^4 + 2*a^2*b^2 - 3*b^4)*cos(x)^4 - a^2*b^2 - 3*b^4 - (a^4 + a^2*b^2 - 6* b^4)*cos(x)^2 + 2*((a^3*b + 3*a*b^3)*cos(x)^3 - (a^3*b + 3*a*b^3)*cos(x))* sin(x))*log(-1/4*cos(x)^2 + 1/4) - 4*(3*(a^3*b - a*b^3)*cos(x)^3 - (2*a^3* b - 3*a*b^3)*cos(x))*sin(x))/(a^5*b^2 - (a^7 - a^5*b^2)*cos(x)^4 + (a^7 - 2*a^5*b^2)*cos(x)^2 - 2*(a^6*b*cos(x)^3 - a^6*b*cos(x))*sin(x))
\[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\int \frac {\csc ^{3}{\left (x \right )}}{\left (a \cos {\left (x \right )} + b \sin {\left (x \right )}\right )^{3}}\, dx \] Input:
integrate(csc(x)**3/(a*cos(x)+b*sin(x))**3,x)
Output:
Integral(csc(x)**3/(a*cos(x) + b*sin(x))**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (113) = 226\).
Time = 0.05 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.63 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {a^{4} - \frac {8 \, a^{3} b \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {2 \, {\left (a^{4} + 22 \, a^{2} b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {4 \, {\left (21 \, a^{3} b + 4 \, a b^{3}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {{\left (15 \, a^{4} - 144 \, a^{2} b^{2} - 112 \, b^{4}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {4 \, {\left (19 \, a^{3} b + 16 \, a b^{3}\right )} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}}{8 \, {\left (\frac {a^{7} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {4 \, a^{6} b \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {4 \, a^{6} b \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {a^{7} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac {2 \, {\left (a^{7} - 2 \, a^{5} b^{2}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}\right )}} - \frac {\frac {12 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a^{4}} - \frac {2 \, {\left (a^{2} + 3 \, b^{2}\right )} \log \left (-a - \frac {2 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{5}} + \frac {2 \, {\left (a^{2} + 3 \, b^{2}\right )} \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{5}} \] Input:
integrate(csc(x)^3/(a*cos(x)+b*sin(x))^3,x, algorithm="maxima")
Output:
-1/8*(a^4 - 8*a^3*b*sin(x)/(cos(x) + 1) - 2*(a^4 + 22*a^2*b^2)*sin(x)^2/(c os(x) + 1)^2 + 4*(21*a^3*b + 4*a*b^3)*sin(x)^3/(cos(x) + 1)^3 - (15*a^4 - 144*a^2*b^2 - 112*b^4)*sin(x)^4/(cos(x) + 1)^4 - 4*(19*a^3*b + 16*a*b^3)*s in(x)^5/(cos(x) + 1)^5)/(a^7*sin(x)^2/(cos(x) + 1)^2 + 4*a^6*b*sin(x)^3/(c os(x) + 1)^3 - 4*a^6*b*sin(x)^5/(cos(x) + 1)^5 + a^7*sin(x)^6/(cos(x) + 1) ^6 - 2*(a^7 - 2*a^5*b^2)*sin(x)^4/(cos(x) + 1)^4) - 1/8*(12*b*sin(x)/(cos( x) + 1) + a*sin(x)^2/(cos(x) + 1)^2)/a^4 - 2*(a^2 + 3*b^2)*log(-a - 2*b*si n(x)/(cos(x) + 1) + a*sin(x)^2/(cos(x) + 1)^2)/a^5 + 2*(a^2 + 3*b^2)*log(s in(x)/(cos(x) + 1))/a^5
Time = 0.16 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.25 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {2 \, {\left (a^{2} + 3 \, b^{2}\right )} \log \left ({\left | \tan \left (x\right ) \right |}\right )}{a^{5}} - \frac {2 \, {\left (a^{2} b + 3 \, b^{3}\right )} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{5} b} - \frac {2 \, a^{4} b \tan \left (x\right )^{3} - 4 \, a^{2} b^{3} \tan \left (x\right )^{3} - 12 \, b^{5} \tan \left (x\right )^{3} + a^{5} \tan \left (x\right )^{2} - 6 \, a^{3} b^{2} \tan \left (x\right )^{2} - 18 \, a b^{4} \tan \left (x\right )^{2} - 4 \, a^{2} b^{3} \tan \left (x\right ) + a^{3} b^{2}}{2 \, {\left (b \tan \left (x\right )^{2} + a \tan \left (x\right )\right )}^{2} a^{4} b^{2}} \] Input:
integrate(csc(x)^3/(a*cos(x)+b*sin(x))^3,x, algorithm="giac")
Output:
2*(a^2 + 3*b^2)*log(abs(tan(x)))/a^5 - 2*(a^2*b + 3*b^3)*log(abs(b*tan(x) + a))/(a^5*b) - 1/2*(2*a^4*b*tan(x)^3 - 4*a^2*b^3*tan(x)^3 - 12*b^5*tan(x) ^3 + a^5*tan(x)^2 - 6*a^3*b^2*tan(x)^2 - 18*a*b^4*tan(x)^2 - 4*a^2*b^3*tan (x) + a^3*b^2)/((b*tan(x)^2 + a*tan(x))^2*a^4*b^2)
Time = 17.19 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.16 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\,\left (2\,a^2+6\,b^2\right )}{a^5}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (42\,a^2\,b+8\,b^3\right )-{\mathrm {tan}\left (\frac {x}{2}\right )}^5\,\left (38\,a^2\,b+32\,b^3\right )-{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (a^3+22\,a\,b^2\right )+\frac {a^3}{2}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^4\,\left (-15\,a^4+144\,a^2\,b^2+112\,b^4\right )}{2\,a}-4\,a^2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}{4\,a^6\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2-{\mathrm {tan}\left (\frac {x}{2}\right )}^4\,\left (8\,a^6-16\,a^4\,b^2\right )+4\,a^6\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+16\,a^5\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3-16\,a^5\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}-\frac {\ln \left (-a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+a\right )\,\left (2\,a^2+6\,b^2\right )}{a^5}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a^3}-\frac {3\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}{2\,a^4} \] Input:
int(1/(sin(x)^3*(a*cos(x) + b*sin(x))^3),x)
Output:
(log(tan(x/2))*(2*a^2 + 6*b^2))/a^5 - (tan(x/2)^3*(42*a^2*b + 8*b^3) - tan (x/2)^5*(38*a^2*b + 32*b^3) - tan(x/2)^2*(22*a*b^2 + a^3) + a^3/2 + (tan(x /2)^4*(112*b^4 - 15*a^4 + 144*a^2*b^2))/(2*a) - 4*a^2*b*tan(x/2))/(4*a^6*t an(x/2)^2 - tan(x/2)^4*(8*a^6 - 16*a^4*b^2) + 4*a^6*tan(x/2)^6 + 16*a^5*b* tan(x/2)^3 - 16*a^5*b*tan(x/2)^5) - (log(a + 2*b*tan(x/2) - a*tan(x/2)^2)* (2*a^2 + 6*b^2))/a^5 - tan(x/2)^2/(8*a^3) - (3*b*tan(x/2))/(2*a^4)
Time = 0.17 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.79 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx =\text {Too large to display} \] Input:
int(csc(x)^3/(a*cos(x)+b*sin(x))^3,x)
Output:
( - 16*cos(x)*log(tan(x/2)**2*a - 2*tan(x/2)*b - a)*sin(x)**3*a**3*b - 48* cos(x)*log(tan(x/2)**2*a - 2*tan(x/2)*b - a)*sin(x)**3*a*b**3 + 16*cos(x)* log(tan(x/2))*sin(x)**3*a**3*b + 48*cos(x)*log(tan(x/2))*sin(x)**3*a*b**3 - 2*cos(x)*sin(x)**3*a**3*b + 8*cos(x)*sin(x)*a**3*b + 8*log(tan(x/2)**2*a - 2*tan(x/2)*b - a)*sin(x)**4*a**4 + 16*log(tan(x/2)**2*a - 2*tan(x/2)*b - a)*sin(x)**4*a**2*b**2 - 24*log(tan(x/2)**2*a - 2*tan(x/2)*b - a)*sin(x) **4*b**4 - 8*log(tan(x/2)**2*a - 2*tan(x/2)*b - a)*sin(x)**2*a**4 - 24*log (tan(x/2)**2*a - 2*tan(x/2)*b - a)*sin(x)**2*a**2*b**2 - 8*log(tan(x/2))*s in(x)**4*a**4 - 16*log(tan(x/2))*sin(x)**4*a**2*b**2 + 24*log(tan(x/2))*si n(x)**4*b**4 + 8*log(tan(x/2))*sin(x)**2*a**4 + 24*log(tan(x/2))*sin(x)**2 *a**2*b**2 - 11*sin(x)**4*a**4 - 25*sin(x)**4*a**2*b**2 - 12*sin(x)**4*b** 4 + 15*sin(x)**2*a**4 + 24*sin(x)**2*a**2*b**2 - 2*a**4)/(4*sin(x)**2*a**5 *(2*cos(x)*sin(x)*a*b - sin(x)**2*a**2 + sin(x)**2*b**2 + a**2))