Integrand size = 14, antiderivative size = 59 \[ \int \frac {\csc (x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {\log (\tan (x))}{a^3}-\frac {\log (a+b \tan (x))}{a^3}+\frac {\frac {1}{a}+\frac {a}{b^2}}{2 (a+b \tan (x))^2}+\frac {\frac {1}{a^2}-\frac {1}{b^2}}{a+b \tan (x)} \]
Time = 0.65 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.63 \[ \int \frac {\csc (x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {a^2 \csc ^2(x)+2 a b \cot (x) (-1+2 \log (\sin (x))-2 \log (a \cos (x)+b \sin (x)))+2 b^2 (-1+\log (\sin (x))-\log (a \cos (x)+b \sin (x)))+2 a^2 \cot ^2(x) (\log (\sin (x))-\log (a \cos (x)+b \sin (x)))}{2 a^3 (b+a \cot (x))^2} \]
(a^2*Csc[x]^2 + 2*a*b*Cot[x]*(-1 + 2*Log[Sin[x]] - 2*Log[a*Cos[x] + b*Sin[ x]]) + 2*b^2*(-1 + Log[Sin[x]] - Log[a*Cos[x] + b*Sin[x]]) + 2*a^2*Cot[x]^ 2*(Log[Sin[x]] - Log[a*Cos[x] + b*Sin[x]]))/(2*a^3*(b + a*Cot[x])^2)
Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, 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.
\(\displaystyle \int \frac {\csc (x)}{(a \cos (x)+b \sin (x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (x) (a \cos (x)+b \sin (x))^3}dx\) |
\(\Big \downarrow \) 3566 |
\(\displaystyle \int \frac {\left (\tan ^2(x)+1\right ) \cot (x)}{(a+b \tan (x))^3}d\tan (x)\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (-\frac {b}{a^3 (a+b \tan (x))}+\frac {\cot (x)}{a^3}+\frac {a^2-b^2}{a^2 b (a+b \tan (x))^2}+\frac {-a^2-b^2}{a b (a+b \tan (x))^3}\right )d\tan (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\log (a+b \tan (x))}{a^3}+\frac {\log (\tan (x))}{a^3}+\frac {\frac {1}{a^2}-\frac {1}{b^2}}{a+b \tan (x)}+\frac {\frac {a}{b^2}+\frac {1}{a}}{2 (a+b \tan (x))^2}\) |
Log[Tan[x]]/a^3 - Log[a + b*Tan[x]]/a^3 + (a^(-1) + a/b^2)/(2*(a + b*Tan[x ])^2) + (a^(-2) - b^(-2))/(a + b*Tan[x])
3.1.26.3.1 Defintions of rubi rules used
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 = 0.70 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24
method | result | size |
default | \(\frac {\ln \left (\tan \left (x \right )\right )}{a^{3}}-\frac {-a^{2}-b^{2}}{2 a \,b^{2} \left (a +b \tan \left (x \right )\right )^{2}}-\frac {a^{2}-b^{2}}{a^{2} b^{2} \left (a +b \tan \left (x \right )\right )}-\frac {\ln \left (a +b \tan \left (x \right )\right )}{a^{3}}\) | \(73\) |
norman | \(\frac {-\frac {2 \left (-a^{2}+3 b^{2}\right ) \tan \left (\frac {x}{2}\right )^{2}}{a^{3}}-\frac {4 b \tan \left (\frac {x}{2}\right )}{a^{2}}+\frac {4 b \tan \left (\frac {x}{2}\right )^{3}}{a^{2}}}{\left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )^{2}}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}-\frac {\ln \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )}{a^{3}}\) | \(103\) |
risch | \(\frac {2 a^{2} {\mathrm e}^{2 i x}-2 b^{2} {\mathrm e}^{2 i x}-4 i a b \,{\mathrm e}^{2 i x}+2 b^{2}-2 i b a}{\left (-i b \,{\mathrm e}^{2 i x}+a \,{\mathrm e}^{2 i x}+i b +a \right )^{2} a^{2} \left (-i b +a \right )}+\frac {\ln \left ({\mathrm e}^{2 i x}-1\right )}{a^{3}}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{3}}\) | \(119\) |
parallelrisch | \(\frac {\left (\left (-2 a^{2}+2 b^{2}\right ) \cos \left (2 x \right )-4 b a \sin \left (2 x \right )-2 a^{2}-2 b^{2}\right ) \ln \left (\frac {-2 a \cos \left (x \right )-2 b \sin \left (x \right )}{\cos \left (x \right )+1}\right )+\left (\left (2 a^{2}-2 b^{2}\right ) \cos \left (2 x \right )+4 b a \sin \left (2 x \right )+2 a^{2}+2 b^{2}\right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+\left (-a^{2}+3 b^{2}\right ) \cos \left (2 x \right )-4 b a \sin \left (2 x \right )+a^{2}-3 b^{2}}{2 a^{3} \left (\left (a^{2}-b^{2}\right ) \cos \left (2 x \right )+2 b a \sin \left (2 x \right )+a^{2}+b^{2}\right )}\) | \(169\) |
ln(tan(x))/a^3-1/2*(-a^2-b^2)/a/b^2/(a+b*tan(x))^2-(a^2-b^2)/a^2/b^2/(a+b* tan(x))-ln(a+b*tan(x))/a^3
Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (57) = 114\).
Time = 0.27 (sec) , antiderivative size = 220, normalized size of antiderivative = 3.73 \[ \int \frac {\csc (x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {4 \, a^{2} b^{2} \cos \left (x\right )^{2} + a^{4} - a^{2} b^{2} - 2 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (x\right ) \sin \left (x\right ) - {\left (a^{2} b^{2} + b^{4} + {\left (a^{4} - b^{4}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{3} b + a b^{3}\right )} \cos \left (x\right ) \sin \left (x\right )\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) + {\left (a^{2} b^{2} + b^{4} + {\left (a^{4} - b^{4}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{3} b + a b^{3}\right )} \cos \left (x\right ) \sin \left (x\right )\right )} \log \left (-\frac {1}{4} \, \cos \left (x\right )^{2} + \frac {1}{4}\right )}{2 \, {\left (a^{5} b^{2} + a^{3} b^{4} + {\left (a^{7} - a^{3} b^{4}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{6} b + a^{4} b^{3}\right )} \cos \left (x\right ) \sin \left (x\right )\right )}} \]
1/2*(4*a^2*b^2*cos(x)^2 + a^4 - a^2*b^2 - 2*(a^3*b - a*b^3)*cos(x)*sin(x) - (a^2*b^2 + b^4 + (a^4 - b^4)*cos(x)^2 + 2*(a^3*b + a*b^3)*cos(x)*sin(x)) *log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) + (a^2*b^2 + b^4 + (a^4 - b^4)*cos(x)^2 + 2*(a^3*b + a*b^3)*cos(x)*sin(x))*log(-1/4*cos(x)^2 + 1/4))/(a^5*b^2 + a^3*b^4 + (a^7 - a^3*b^4)*cos(x)^2 + 2*(a^6*b + a^4*b^3 )*cos(x)*sin(x))
\[ \int \frac {\csc (x)}{(a \cos (x)+b \sin (x))^3} \, dx=\int \frac {\csc {\left (x \right )}}{\left (a \cos {\left (x \right )} + b \sin {\left (x \right )}\right )^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (57) = 114\).
Time = 0.23 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.92 \[ \int \frac {\csc (x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {2 \, {\left (\frac {2 \, a b \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {2 \, a b \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {{\left (a^{2} - 3 \, b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}}{a^{5} + \frac {4 \, a^{4} b \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {4 \, a^{4} b \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {a^{5} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {2 \, {\left (a^{5} - 2 \, a^{3} b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}} - \frac {\log \left (-a - \frac {2 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{3}} + \frac {\log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} \]
-2*(2*a*b*sin(x)/(cos(x) + 1) - 2*a*b*sin(x)^3/(cos(x) + 1)^3 - (a^2 - 3*b ^2)*sin(x)^2/(cos(x) + 1)^2)/(a^5 + 4*a^4*b*sin(x)/(cos(x) + 1) - 4*a^4*b* sin(x)^3/(cos(x) + 1)^3 + a^5*sin(x)^4/(cos(x) + 1)^4 - 2*(a^5 - 2*a^3*b^2 )*sin(x)^2/(cos(x) + 1)^2) - log(-a - 2*b*sin(x)/(cos(x) + 1) + a*sin(x)^2 /(cos(x) + 1)^2)/a^3 + log(sin(x)/(cos(x) + 1))/a^3
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.31 \[ \int \frac {\csc (x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {\log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{3}} + \frac {\log \left ({\left | \tan \left (x\right ) \right |}\right )}{a^{3}} + \frac {3 \, b^{4} \tan \left (x\right )^{2} - 2 \, a^{3} b \tan \left (x\right ) + 8 \, a b^{3} \tan \left (x\right ) - a^{4} + 6 \, a^{2} b^{2}}{2 \, {\left (b \tan \left (x\right ) + a\right )}^{2} a^{3} b^{2}} \]
-log(abs(b*tan(x) + a))/a^3 + log(abs(tan(x)))/a^3 + 1/2*(3*b^4*tan(x)^2 - 2*a^3*b*tan(x) + 8*a*b^3*tan(x) - a^4 + 6*a^2*b^2)/((b*tan(x) + a)^2*a^3* b^2)
Time = 21.56 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.22 \[ \int \frac {\csc (x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^3}-\frac {\ln \left (-a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+a\right )}{a^3}+\frac {\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (a^2-3\,b^2\right )}{a^3}+\frac {4\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{a^2}-\frac {4\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2}}{a^2-{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (2\,a^2-4\,b^2\right )+a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+4\,a\,b\,\mathrm {tan}\left (\frac {x}{2}\right )-4\,a\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3} \]