Integrand size = 29, antiderivative size = 131 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 b \csc (c+d x)}{a^3 d}-\frac {\csc ^2(c+d x)}{2 a^2 d}-\frac {\left (2 a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac {\left (a^4+2 a^2 b^2-3 b^4\right ) \log (a+b \sin (c+d x))}{a^4 b^2 d}+\frac {\left (a^2-b^2\right )^2}{a^3 b^2 d (a+b \sin (c+d x))} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2916, 12, 908} \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 b \csc (c+d x)}{a^3 d}-\frac {\csc ^2(c+d x)}{2 a^2 d}-\frac {\left (2 a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac {\left (a^4+2 a^2 b^2-3 b^4\right ) \log (a+b \sin (c+d x))}{a^4 b^2 d}+\frac {\left (a^2-b^2\right )^2}{a^3 b^2 d (a+b \sin (c+d x))} \]
[In]
[Out]
Rule 12
Rule 908
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b^3 \left (b^2-x^2\right )^2}{x^3 (a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{x^3 (a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^2 d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {b^4}{a^2 x^3}-\frac {2 b^4}{a^3 x^2}+\frac {-2 a^2 b^2+3 b^4}{a^4 x}-\frac {\left (a^2-b^2\right )^2}{a^3 (a+x)^2}+\frac {a^4+2 a^2 b^2-3 b^4}{a^4 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^2 d} \\ & = \frac {2 b \csc (c+d x)}{a^3 d}-\frac {\csc ^2(c+d x)}{2 a^2 d}-\frac {\left (2 a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac {\left (a^4+2 a^2 b^2-3 b^4\right ) \log (a+b \sin (c+d x))}{a^4 b^2 d}+\frac {\left (a^2-b^2\right )^2}{a^3 b^2 d (a+b \sin (c+d x))} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.89 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {4 a b \csc (c+d x)-a^2 \csc ^2(c+d x)-2 \left (2 a^2-3 b^2\right ) \log (\sin (c+d x))+\frac {2 \left (a^4+2 a^2 b^2-3 b^4\right ) \log (a+b \sin (c+d x))}{b^2}+\frac {2 a \left (a^2-b^2\right )^2}{b^2 (a+b \sin (c+d x))}}{2 a^4 d} \]
[In]
[Out]
Time = 0.84 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {-\frac {-a^{4}+2 a^{2} b^{2}-b^{4}}{a^{3} b^{2} \left (a +b \sin \left (d x +c \right )\right )}+\frac {\left (a^{4}+2 a^{2} b^{2}-3 b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{a^{4} b^{2}}-\frac {1}{2 a^{2} \sin \left (d x +c \right )^{2}}+\frac {\left (-2 a^{2}+3 b^{2}\right ) \ln \left (\sin \left (d x +c \right )\right )}{a^{4}}+\frac {2 b}{a^{3} \sin \left (d x +c \right )}}{d}\) | \(129\) |
default | \(\frac {-\frac {-a^{4}+2 a^{2} b^{2}-b^{4}}{a^{3} b^{2} \left (a +b \sin \left (d x +c \right )\right )}+\frac {\left (a^{4}+2 a^{2} b^{2}-3 b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{a^{4} b^{2}}-\frac {1}{2 a^{2} \sin \left (d x +c \right )^{2}}+\frac {\left (-2 a^{2}+3 b^{2}\right ) \ln \left (\sin \left (d x +c \right )\right )}{a^{4}}+\frac {2 b}{a^{3} \sin \left (d x +c \right )}}{d}\) | \(129\) |
parallelrisch | \(\frac {\left (a -b \right ) \left (a +b \right ) \left (a^{2}+3 b^{2}\right ) \left (a +b \sin \left (d x +c \right )\right ) \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-a^{4} \left (a +b \sin \left (d x +c \right )\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b^{2} \left (a^{2}-\frac {3 b^{2}}{2}\right ) \left (a +b \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {b^{2} a^{2} \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+22\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {3 a \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3} \left (3+\cos \left (2 d x +2 c \right )\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}+\frac {11 a^{2} b^{2} \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+a^{4}+3 b^{4}\right ) a}{a^{4} b^{2} d \left (a +b \sin \left (d x +c \right )\right )}\) | \(237\) |
norman | \(\frac {-\frac {1}{8 a d}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {5 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {3 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2} d}+\frac {3 b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} d}-\frac {\left (4 a^{4}-15 a^{2} b^{2}+12 b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4} b d}-\frac {\left (8 a^{4}-33 a^{2} b^{2}+24 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4} b d}-\frac {\left (8 a^{4}-33 a^{2} b^{2}+24 b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4} b d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}+\frac {\left (a^{4}+2 a^{2} b^{2}-3 b^{4}\right ) \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{b^{2} a^{4} d}-\frac {\ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{2} d}-\frac {\left (2 a^{2}-3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4} d}\) | \(391\) |
risch | \(-\frac {i x}{b^{2}}-\frac {2 i c}{b^{2} d}+\frac {6 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+2 a^{4} {\mathrm e}^{5 i \left (d x +c \right )}-4 a^{2} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+6 b^{4} {\mathrm e}^{5 i \left (d x +c \right )}-6 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-4 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}+12 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-12 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}+2 a^{4} {\mathrm e}^{i \left (d x +c \right )}-4 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}+6 b^{4} {\mathrm e}^{i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} b^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right ) a^{3} d}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{a^{4} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{b^{2} d}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{2} d}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{4} d}\) | \(394\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (129) = 258\).
Time = 0.39 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.56 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {3 \, a^{2} b^{3} \sin \left (d x + c\right ) + 2 \, a^{5} - 5 \, a^{3} b^{2} + 6 \, a b^{4} - 2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} - 3 \, a b^{4} - {\left (a^{5} + 2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{4} b + 2 \, a^{2} b^{3} - 3 \, b^{5} - {\left (a^{4} b + 2 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \, {\left (2 \, a^{3} b^{2} - 3 \, a b^{4} - {\left (2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, a^{2} b^{3} - 3 \, b^{5} - {\left (2 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right )}{2 \, {\left (a^{5} b^{2} d \cos \left (d x + c\right )^{2} - a^{5} b^{2} d + {\left (a^{4} b^{3} d \cos \left (d x + c\right )^{2} - a^{4} b^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {3 \, a b^{3} \sin \left (d x + c\right ) - a^{2} b^{2} + 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + 3 \, b^{4}\right )} \sin \left (d x + c\right )^{2}}{a^{3} b^{3} \sin \left (d x + c\right )^{3} + a^{4} b^{2} \sin \left (d x + c\right )^{2}} - \frac {2 \, {\left (2 \, a^{2} - 3 \, b^{2}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{4}} + \frac {2 \, {\left (a^{4} + 2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{4} b^{2}}}{2 \, d} \]
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.45 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (2 \, a^{2} - 3 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac {2 \, {\left (a^{4} + 2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{4} b^{2}} + \frac {2 \, {\left (a^{4} \sin \left (d x + c\right ) + 2 \, a^{2} b^{2} \sin \left (d x + c\right ) - 3 \, b^{4} \sin \left (d x + c\right ) + 4 \, a^{3} b - 4 \, a b^{3}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} a^{4} b} - \frac {6 \, a^{2} \sin \left (d x + c\right )^{2} - 9 \, b^{2} \sin \left (d x + c\right )^{2} + 4 \, a b \sin \left (d x + c\right ) - a^{2}}{a^{4} \sin \left (d x + c\right )^{2}}}{2 \, d} \]
[In]
[Out]
Time = 13.80 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.14 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^3\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{2}-8\,b^2\right )+\frac {a^2}{2}-3\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,a^4-5\,a^2\,b^2+2\,b^4\right )}{a\,b}}{d\,\left (4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+8\,b\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{b^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^2\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a^2-3\,b^2\right )}{a^4\,d}+\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (a^4+2\,a^2\,b^2-3\,b^4\right )}{a^4\,b^2\,d} \]
[In]
[Out]