Integrand size = 27, antiderivative size = 61 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \cot ^6(c+d x)}{6 d}-\frac {b \csc (c+d x)}{d}+\frac {2 b \csc ^3(c+d x)}{3 d}-\frac {b \csc ^5(c+d x)}{5 d} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2913, 2687, 30, 2686, 200} \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \cot ^6(c+d x)}{6 d}-\frac {b \csc ^5(c+d x)}{5 d}+\frac {2 b \csc ^3(c+d x)}{3 d}-\frac {b \csc (c+d x)}{d} \]
[In]
[Out]
Rule 30
Rule 200
Rule 2686
Rule 2687
Rule 2913
Rubi steps \begin{align*} \text {integral}& = a \int \cot ^5(c+d x) \csc ^2(c+d x) \, dx+b \int \cot ^5(c+d x) \csc (c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int x^5 \, dx,x,-\cot (c+d x)\right )}{d}-\frac {b \text {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a \cot ^6(c+d x)}{6 d}-\frac {b \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a \cot ^6(c+d x)}{6 d}-\frac {b \csc (c+d x)}{d}+\frac {2 b \csc ^3(c+d x)}{3 d}-\frac {b \csc ^5(c+d x)}{5 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \cot ^6(c+d x)}{6 d}-\frac {b \csc (c+d x)}{d}+\frac {2 b \csc ^3(c+d x)}{3 d}-\frac {b \csc ^5(c+d x)}{5 d} \]
[In]
[Out]
Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.15
method | result | size |
derivativedivides | \(-\frac {\frac {\left (\csc ^{6}\left (d x +c \right )\right ) a}{6}+\frac {b \left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right ) a}{2}-\frac {2 b \left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {a \left (\csc ^{2}\left (d x +c \right )\right )}{2}+\csc \left (d x +c \right ) b}{d}\) | \(70\) |
default | \(-\frac {\frac {\left (\csc ^{6}\left (d x +c \right )\right ) a}{6}+\frac {b \left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right ) a}{2}-\frac {2 b \left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {a \left (\csc ^{2}\left (d x +c \right )\right )}{2}+\csc \left (d x +c \right ) b}{d}\) | \(70\) |
risch | \(-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} \left (15 i a \,{\mathrm e}^{9 i \left (d x +c \right )}+15 b \,{\mathrm e}^{10 i \left (d x +c \right )}-35 b \,{\mathrm e}^{8 i \left (d x +c \right )}+50 i a \,{\mathrm e}^{5 i \left (d x +c \right )}+78 b \,{\mathrm e}^{6 i \left (d x +c \right )}-78 b \,{\mathrm e}^{4 i \left (d x +c \right )}+15 i a \,{\mathrm e}^{i \left (d x +c \right )}+35 b \,{\mathrm e}^{2 i \left (d x +c \right )}-15 b \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}\) | \(148\) |
parallelrisch | \(\frac {-5 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -5 a \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 b \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+30 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +30 a \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+100 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+100 b \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-75 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -75 a \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-600 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-600 b \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{1920 d}\) | \(171\) |
norman | \(\frac {-\frac {a}{384 d}+\frac {5 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-\frac {3 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d}-\frac {3 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d}+\frac {5 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-\frac {a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d}+\frac {11 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}-\frac {25 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {5 b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {25 b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}+\frac {11 b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}-\frac {b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(237\) |
[In]
[Out]
none
Time = 0.37 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.64 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {15 \, a \cos \left (d x + c\right )^{4} - 15 \, a \cos \left (d x + c\right )^{2} + 2 \, {\left (15 \, b \cos \left (d x + c\right )^{4} - 20 \, b \cos \left (d x + c\right )^{2} + 8 \, b\right )} \sin \left (d x + c\right ) + 5 \, a}{30 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
[In]
[Out]
Timed out. \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.15 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {30 \, b \sin \left (d x + c\right )^{5} + 15 \, a \sin \left (d x + c\right )^{4} - 20 \, b \sin \left (d x + c\right )^{3} - 15 \, a \sin \left (d x + c\right )^{2} + 6 \, b \sin \left (d x + c\right ) + 5 \, a}{30 \, d \sin \left (d x + c\right )^{6}} \]
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.15 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {30 \, b \sin \left (d x + c\right )^{5} + 15 \, a \sin \left (d x + c\right )^{4} - 20 \, b \sin \left (d x + c\right )^{3} - 15 \, a \sin \left (d x + c\right )^{2} + 6 \, b \sin \left (d x + c\right ) + 5 \, a}{30 \, d \sin \left (d x + c\right )^{6}} \]
[In]
[Out]
Time = 11.56 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.13 \[ \int \cot ^5(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b\,{\sin \left (c+d\,x\right )}^5+\frac {a\,{\sin \left (c+d\,x\right )}^4}{2}-\frac {2\,b\,{\sin \left (c+d\,x\right )}^3}{3}-\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}+\frac {b\,\sin \left (c+d\,x\right )}{5}+\frac {a}{6}}{d\,{\sin \left (c+d\,x\right )}^6} \]
[In]
[Out]