Integrand size = 27, antiderivative size = 86 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc (c+d x)}{d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {2 a \log (\sin (c+d x))}{d}-\frac {2 a \sin (c+d x)}{d}+\frac {a \sin ^2(c+d x)}{2 d}+\frac {a \sin ^3(c+d x)}{3 d} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2915, 12, 90} \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^2(c+d x)}{2 d}-\frac {2 a \sin (c+d x)}{d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc (c+d x)}{d}-\frac {2 a \log (\sin (c+d x))}{d} \]
[In]
[Out]
Rule 12
Rule 90
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^3 (a-x)^2 (a+x)^3}{x^3} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {(a-x)^2 (a+x)^3}{x^3} \, dx,x,a \sin (c+d x)\right )}{a^2 d} \\ & = \frac {\text {Subst}\left (\int \left (-2 a^2+\frac {a^5}{x^3}+\frac {a^4}{x^2}-\frac {2 a^3}{x}+a x+x^2\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d} \\ & = -\frac {a \csc (c+d x)}{d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {2 a \log (\sin (c+d x))}{d}-\frac {2 a \sin (c+d x)}{d}+\frac {a \sin ^2(c+d x)}{2 d}+\frac {a \sin ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc (c+d x)}{d}-\frac {2 a \sin (c+d x)}{d}+\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \left (\csc ^2(c+d x)+4 \log (\sin (c+d x))-\sin ^2(c+d x)\right )}{2 d} \]
[In]
[Out]
Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.22
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+a \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(105\) |
default | \(\frac {a \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+a \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(105\) |
parallelrisch | \(\frac {\left (192 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-12 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+60\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )+\cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+20 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+20 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-90 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-96 \cos \left (d x +c \right )+24 \cos \left (2 d x +2 c \right )+12\right ) \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a}{96 d}\) | \(167\) |
risch | \(2 i a x +\frac {i a \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {7 i a \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {7 i a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {i a \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}+\frac {4 i a c}{d}-\frac {2 i a \left (i {\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {2 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(180\) |
norman | \(\frac {-\frac {a}{8 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {6 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {25 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {6 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {21 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {21 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {2 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(206\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.19 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {6 \, a \cos \left (d x + c\right )^{4} - 9 \, a \cos \left (d x + c\right )^{2} + 24 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 4 \, {\left (a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} - 8 \, a\right )} \sin \left (d x + c\right ) - 3 \, a}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
[In]
[Out]
Timed out. \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.79 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {2 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - 12 \, a \log \left (\sin \left (d x + c\right )\right ) - 12 \, a \sin \left (d x + c\right ) - \frac {3 \, {\left (2 \, a \sin \left (d x + c\right ) + a\right )}}{\sin \left (d x + c\right )^{2}}}{6 \, d} \]
[In]
[Out]
none
Time = 0.76 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {2 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - 12 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 12 \, a \sin \left (d x + c\right ) + \frac {3 \, {\left (6 \, a \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right ) - a\right )}}{\sin \left (d x + c\right )^{2}}}{6 \, d} \]
[In]
[Out]
Time = 10.05 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.66 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {2\,a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {18\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {15\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2}+\frac {82\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}-\frac {13\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+22\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {2\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
[In]
[Out]