Integrand size = 27, antiderivative size = 110 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {5 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a \cot (c+d x)}{d}+\frac {5 a \sec (c+d x)}{2 d}+\frac {5 a \sec ^3(c+d x)}{6 d}-\frac {a \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}+\frac {2 a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2917, 2702, 294, 308, 213, 2700, 276} \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {5 a \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {2 a \tan (c+d x)}{d}-\frac {a \cot (c+d x)}{d}+\frac {5 a \sec ^3(c+d x)}{6 d}+\frac {5 a \sec (c+d x)}{2 d}-\frac {a \csc ^2(c+d x) \sec ^3(c+d x)}{2 d} \]
[In]
[Out]
Rule 213
Rule 276
Rule 294
Rule 308
Rule 2700
Rule 2702
Rule 2917
Rubi steps \begin{align*} \text {integral}& = a \int \csc ^2(c+d x) \sec ^4(c+d x) \, dx+a \int \csc ^3(c+d x) \sec ^4(c+d x) \, dx \\ & = \frac {a \text {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{d}+\frac {a \text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {a \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}+\frac {a \text {Subst}\left (\int \left (2+\frac {1}{x^2}+x^2\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {(5 a) \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d} \\ & = -\frac {a \cot (c+d x)}{d}-\frac {a \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}+\frac {2 a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {(5 a) \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{2 d} \\ & = -\frac {a \cot (c+d x)}{d}+\frac {5 a \sec (c+d x)}{2 d}+\frac {5 a \sec ^3(c+d x)}{6 d}-\frac {a \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}+\frac {2 a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {(5 a) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d} \\ & = -\frac {5 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a \cot (c+d x)}{d}+\frac {5 a \sec (c+d x)}{2 d}+\frac {5 a \sec ^3(c+d x)}{6 d}-\frac {a \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}+\frac {2 a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(359\) vs. \(2(110)=220\).
Time = 6.19 (sec) , antiderivative size = 359, normalized size of antiderivative = 3.26 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot (c+d x)}{d}-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a}{12 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {a \sin \left (\frac {1}{2} (c+d x)\right )}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {13 a \sin \left (\frac {1}{2} (c+d x)\right )}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {a \sin \left (\frac {1}{2} (c+d x)\right )}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {a}{12 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {13 a \sin \left (\frac {1}{2} (c+d x)\right )}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {5 a \tan (c+d x)}{3 d}+\frac {a \sec ^2(c+d x) \tan (c+d x)}{3 d} \]
[In]
[Out]
Time = 0.41 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {a \left (\frac {1}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+a \left (\frac {1}{3 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5}{6 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {5}{2 \cos \left (d x +c \right )}+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(120\) |
default | \(\frac {a \left (\frac {1}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+a \left (\frac {1}{3 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5}{6 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {5}{2 \cos \left (d x +c \right )}+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(120\) |
parallelrisch | \(\frac {\left (20 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )+2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-50 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+30 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {170 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3}-\frac {160}{3}\right ) a}{8 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(154\) |
risch | \(\frac {a \left (-30 i {\mathrm e}^{6 i \left (d x +c \right )}+15 \,{\mathrm e}^{7 i \left (d x +c \right )}+20 i {\mathrm e}^{4 i \left (d x +c \right )}-25 \,{\mathrm e}^{5 i \left (d x +c \right )}+2 i {\mathrm e}^{2 i \left (d x +c \right )}-7 \,{\mathrm e}^{3 i \left (d x +c \right )}-16 i+17 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) d}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}\) | \(168\) |
norman | \(\frac {\frac {a}{8 d}+\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {11 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {7 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {5 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {7 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {11 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {31 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {53 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {89 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {5 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(252\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (100) = 200\).
Time = 0.29 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.02 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {32 \, a \cos \left (d x + c\right )^{4} - 18 \, a \cos \left (d x + c\right )^{2} - 15 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (a \cos \left (d x + c\right )^{2} + 2 \, a\right )} \sin \left (d x + c\right ) - 8 \, a}{12 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right ) - {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
[In]
[Out]
Timed out. \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.96 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {4 \, {\left (\tan \left (d x + c\right )^{3} - \frac {3}{\tan \left (d x + c\right )} + 6 \, \tan \left (d x + c\right )\right )} a + a {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \]
[In]
[Out]
none
Time = 0.45 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.35 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 60 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {12 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1} - \frac {3 \, {\left (30 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - \frac {4 \, {\left (27 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 25 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{24 \, d} \]
[In]
[Out]
Time = 9.88 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.64 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {-18\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {23\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+\frac {67\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {68\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a}{2}}{d\,\left (-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {5\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d} \]
[In]
[Out]