Integrand size = 27, antiderivative size = 101 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=\frac {11 a^3 x}{2}-\frac {3 a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {19 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d} \]
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Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2951, 2729, 2727, 2718, 2715, 8} \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=-\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {19 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {11 a^3 x}{2} \]
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Rule 8
Rule 2715
Rule 2718
Rule 2727
Rule 2729
Rule 2951
Rubi steps \begin{align*} \text {integral}& = a^4 \int \left (\frac {5}{a}+\frac {2}{a (-1+\sin (c+d x))^2}+\frac {7}{a (-1+\sin (c+d x))}+\frac {3 \sin (c+d x)}{a}+\frac {\sin ^2(c+d x)}{a}\right ) \, dx \\ & = 5 a^3 x+a^3 \int \sin ^2(c+d x) \, dx+\left (2 a^3\right ) \int \frac {1}{(-1+\sin (c+d x))^2} \, dx+\left (3 a^3\right ) \int \sin (c+d x) \, dx+\left (7 a^3\right ) \int \frac {1}{-1+\sin (c+d x)} \, dx \\ & = 5 a^3 x-\frac {3 a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {7 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} a^3 \int 1 \, dx-\frac {1}{3} \left (2 a^3\right ) \int \frac {1}{-1+\sin (c+d x)} \, dx \\ & = \frac {11 a^3 x}{2}-\frac {3 a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {19 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d} \\ \end{align*}
Time = 6.28 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.57 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=-\frac {a^3 \left (-3 (89+132 c+132 d x) \cos \left (\frac {1}{2} (c+d x)\right )+(403+132 c+132 d x) \cos \left (\frac {3}{2} (c+d x)\right )+3 \left (-9 \cos \left (\frac {5}{2} (c+d x)\right )+\cos \left (\frac {7}{2} (c+d x)\right )+2 (86+88 c+88 d x+(-43+44 c+44 d x) \cos (c+d x)-10 \cos (2 (c+d x))-\cos (3 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )\right )}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
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Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.24
method | result | size |
risch | \(\frac {11 a^{3} x}{2}+\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {3 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {3 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {2 a^{3} \left (-36 i {\mathrm e}^{i \left (d x +c \right )}+21 \,{\mathrm e}^{2 i \left (d x +c \right )}-19\right )}{3 d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3}}\) | \(125\) |
parallelrisch | \(\frac {a^{3} \left (132 d x \cos \left (3 d x +3 c \right )+396 d x \cos \left (d x +c \right )-208 \cos \left (3 d x +3 c \right )-480 \cos \left (2 d x +2 c \right )-161 \sin \left (3 d x +3 c \right )-624 \cos \left (d x +c \right )-30 \sin \left (d x +c \right )-3 \sin \left (5 d x +5 c \right )-36 \cos \left (4 d x +4 c \right )-316\right )}{24 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(125\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 a^{3} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )}{d}\) | \(246\) |
default | \(\frac {a^{3} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 a^{3} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )}{d}\) | \(246\) |
norman | \(\frac {-\frac {11 a^{3} x}{2}+\frac {52 a^{3}}{3 d}+\frac {11 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {11 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {50 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {128 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {50 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {11 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {11 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {33 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {33 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {11 a^{3} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {4 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {56 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\) | \(267\) |
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Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (89) = 178\).
Time = 0.27 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.94 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=\frac {3 \, a^{3} \cos \left (d x + c\right )^{4} - 12 \, a^{3} \cos \left (d x + c\right )^{3} - 66 \, a^{3} d x - 4 \, a^{3} + {\left (33 \, a^{3} d x + 53 \, a^{3}\right )} \cos \left (d x + c\right )^{2} - {\left (33 \, a^{3} d x - 64 \, a^{3}\right )} \cos \left (d x + c\right ) - {\left (3 \, a^{3} \cos \left (d x + c\right )^{3} - 66 \, a^{3} d x + 15 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \, a^{3} - {\left (33 \, a^{3} d x - 68 \, a^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
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Timed out. \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.44 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=\frac {{\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{3} + 6 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{3} - 6 \, a^{3} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - \frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{3}}{\cos \left (d x + c\right )^{3}}}{6 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.34 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=\frac {33 \, {\left (d x + c\right )} a^{3} + \frac {6 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac {4 \, {\left (15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \]
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Time = 16.02 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.84 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx=\frac {11\,a^3\,x}{2}+\frac {\frac {11\,a^3\,\left (c+d\,x\right )}{2}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {33\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (99\,c+99\,d\,x-246\right )}{6}\right )-\frac {a^3\,\left (33\,c+33\,d\,x-104\right )}{6}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {33\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (99\,c+99\,d\,x-66\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {55\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (165\,c+165\,d\,x-198\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {55\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (165\,c+165\,d\,x-322\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {77\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (231\,c+231\,d\,x-308\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {77\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (231\,c+231\,d\,x-420\right )}{6}\right )}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
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