Integrand size = 25, antiderivative size = 67 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=-\frac {9 a^3 x}{2}+\frac {6 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {\sec (c+d x) (a+a \sin (c+d x))^3}{d} \]
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Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2934, 2723} \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=\frac {6 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {9 a^3 x}{2}+\frac {\sec (c+d x) (a \sin (c+d x)+a)^3}{d} \]
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Rule 2723
Rule 2934
Rubi steps \begin{align*} \text {integral}& = \frac {\sec (c+d x) (a+a \sin (c+d x))^3}{d}-(3 a) \int (a+a \sin (c+d x))^2 \, dx \\ & = -\frac {9 a^3 x}{2}+\frac {6 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {\sec (c+d x) (a+a \sin (c+d x))^3}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(145\) vs. \(2(67)=134\).
Time = 5.61 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.16 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=-\frac {a^3 (1+\sin (c+d x))^3 \left (\cos \left (\frac {1}{2} (c+d x)\right ) (18 (c+d x)-12 \cos (c+d x)-\sin (2 (c+d x)))+\sin \left (\frac {1}{2} (c+d x)\right ) (-2 (16+9 c+9 d x)+12 \cos (c+d x)+\sin (2 (c+d x)))\right )}{4 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
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Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.21
method | result | size |
risch | \(-\frac {9 a^{3} x}{2}+\frac {3 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {3 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {8 a^{3}}{d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}+\frac {a^{3} \sin \left (2 d x +2 c \right )}{4 d}\) | \(81\) |
parallelrisch | \(\frac {\left (\left (-36 d x +72\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+36 d x \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+11 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+\cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+16 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-11 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )\right ) a^{3}}{8 d \left (-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(113\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+3 a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (\tan \left (d x +c \right )-d x -c \right )+\frac {a^{3}}{\cos \left (d x +c \right )}}{d}\) | \(130\) |
default | \(\frac {a^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+3 a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (\tan \left (d x +c \right )-d x -c \right )+\frac {a^{3}}{\cos \left (d x +c \right )}}{d}\) | \(130\) |
norman | \(\frac {\frac {9 a^{3} x}{2}-\frac {14 a^{3}}{d}-\frac {9 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {23 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {18 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {23 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {9 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+9 a^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {9 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {2 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {30 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\) | \(229\) |
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Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.87 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=\frac {a^{3} \cos \left (d x + c\right )^{3} - 9 \, a^{3} d x + 6 \, a^{3} \cos \left (d x + c\right )^{2} + 8 \, a^{3} - {\left (9 \, a^{3} d x - 13 \, a^{3}\right )} \cos \left (d x + c\right ) + {\left (9 \, a^{3} d x + a^{3} \cos \left (d x + c\right )^{2} - 5 \, a^{3} \cos \left (d x + c\right ) + 8 \, a^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\right )}} \]
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\[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=a^{3} \left (\int \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 \sin ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.45 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=-\frac {{\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{3} + 6 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{3} - 6 \, a^{3} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} - \frac {2 \, a^{3}}{\cos \left (d x + c\right )}}{2 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.52 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=-\frac {9 \, {\left (d x + c\right )} a^{3} + \frac {16 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1} + \frac {2 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
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Time = 12.04 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.73 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=-\frac {9\,a^3\,x}{2}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {a^3\,\left (9\,d\,x-10\right )}{2}-\frac {9\,a^3\,d\,x}{2}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^3\,\left (9\,d\,x-18\right )}{2}-\frac {9\,a^3\,d\,x}{2}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {a^3\,\left (18\,d\,x-14\right )}{2}-9\,a^3\,d\,x\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3\,\left (18\,d\,x-42\right )}{2}-9\,a^3\,d\,x\right )-\frac {a^3\,\left (9\,d\,x-28\right )}{2}+\frac {9\,a^3\,d\,x}{2}}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
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