Integrand size = 21, antiderivative size = 101 \[ \int (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx=\frac {7 a^2 x}{2}-\frac {2 a^2 \cos (c+d x)}{d}+\frac {a^2 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {11 a^2 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 d} \]
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Time = 0.16 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2787, 2844, 3056, 2813} \[ \int (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx=\frac {a^4 \sin ^3(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {16 a^2 \cos (c+d x)}{3 d}-\frac {8 a^2 \sin ^2(c+d x) \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac {7 a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {7 a^2 x}{2} \]
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Rule 2787
Rule 2813
Rule 2844
Rule 3056
Rubi steps \begin{align*} \text {integral}& = a^4 \int \frac {\sin ^4(c+d x)}{(a-a \sin (c+d x))^2} \, dx \\ & = \frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac {1}{3} a^2 \int \frac {\sin ^2(c+d x) (-3 a-5 a \sin (c+d x))}{a-a \sin (c+d x)} \, dx \\ & = -\frac {8 a^2 \cos (c+d x) \sin ^2(c+d x)}{3 d (1-\sin (c+d x))}+\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {1}{3} \int \sin (c+d x) \left (-16 a^2-21 a^2 \sin (c+d x)\right ) \, dx \\ & = \frac {7 a^2 x}{2}-\frac {16 a^2 \cos (c+d x)}{3 d}-\frac {7 a^2 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {8 a^2 \cos (c+d x) \sin ^2(c+d x)}{3 d (1-\sin (c+d x))}+\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{3 d (a-a \sin (c+d x))^2} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.57 \[ \int (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx=-\frac {a^2 \left (-21 (7+12 c+12 d x) \cos \left (\frac {1}{2} (c+d x)\right )+(239+84 c+84 d x) \cos \left (\frac {3}{2} (c+d x)\right )+3 \left (-5 \cos \left (\frac {5}{2} (c+d x)\right )+\cos \left (\frac {7}{2} (c+d x)\right )+2 (50+56 c+56 d x+(-27+28 c+28 d x) \cos (c+d x)-6 \cos (2 (c+d x))-\cos (3 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )\right )}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
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Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.31
method | result | size |
risch | \(\frac {7 a^{2} x}{2}+\frac {i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {a^{2} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{d}-\frac {i a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {2 \left (-21 i a^{2} {\mathrm e}^{i \left (d x +c \right )}+12 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-11 a^{2}\right )}{3 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}\) | \(132\) |
derivativedivides | \(\frac {a^{2} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+2 a^{2} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+a^{2} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )}{d}\) | \(186\) |
default | \(\frac {a^{2} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+2 a^{2} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+a^{2} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )}{d}\) | \(186\) |
parallelrisch | \(-\frac {a^{2} \left (-84 d x \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-252 d x \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+252 d x \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-84 d x \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+3 \sin \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+15 \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+279 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+267 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-501 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+15 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-3 \cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )-23 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )\right )}{24 d \left (\sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )\right )}\) | \(193\) |
norman | \(\frac {-\frac {7 a^{2} x}{2}+\frac {32 a^{2}}{3 d}+\frac {7 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {28 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {50 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {28 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {7 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {7 a^{2} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+7 a^{2} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {7 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {7 a^{2} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {32 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {64 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\) | \(263\) |
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Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (89) = 178\).
Time = 0.26 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.94 \[ \int (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx=\frac {3 \, a^{2} \cos \left (d x + c\right )^{4} - 6 \, a^{2} \cos \left (d x + c\right )^{3} - 42 \, a^{2} d x + {\left (21 \, a^{2} d x + 31 \, a^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - {\left (21 \, a^{2} d x - 38 \, a^{2}\right )} \cos \left (d x + c\right ) - {\left (3 \, a^{2} \cos \left (d x + c\right )^{3} - 42 \, a^{2} d x + 9 \, a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} - {\left (21 \, a^{2} d x - 40 \, a^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
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Timed out. \[ \int (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx=\text {Timed out} \]
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Time = 0.33 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.19 \[ \int (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx=\frac {{\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{2} + 2 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{2} - 4 \, a^{2} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )}}{6 \, d} \]
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Time = 0.41 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.34 \[ \int (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx=\frac {21 \, {\left (d x + c\right )} a^{2} + \frac {6 \, {\left (a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac {4 \, {\left (9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 10 \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \]
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Time = 15.82 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.84 \[ \int (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx=\frac {7\,a^2\,x}{2}+\frac {\frac {7\,a^2\,\left (c+d\,x\right )}{2}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {21\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (63\,c+63\,d\,x-150\right )}{6}\right )-\frac {a^2\,\left (21\,c+21\,d\,x-64\right )}{6}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {21\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (63\,c+63\,d\,x-42\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {35\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (105\,c+105\,d\,x-126\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {35\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (105\,c+105\,d\,x-194\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {49\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (147\,c+147\,d\,x-196\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {49\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (147\,c+147\,d\,x-252\right )}{6}\right )}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
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