Integrand size = 27, antiderivative size = 105 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{a}+\frac {\sec (c+d x)}{a d}-\frac {2 \sec ^3(c+d x)}{3 a d}+\frac {\sec ^5(c+d x)}{5 a d}-\frac {\tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d}-\frac {\tan ^5(c+d x)}{5 a d} \]
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Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2918, 2686, 200, 3554, 8} \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\tan ^5(c+d x)}{5 a d}+\frac {\tan ^3(c+d x)}{3 a d}-\frac {\tan (c+d x)}{a d}+\frac {\sec ^5(c+d x)}{5 a d}-\frac {2 \sec ^3(c+d x)}{3 a d}+\frac {\sec (c+d x)}{a d}+\frac {x}{a} \]
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Rule 8
Rule 200
Rule 2686
Rule 2918
Rule 3554
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec (c+d x) \tan ^5(c+d x) \, dx}{a}-\frac {\int \tan ^6(c+d x) \, dx}{a} \\ & = -\frac {\tan ^5(c+d x)}{5 a d}+\frac {\int \tan ^4(c+d x) \, dx}{a}+\frac {\text {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a d} \\ & = \frac {\tan ^3(c+d x)}{3 a d}-\frac {\tan ^5(c+d x)}{5 a d}-\frac {\int \tan ^2(c+d x) \, dx}{a}+\frac {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\sec (c+d x)\right )}{a d} \\ & = \frac {\sec (c+d x)}{a d}-\frac {2 \sec ^3(c+d x)}{3 a d}+\frac {\sec ^5(c+d x)}{5 a d}-\frac {\tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d}-\frac {\tan ^5(c+d x)}{5 a d}+\frac {\int 1 \, dx}{a} \\ & = \frac {x}{a}+\frac {\sec (c+d x)}{a d}-\frac {2 \sec ^3(c+d x)}{3 a d}+\frac {\sec ^5(c+d x)}{5 a d}-\frac {\tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d}-\frac {\tan ^5(c+d x)}{5 a d} \\ \end{align*}
Time = 0.90 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.82 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sec ^3(c+d x) \left (-25+\left (\frac {267}{4}-90 c-90 d x\right ) \cos (c+d x)-16 \cos (2 (c+d x))+\frac {89}{4} \cos (3 (c+d x))-30 c \cos (3 (c+d x))-30 d x \cos (3 (c+d x))-23 \cos (4 (c+d x))+8 \sin (c+d x)+\frac {89}{4} \sin (2 (c+d x))-30 c \sin (2 (c+d x))-30 d x \sin (2 (c+d x))+16 \sin (3 (c+d x))+\frac {89}{8} \sin (4 (c+d x))-15 c \sin (4 (c+d x))-15 d x \sin (4 (c+d x))\right )}{120 a d (1+\sin (c+d x))} \]
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Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.20
method | result | size |
risch | \(\frac {x}{a}+\frac {-2 i {\mathrm e}^{6 i \left (d x +c \right )}+2 \,{\mathrm e}^{7 i \left (d x +c \right )}+\frac {10 i {\mathrm e}^{4 i \left (d x +c \right )}}{3}+\frac {26 \,{\mathrm e}^{5 i \left (d x +c \right )}}{3}+\frac {62 i {\mathrm e}^{2 i \left (d x +c \right )}}{15}+\frac {146 \,{\mathrm e}^{3 i \left (d x +c \right )}}{15}+\frac {46 i}{15}+\frac {62 \,{\mathrm e}^{i \left (d x +c \right )}}{15}}{\left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} d a}\) | \(126\) |
derivativedivides | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {5}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {11}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d a}\) | \(127\) |
default | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {5}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {11}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d a}\) | \(127\) |
parallelrisch | \(\frac {15 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d +\left (30 d x +30\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-30 d x +60\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-90 d x -70\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-200 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (90 d x +26\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 d x +92\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-30 d x -2\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-15 d x -16}{15 d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(175\) |
norman | \(\frac {\frac {x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x}{a}-\frac {44 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}-\frac {2 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}+\frac {4 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {2 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {2 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {4 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {2 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {16}{15 a d}+\frac {4 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {8 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {28 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {36 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{15 d a}+\frac {76 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(376\) |
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Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.93 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {15 \, d x \cos \left (d x + c\right )^{3} + 23 \, \cos \left (d x + c\right )^{4} - 19 \, \cos \left (d x + c\right )^{2} + {\left (15 \, d x \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 4}{15 \, {\left (a d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{3}\right )}} \]
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Timed out. \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (97) = 194\).
Time = 0.31 (sec) , antiderivative size = 318, normalized size of antiderivative = 3.03 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 \, {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {46 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {13 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {100 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {35 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {30 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + 8}{a + \frac {2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {6 \, a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {2 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {2 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}\right )}}{15 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.24 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {120 \, {\left (d x + c\right )}}{a} + \frac {5 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17\right )}}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {3 \, {\left (55 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 260 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 450 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 300 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 71\right )}}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{120 \, d} \]
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Time = 12.95 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.25 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{a}-\frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+\frac {40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-\frac {26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{15}-\frac {92\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}+\frac {16}{15}}{a\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5} \]
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