Integrand size = 21, antiderivative size = 69 \[ \int \frac {\tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sec (c+d x)}{a d}+\frac {2 \sec ^3(c+d x)}{3 a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {\tan ^5(c+d x)}{5 a d} \]
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Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2785, 2687, 30, 2686, 200} \[ \int \frac {\tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\tan ^5(c+d x)}{5 a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {2 \sec ^3(c+d x)}{3 a d}-\frac {\sec (c+d x)}{a d} \]
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Rule 30
Rule 200
Rule 2686
Rule 2687
Rule 2785
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^2(c+d x) \tan ^4(c+d x) \, dx}{a}-\frac {\int \sec (c+d x) \tan ^5(c+d x) \, dx}{a} \\ & = \frac {\text {Subst}\left (\int x^4 \, dx,x,\tan (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a d} \\ & = \frac {\tan ^5(c+d x)}{5 a d}-\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 ^5(c+d x)}{5 a d} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.54 \[ \int \frac {\tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sec ^3(c+d x) (200-534 \cos (c+d x)+288 \cos (2 (c+d x))-178 \cos (3 (c+d x))+24 \cos (4 (c+d x))-64 \sin (c+d x)-178 \sin (2 (c+d x))+192 \sin (3 (c+d x))-89 \sin (4 (c+d x)))}{960 a d (1+\sin (c+d x))} \]
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Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07
method | result | size |
parallelrisch | \(\frac {\frac {16}{15}-\frac {32 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {32 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}+\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{15}}{d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(74\) |
norman | \(\frac {\frac {16}{15 a d}+\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{15 d a}-\frac {32 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}-\frac {32 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(92\) |
risch | \(-\frac {2 \left (25 i {\mathrm e}^{4 i \left (d x +c \right )}+5 \,{\mathrm e}^{5 i \left (d x +c \right )}+21 i {\mathrm e}^{2 i \left (d x +c \right )}+13 \,{\mathrm e}^{3 i \left (d x +c \right )}+15 i {\mathrm e}^{6 i \left (d x +c \right )}+15 \,{\mathrm e}^{7 i \left (d x +c \right )}-9 \,{\mathrm e}^{i \left (d x +c \right )}+3 i\right )}{15 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d a}\) | \(120\) |
derivativedivides | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {3}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d a}\) | \(130\) |
default | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {3}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d a}\) | \(130\) |
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Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - 1}{15 \, {\left (a d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{3}\right )}} \]
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\[ \int \frac {\tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\sin ^{4}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (63) = 126\).
Time = 0.23 (sec) , antiderivative size = 214, normalized size of antiderivative = 3.10 \[ \int \frac {\tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {16 \, {\left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + 1\right )}}{15 \, {\left (a + \frac {2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {6 \, a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {2 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {2 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d} \]
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Time = 0.33 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.74 \[ \int \frac {\tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {5 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11\right )}}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} - \frac {45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 490 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 73}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{120 \, d} \]
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Time = 10.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.06 \[ \int \frac {\tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {16\,\left (-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{15\,a\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5} \]
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