Integrand size = 15, antiderivative size = 81 \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=\frac {(A+B) \cos (x)}{7 (1-\sin (x))^4}+\frac {(3 A-4 B) \cos (x)}{35 (1-\sin (x))^3}+\frac {2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))^2}+\frac {2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))} \]
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Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2829, 2729, 2727} \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=\frac {2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))}+\frac {2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))^2}+\frac {(3 A-4 B) \cos (x)}{35 (1-\sin (x))^3}+\frac {(A+B) \cos (x)}{7 (1-\sin (x))^4} \]
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Rule 2727
Rule 2729
Rule 2829
Rubi steps \begin{align*} \text {integral}& = \frac {(A+B) \cos (x)}{7 (1-\sin (x))^4}+\frac {1}{7} (3 A-4 B) \int \frac {1}{(1-\sin (x))^3} \, dx \\ & = \frac {(A+B) \cos (x)}{7 (1-\sin (x))^4}+\frac {(3 A-4 B) \cos (x)}{35 (1-\sin (x))^3}+\frac {1}{35} (2 (3 A-4 B)) \int \frac {1}{(1-\sin (x))^2} \, dx \\ & = \frac {(A+B) \cos (x)}{7 (1-\sin (x))^4}+\frac {(3 A-4 B) \cos (x)}{35 (1-\sin (x))^3}+\frac {2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))^2}+\frac {1}{105} (2 (3 A-4 B)) \int \frac {1}{1-\sin (x)} \, dx \\ & = \frac {(A+B) \cos (x)}{7 (1-\sin (x))^4}+\frac {(3 A-4 B) \cos (x)}{35 (1-\sin (x))^3}+\frac {2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))^2}+\frac {2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.67 \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=\frac {\cos (x) \left (36 A-13 B+(-39 A+52 B) \sin (x)+8 (3 A-4 B) \sin ^2(x)+(-6 A+8 B) \sin ^3(x)\right )}{105 (-1+\sin (x))^4} \]
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Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.99
method | result | size |
risch | \(\frac {\frac {8 B \,{\mathrm e}^{4 i x}}{3}-\frac {8 i B \,{\mathrm e}^{3 i x}}{3}+\frac {12 A \,{\mathrm e}^{2 i x}}{5}-\frac {4 i A \,{\mathrm e}^{i x}}{5}+4 i A \,{\mathrm e}^{3 i x}-\frac {16 B \,{\mathrm e}^{2 i x}}{5}+\frac {16 i B \,{\mathrm e}^{i x}}{15}-\frac {4 A}{35}+\frac {16 B}{105}}{\left ({\mathrm e}^{i x}-i\right )^{7}}\) | \(80\) |
parallelrisch | \(\frac {-210 A \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+\left (630 A -210 B \right ) \left (\tan ^{5}\left (\frac {x}{2}\right )\right )+\left (-1260 A +350 B \right ) \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\left (1260 A -560 B \right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+\left (-882 A +336 B \right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\left (294 A -182 B \right ) \tan \left (\frac {x}{2}\right )-72 A +26 B}{105 \left (\tan \left (\frac {x}{2}\right )-1\right )^{7}}\) | \(95\) |
default | \(-\frac {32 A +24 B}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {2 \left (36 A +32 B \right )}{5 \left (\tan \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {2 A}{\tan \left (\frac {x}{2}\right )-1}-\frac {2 \left (8 A +8 B \right )}{7 \left (\tan \left (\frac {x}{2}\right )-1\right )^{7}}-\frac {2 \left (18 A +10 B \right )}{3 \left (\tan \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {6 A +2 B}{\left (\tan \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {24 A +24 B}{3 \left (\tan \left (\frac {x}{2}\right )-1\right )^{6}}\) | \(115\) |
norman | \(\frac {-2 A \left (\tan ^{8}\left (\frac {x}{2}\right )\right )+\left (-\frac {318 A}{35}+\frac {362 B}{105}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\left (-\frac {102 A}{5}+\frac {98 B}{15}\right ) \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\left (\frac {14 A}{5}-\frac {26 B}{15}\right ) \tan \left (\frac {x}{2}\right )+\left (\frac {74 A}{5}-\frac {106 B}{15}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+\left (-14 A +\frac {10 B}{3}\right ) \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+\left (6 A -2 B \right ) \left (\tan ^{7}\left (\frac {x}{2}\right )\right )+\left (18 A -\frac {22 B}{3}\right ) \left (\tan ^{5}\left (\frac {x}{2}\right )\right )-\frac {24 A}{35}+\frac {26 B}{105}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right )^{7}}\) | \(132\) |
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Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (65) = 130\).
Time = 0.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.85 \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=-\frac {2 \, {\left (3 \, A - 4 \, B\right )} \cos \left (x\right )^{4} + 8 \, {\left (3 \, A - 4 \, B\right )} \cos \left (x\right )^{3} - 9 \, {\left (3 \, A - 4 \, B\right )} \cos \left (x\right )^{2} - 15 \, {\left (4 \, A - 3 \, B\right )} \cos \left (x\right ) - {\left (2 \, {\left (3 \, A - 4 \, B\right )} \cos \left (x\right )^{3} - 6 \, {\left (3 \, A - 4 \, B\right )} \cos \left (x\right )^{2} - 15 \, {\left (3 \, A - 4 \, B\right )} \cos \left (x\right ) + 15 \, A + 15 \, B\right )} \sin \left (x\right ) - 15 \, A - 15 \, B}{105 \, {\left (\cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 8 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{3} + 4 \, \cos \left (x\right )^{2} - 4 \, \cos \left (x\right ) - 8\right )} \sin \left (x\right ) + 4 \, \cos \left (x\right ) + 8\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 887 vs. \(2 (75) = 150\).
Time = 2.69 (sec) , antiderivative size = 887, normalized size of antiderivative = 10.95 \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (65) = 130\).
Time = 0.20 (sec) , antiderivative size = 309, normalized size of antiderivative = 3.81 \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=-\frac {2 \, B {\left (\frac {91 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {168 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {280 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {175 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - 13\right )}}{105 \, {\left (\frac {7 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {21 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {35 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {35 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {21 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {7 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {\sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} - 1\right )}} + \frac {2 \, A {\left (\frac {49 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {147 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {210 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {210 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {35 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - 12\right )}}{35 \, {\left (\frac {7 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {21 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {35 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {35 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {21 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {7 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {\sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} - 1\right )}} \]
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Time = 0.45 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.38 \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=-\frac {2 \, {\left (105 \, A \tan \left (\frac {1}{2} \, x\right )^{6} - 315 \, A \tan \left (\frac {1}{2} \, x\right )^{5} + 105 \, B \tan \left (\frac {1}{2} \, x\right )^{5} + 630 \, A \tan \left (\frac {1}{2} \, x\right )^{4} - 175 \, B \tan \left (\frac {1}{2} \, x\right )^{4} - 630 \, A \tan \left (\frac {1}{2} \, x\right )^{3} + 280 \, B \tan \left (\frac {1}{2} \, x\right )^{3} + 441 \, A \tan \left (\frac {1}{2} \, x\right )^{2} - 168 \, B \tan \left (\frac {1}{2} \, x\right )^{2} - 147 \, A \tan \left (\frac {1}{2} \, x\right ) + 91 \, B \tan \left (\frac {1}{2} \, x\right ) + 36 \, A - 13 \, B\right )}}{105 \, {\left (\tan \left (\frac {1}{2} \, x\right ) - 1\right )}^{7}} \]
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Time = 6.73 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.20 \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=-\frac {2\,A\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+\left (2\,B-6\,A\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+\left (12\,A-\frac {10\,B}{3}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+\left (\frac {16\,B}{3}-12\,A\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\left (\frac {42\,A}{5}-\frac {16\,B}{5}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\left (\frac {26\,B}{15}-\frac {14\,A}{5}\right )\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {24\,A}{35}-\frac {26\,B}{105}}{{\left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right )}^7} \]
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