Integrand size = 13, antiderivative size = 75 \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx=-\frac {(A-B) \cos (x)}{7 (1+\sin (x))^4}-\frac {(3 A+4 B) \cos (x)}{35 (1+\sin (x))^3}-\frac {2 (3 A+4 B) \cos (x)}{105 (1+\sin (x))^2}-\frac {2 (3 A+4 B) \cos (x)}{105 (1+\sin (x))} \]
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Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2829, 2729, 2727} \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx=-\frac {2 (3 A+4 B) \cos (x)}{105 (\sin (x)+1)}-\frac {2 (3 A+4 B) \cos (x)}{105 (\sin (x)+1)^2}-\frac {(3 A+4 B) \cos (x)}{35 (\sin (x)+1)^3}-\frac {(A-B) \cos (x)}{7 (\sin (x)+1)^4} \]
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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 = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.73 \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx=-\frac {\cos (x) \left (36 A+13 B+13 (3 A+4 B) \sin (x)+8 (3 A+4 B) \sin ^2(x)+(6 A+8 B) \sin ^3(x)\right )}{105 (1+\sin (x))^4} \]
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Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.07
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 | \(-\frac {-32 A +24 B}{2 \left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {-24 A +24 B}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{6}}-\frac {2 \left (8 A -8 B \right )}{7 \left (\tan \left (\frac {x}{2}\right )+1\right )^{7}}-\frac {2 A}{\tan \left (\frac {x}{2}\right )+1}-\frac {2 \left (18 A -10 B \right )}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {2 \left (36 A -32 B \right )}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {-6 A +2 B}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}\) | \(115\) |
norman | \(\frac {-2 A \left (\tan ^{8}\left (\frac {x}{2}\right )\right )+\left (-\frac {318 A}{35}-\frac {362 B}{105}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\left (-\frac {102 A}{5}-\frac {98 B}{15}\right ) \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\left (-\frac {74 A}{5}-\frac {106 B}{15}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+\left (-\frac {14 A}{5}-\frac {26 B}{15}\right ) \tan \left (\frac {x}{2}\right )+\left (-6 A -2 B \right ) \left (\tan ^{7}\left (\frac {x}{2}\right )\right )+\left (-14 A -\frac {10 B}{3}\right ) \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+\left (-18 A -\frac {22 B}{3}\right ) \left (\tan ^{5}\left (\frac {x}{2}\right )\right )-\frac {24 A}{35}-\frac {26 B}{105}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )^{7}}\) | \(132\) |
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Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (67) = 134\).
Time = 0.25 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.00 \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx=\frac {2 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{4} + 8 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{3} - 9 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{2} - 15 \, {\left (4 \, A + 3 \, B\right )} \cos \left (x\right ) + {\left (2 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{3} - 6 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{2} - 15 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right ) + 15 \, A - 15 \, B\right )} \sin \left (x\right ) - 15 \, A + 15 \, B}{105 \, {\left (\cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 8 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{3} + 4 \, \cos \left (x\right )^{2} - 4 \, \cos \left (x\right ) - 8\right )} \sin \left (x\right ) + 4 \, \cos \left (x\right ) + 8\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 889 vs. \(2 (75) = 150\).
Time = 2.67 (sec) , antiderivative size = 889, normalized size of antiderivative = 11.85 \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (67) = 134\).
Time = 0.21 (sec) , antiderivative size = 309, normalized size of antiderivative = 4.12 \[ \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 = 0.39 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.49 \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx=-\frac {2 \, {\left (105 \, A \tan \left (\frac {1}{2} \, x\right )^{6} + 315 \, A \tan \left (\frac {1}{2} \, x\right )^{5} + 105 \, B \tan \left (\frac {1}{2} \, x\right )^{5} + 630 \, A \tan \left (\frac {1}{2} \, x\right )^{4} + 175 \, B \tan \left (\frac {1}{2} \, x\right )^{4} + 630 \, A \tan \left (\frac {1}{2} \, x\right )^{3} + 280 \, B \tan \left (\frac {1}{2} \, x\right )^{3} + 441 \, A \tan \left (\frac {1}{2} \, x\right )^{2} + 168 \, B \tan \left (\frac {1}{2} \, x\right )^{2} + 147 \, A \tan \left (\frac {1}{2} \, x\right ) + 91 \, B \tan \left (\frac {1}{2} \, x\right ) + 36 \, A + 13 \, B\right )}}{105 \, {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{7}} \]
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Time = 6.81 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.25 \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx=-\frac {2\,A\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+\left (6\,A+2\,B\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+\left (12\,A+\frac {10\,B}{3}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+\left (12\,A+\frac {16\,B}{3}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\left (\frac {42\,A}{5}+\frac {16\,B}{5}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\left (\frac {14\,A}{5}+\frac {26\,B}{15}\right )\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {24\,A}{35}+\frac {26\,B}{105}}{{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^7} \]
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