Integrand size = 32, antiderivative size = 129 \[ \int \frac {\sin ^4(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=-\frac {19 A x}{2 a^3}-\frac {4 A \cos (c+d x)}{a^3 d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac {41 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}-\frac {199 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))} \]
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Time = 0.17 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3045, 2718, 2715, 8, 2729, 2727} \[ \int \frac {\sin ^4(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=-\frac {4 A \cos (c+d x)}{a^3 d}+\frac {A \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {199 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)}+\frac {41 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)^2}-\frac {2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}-\frac {19 A x}{2 a^3} \]
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Rule 8
Rule 2715
Rule 2718
Rule 2727
Rule 2729
Rule 3045
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {9 A}{a^3}+\frac {4 A \sin (c+d x)}{a^3}-\frac {A \sin ^2(c+d x)}{a^3}+\frac {2 A}{a^3 (1+\sin (c+d x))^3}-\frac {9 A}{a^3 (1+\sin (c+d x))^2}+\frac {16 A}{a^3 (1+\sin (c+d x))}\right ) \, dx \\ & = -\frac {9 A x}{a^3}-\frac {A \int \sin ^2(c+d x) \, dx}{a^3}+\frac {(2 A) \int \frac {1}{(1+\sin (c+d x))^3} \, dx}{a^3}+\frac {(4 A) \int \sin (c+d x) \, dx}{a^3}-\frac {(9 A) \int \frac {1}{(1+\sin (c+d x))^2} \, dx}{a^3}+\frac {(16 A) \int \frac {1}{1+\sin (c+d x)} \, dx}{a^3} \\ & = -\frac {9 A x}{a^3}-\frac {4 A \cos (c+d x)}{a^3 d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac {3 A \cos (c+d x)}{a^3 d (1+\sin (c+d x))^2}-\frac {16 A \cos (c+d x)}{a^3 d (1+\sin (c+d x))}-\frac {A \int 1 \, dx}{2 a^3}+\frac {(4 A) \int \frac {1}{(1+\sin (c+d x))^2} \, dx}{5 a^3}-\frac {(3 A) \int \frac {1}{1+\sin (c+d x)} \, dx}{a^3} \\ & = -\frac {19 A x}{2 a^3}-\frac {4 A \cos (c+d x)}{a^3 d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac {41 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}-\frac {13 A \cos (c+d x)}{a^3 d (1+\sin (c+d x))}+\frac {(4 A) \int \frac {1}{1+\sin (c+d x)} \, dx}{15 a^3} \\ & = -\frac {19 A x}{2 a^3}-\frac {4 A \cos (c+d x)}{a^3 d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac {41 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}-\frac {199 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 3.80 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.64 \[ \int \frac {\sin ^4(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=\frac {A \sec (c+d x) \sqrt {1-\sin (c+d x)} \left (140 \sqrt {2} \sqrt {a} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},\frac {1}{2} (1+\sin (c+d x))\right ) (1+\sin (c+d x))-360 \arcsin \left (\frac {\sqrt {a (1+\sin (c+d x))}}{\sqrt {2} \sqrt {a}}\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 \sqrt {a (1+\sin (c+d x))}+\sqrt {a} \sqrt {1-\sin (c+d x)} \left (-308-639 \sin (c+d x)-433 \sin ^2(c+d x)-75 \sin ^3(c+d x)+15 \sin ^4(c+d x)\right )\right )}{30 a^{7/2} d (1+\sin (c+d x))^2} \]
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Time = 1.04 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {32 A \left (-\frac {1}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {5}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {9}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+4}{16 \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {19 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}\right )}{d \,a^{3}}\) | \(154\) |
default | \(\frac {32 A \left (-\frac {1}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {5}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {9}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+4}{16 \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {19 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}\right )}{d \,a^{3}}\) | \(154\) |
risch | \(-\frac {19 A x}{2 a^{3}}-\frac {i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}-\frac {2 A \,{\mathrm e}^{i \left (d x +c \right )}}{a^{3} d}-\frac {2 A \,{\mathrm e}^{-i \left (d x +c \right )}}{a^{3} d}+\frac {i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}-\frac {2 \left (825 i A \,{\mathrm e}^{3 i \left (d x +c \right )}+240 A \,{\mathrm e}^{4 i \left (d x +c \right )}-755 i A \,{\mathrm e}^{i \left (d x +c \right )}-1165 A \,{\mathrm e}^{2 i \left (d x +c \right )}+199 A \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5}}\) | \(159\) |
parallelrisch | \(\frac {\left (\left (76 d x -120\right ) \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+\left (380 d x -128\right ) \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+\left (76 d x +\frac {3484}{15}\right ) \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+\left (-760 d x -304\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-380 d x \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-760 d x \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )-\frac {2068 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )}{3}+11 \sin \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )-\sin \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )-\frac {2456 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{3}+11 \cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )\right ) A}{8 d \,a^{3} \left (-\sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+5 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-5 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-\cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+10 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(234\) |
norman | \(\frac {-\frac {19 A x \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {95 A \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {19 A \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {391 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a d}-\frac {1919 A x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {448 A}{15 a d}-\frac {1900 A \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {836 A \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {2565 A x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {665 A x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {353 A \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {285 A x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {2945 A x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {95 A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {19 A x}{2 a}-\frac {2945 A x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {1235 A x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {665 A x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {7192 A \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {2232 A \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {5117 A \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {1919 A x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {2565 A x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {6979 A \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {2300 A \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {1308 A \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {285 A x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {5751 A \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d}-\frac {95 A x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {5599 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {1235 A x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(596\) |
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Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (119) = 238\).
Time = 0.26 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.92 \[ \int \frac {\sin ^4(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=-\frac {15 \, A \cos \left (d x + c\right )^{5} + 90 \, A \cos \left (d x + c\right )^{4} + {\left (285 \, A d x + 683 \, A\right )} \cos \left (d x + c\right )^{3} - 1140 \, A d x + {\left (855 \, A d x - 526 \, A\right )} \cos \left (d x + c\right )^{2} - 6 \, {\left (95 \, A d x + 191 \, A\right )} \cos \left (d x + c\right ) - {\left (15 \, A \cos \left (d x + c\right )^{4} - 75 \, A \cos \left (d x + c\right )^{3} + 1140 \, A d x - 19 \, {\left (15 \, A d x - 32 \, A\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left (95 \, A d x + 189 \, A\right )} \cos \left (d x + c\right ) - 12 \, A\right )} \sin \left (d x + c\right ) - 12 \, A}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 3614 vs. \(2 (126) = 252\).
Time = 22.20 (sec) , antiderivative size = 3614, normalized size of antiderivative = 28.02 \[ \int \frac {\sin ^4(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (119) = 238\).
Time = 0.35 (sec) , antiderivative size = 715, normalized size of antiderivative = 5.54 \[ \int \frac {\sin ^4(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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Time = 0.32 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.21 \[ \int \frac {\sin ^4(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {285 \, {\left (d x + c\right )} A}{a^{3}} + \frac {30 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, A\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}} + \frac {4 \, {\left (135 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 615 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1025 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 685 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 164 \, A\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{30 \, d} \]
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Time = 16.44 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.53 \[ \int \frac {\sin ^4(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=\frac {\left (\frac {95\,A\,\left (c+d\,x\right )}{2}-\frac {A\,\left (1425\,c+1425\,d\,x+570\right )}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\left (114\,A\,\left (c+d\,x\right )-\frac {A\,\left (3420\,c+3420\,d\,x+2850\right )}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (190\,A\,\left (c+d\,x\right )-\frac {A\,\left (5700\,c+5700\,d\,x+6650\right )}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (247\,A\,\left (c+d\,x\right )-\frac {A\,\left (7410\,c+7410\,d\,x+10450\right )}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (247\,A\,\left (c+d\,x\right )-\frac {A\,\left (7410\,c+7410\,d\,x+12846\right )}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (190\,A\,\left (c+d\,x\right )-\frac {A\,\left (5700\,c+5700\,d\,x+11270\right )}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (114\,A\,\left (c+d\,x\right )-\frac {A\,\left (3420\,c+3420\,d\,x+7902\right )}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\left (\frac {95\,A\,\left (c+d\,x\right )}{2}-\frac {A\,\left (1425\,c+1425\,d\,x+3910\right )}{30}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {19\,A\,\left (c+d\,x\right )}{2}-\frac {A\,\left (285\,c+285\,d\,x+896\right )}{30}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2}-\frac {19\,A\,x}{2\,a^3} \]
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