Integrand size = 36, antiderivative size = 151 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {a^3 B x}{c^4}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {2 a^3 B c \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}+\frac {2 a^3 B c^2 \cos ^3(e+f x)}{3 f \left (c^2-c^2 \sin (e+f x)\right )^3}-\frac {2 a^3 B \cos (e+f x)}{f \left (c^4-c^4 \sin (e+f x)\right )} \]
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Time = 0.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3046, 2938, 2759, 8} \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {a^3 c^3 (A+B) \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {2 a^3 B \cos (e+f x)}{f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {a^3 B x}{c^4}+\frac {2 a^3 B c^2 \cos ^3(e+f x)}{3 f \left (c^2-c^2 \sin (e+f x)\right )^3}-\frac {2 a^3 B c \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5} \]
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Rule 8
Rule 2759
Rule 2938
Rule 3046
Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\left (a^3 B c^2\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^6} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {2 a^3 B c \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}+\left (a^3 B\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^4} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {2 a^3 B c \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}+\frac {2 a^3 B \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^3}-\frac {\left (a^3 B\right ) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{c^2} \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {2 a^3 B c \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}+\frac {2 a^3 B \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^3}-\frac {2 a^3 B \cos (e+f x)}{f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {\left (a^3 B\right ) \int 1 \, dx}{c^4} \\ & = \frac {a^3 B x}{c^4}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {2 a^3 B c \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}+\frac {2 a^3 B \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^3}-\frac {2 a^3 B \cos (e+f x)}{f \left (c^4-c^4 \sin (e+f x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(356\) vs. \(2(151)=302\).
Time = 11.47 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.36 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (120 (A+B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-12 (15 A+29 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+2 (45 A+199 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+105 B (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7+240 (A+B) \sin \left (\frac {1}{2} (e+f x)\right )-24 (15 A+29 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {1}{2} (e+f x)\right )+4 (45 A+199 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sin \left (\frac {1}{2} (e+f x)\right )-2 (15 A+337 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3}{105 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^4} \]
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Time = 1.03 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.17
method | result | size |
derivativedivides | \(\frac {2 a^{3} \left (B \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {A -B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {12 A +4 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {60 A +20 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {64 A +64 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {160 A +96 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {192 A +192 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {240 A +208 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\right )}{f \,c^{4}}\) | \(177\) |
default | \(\frac {2 a^{3} \left (B \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {A -B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {12 A +4 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {60 A +20 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {64 A +64 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {160 A +96 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {192 A +192 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {240 A +208 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\right )}{f \,c^{4}}\) | \(177\) |
parallelrisch | \(-\frac {2 \left (-\frac {B \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) f x}{2}+\left (\frac {7}{2} f x B +A -B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+B \left (-\frac {21 f x}{2}+8\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {35}{2} f x B +5 A -\frac {55}{3} B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+B \left (-\frac {35 f x}{2}+\frac {112}{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {21}{2} f x B +3 A -\frac {127}{5} B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+B \left (-\frac {7 f x}{2}+\frac {152}{15}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {f x B}{2}+\frac {A}{7}-\frac {167 B}{105}\right ) a^{3}}{f \,c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(178\) |
risch | \(\frac {a^{3} B x}{c^{4}}-\frac {2 \left (-337 B \,a^{3}-15 A \,a^{3}-2520 i B \,a^{3} {\mathrm e}^{5 i \left (f x +e \right )}+6160 i B \,a^{3} {\mathrm e}^{3 i \left (f x +e \right )}-1624 i B \,a^{3} {\mathrm e}^{i \left (f x +e \right )}+105 A \,a^{3} {\mathrm e}^{6 i \left (f x +e \right )}+735 B \,a^{3} {\mathrm e}^{6 i \left (f x +e \right )}-525 A \,a^{3} {\mathrm e}^{4 i \left (f x +e \right )}-5635 B \,a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+315 A \,a^{3} {\mathrm e}^{2 i \left (f x +e \right )}+4557 B \,a^{3} {\mathrm e}^{2 i \left (f x +e \right )}\right )}{105 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7} f \,c^{4}}\) | \(184\) |
norman | \(\frac {-\frac {2336 B \,a^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}-\frac {6544 B \,a^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}-\frac {a^{3} x B}{c}-\frac {304 B \,a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{15 c f}+\frac {7 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c}-\frac {\left (870 A \,a^{3}-3142 B \,a^{3}\right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}-\frac {\left (2 A \,a^{3}-2 B \,a^{3}\right ) \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {\left (150 A \,a^{3}-1334 B \,a^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 c f}-\frac {\left (750 A \,a^{3}-5438 B \,a^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 c f}+\frac {a^{3} x B \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {25 a^{3} x B \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {63 a^{3} x B \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {125 a^{3} x B \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {203 a^{3} x B \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {8896 B \,a^{3} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}-\frac {6224 B \,a^{3} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}-\frac {16 B \,a^{3} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {416 B \,a^{3} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {277 a^{3} x B \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {323 a^{3} x B \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {323 a^{3} x B \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {277 a^{3} x B \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {203 a^{3} x B \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {125 a^{3} x B \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {63 a^{3} x B \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {25 a^{3} x B \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {7 a^{3} x B \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {\left (54 A \,a^{3}-134 B \,a^{3}\right ) \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {\left (1662 A \,a^{3}-9790 B \,a^{3}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 c f}-\frac {\left (1938 A \,a^{3}-9122 B \,a^{3}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 c f}-\frac {30 A \,a^{3}-334 B \,a^{3}}{105 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4} c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(760\) |
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Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (149) = 298\).
Time = 0.27 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.40 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {840 \, B a^{3} f x + {\left (105 \, B a^{3} f x + {\left (15 \, A + 337 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{4} + 120 \, {\left (A + B\right )} a^{3} - {\left (315 \, B a^{3} f x + {\left (45 \, A - 613 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{3} - 24 \, {\left (35 \, B a^{3} f x + {\left (5 \, A + 26 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{2} + 60 \, {\left (7 \, B a^{3} f x + {\left (A - 13 \, B\right )} a^{3}\right )} \cos \left (f x + e\right ) - {\left (840 \, B a^{3} f x - 120 \, {\left (A + B\right )} a^{3} - {\left (105 \, B a^{3} f x - {\left (15 \, A + 337 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{3} - 12 \, {\left (35 \, B a^{3} f x - {\left (5 \, A - 23 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{2} + 60 \, {\left (7 \, B a^{3} f x - {\left (A + 15 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{105 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2951 vs. \(2 (141) = 282\).
Time = 23.73 (sec) , antiderivative size = 2951, normalized size of antiderivative = 19.54 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2118 vs. \(2 (149) = 298\).
Time = 0.39 (sec) , antiderivative size = 2118, normalized size of antiderivative = 14.03 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \]
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Time = 0.50 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.34 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {\frac {105 \, {\left (f x + e\right )} B a^{3}}{c^{4}} - \frac {2 \, {\left (105 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 105 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 840 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 525 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1925 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 3920 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 315 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2667 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1064 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, A a^{3} - 167 \, B a^{3}\right )}}{c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}}}{105 \, f} \]
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Time = 16.54 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.09 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {B\,a^3\,x}{c^4}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (\frac {a^3\,\left (1680\,B-2205\,B\,\left (e+f\,x\right )\right )}{105}+21\,B\,a^3\,\left (e+f\,x\right )\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {a^3\,\left (7840\,B-3675\,B\,\left (e+f\,x\right )\right )}{105}+35\,B\,a^3\,\left (e+f\,x\right )\right )+\frac {a^3\,\left (30\,A-334\,B+105\,B\,\left (e+f\,x\right )\right )}{105}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {a^3\,\left (210\,A-210\,B+735\,B\,\left (e+f\,x\right )\right )}{105}-7\,B\,a^3\,\left (e+f\,x\right )\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {a^3\,\left (630\,A-5334\,B+2205\,B\,\left (e+f\,x\right )\right )}{105}-21\,B\,a^3\,\left (e+f\,x\right )\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {a^3\,\left (1050\,A-3850\,B+3675\,B\,\left (e+f\,x\right )\right )}{105}-35\,B\,a^3\,\left (e+f\,x\right )\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {a^3\,\left (2128\,B-735\,B\,\left (e+f\,x\right )\right )}{105}+7\,B\,a^3\,\left (e+f\,x\right )\right )-B\,a^3\,\left (e+f\,x\right )}{c^4\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^7} \]
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