Integrand size = 26, antiderivative size = 69 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=\frac {a^3 c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}+\frac {a^3 c^2 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^7} \]
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Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2751, 2750} \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=\frac {a^3 c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}+\frac {a^3 c^2 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^7} \]
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Rule 2750
Rule 2751
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^8} \, dx \\ & = \frac {a^3 c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}+\frac {1}{9} \left (a^3 c^2\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx \\ & = \frac {a^3 c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}+\frac {a^3 c^2 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^7} \\ \end{align*}
Time = 2.41 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.96 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=\frac {a^3 \left (315 \cos \left (\frac {1}{2} (e+f x)\right )-189 \cos \left (\frac {3}{2} (e+f x)\right )-63 \cos \left (\frac {5}{2} (e+f x)\right )+9 \cos \left (\frac {7}{2} (e+f x)\right )+189 \sin \left (\frac {1}{2} (e+f x)\right )+105 \sin \left (\frac {3}{2} (e+f x)\right )-27 \sin \left (\frac {5}{2} (e+f x)\right )-\sin \left (\frac {9}{2} (e+f x)\right )\right )}{504 c^5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \]
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Result contains complex when optimal does not.
Time = 1.14 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.59
method | result | size |
risch | \(\frac {2 i a^{3} \left (105 i {\mathrm e}^{6 i \left (f x +e \right )}+63 \,{\mathrm e}^{7 i \left (f x +e \right )}-189 i {\mathrm e}^{4 i \left (f x +e \right )}-315 \,{\mathrm e}^{5 i \left (f x +e \right )}+27 i {\mathrm e}^{2 i \left (f x +e \right )}+189 \,{\mathrm e}^{3 i \left (f x +e \right )}+i-9 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{63 f \,c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9}}\) | \(110\) |
parallelrisch | \(-\frac {2 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )-\left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {23 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-5 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+11 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-3 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {25 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{7}+\frac {8}{63}\right ) a^{3}}{f \,c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(127\) |
derivativedivides | \(\frac {2 a^{3} \left (-\frac {928}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {128}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {7}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {136}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {76}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {496}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {86}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{5}}\) | \(148\) |
default | \(\frac {2 a^{3} \left (-\frac {928}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {128}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {7}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {136}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {76}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {496}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {86}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{5}}\) | \(148\) |
norman | \(\frac {\frac {2 a^{3} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {16 a^{3} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {42 a^{3} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {16 a^{3}}{63 c f}-\frac {2 a^{3} \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {2 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{7 c f}+\frac {48 a^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {64 a^{3} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {74 a^{3} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {166 a^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 c f}+\frac {202 a^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}+\frac {352 a^{3} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {848 a^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {928 a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 c f}-\frac {6490 a^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{63 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(351\) |
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Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (67) = 134\).
Time = 0.26 (sec) , antiderivative size = 276, normalized size of antiderivative = 4.00 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=-\frac {a^{3} \cos \left (f x + e\right )^{5} - 4 \, a^{3} \cos \left (f x + e\right )^{4} + 19 \, a^{3} \cos \left (f x + e\right )^{3} + 52 \, a^{3} \cos \left (f x + e\right )^{2} - 28 \, a^{3} \cos \left (f x + e\right ) - 56 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{4} + 5 \, a^{3} \cos \left (f x + e\right )^{3} + 24 \, a^{3} \cos \left (f x + e\right )^{2} - 28 \, a^{3} \cos \left (f x + e\right ) - 56 \, a^{3}\right )} \sin \left (f x + e\right )}{63 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f - {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1717 vs. \(2 (60) = 120\).
Time = 25.07 (sec) , antiderivative size = 1717, normalized size of antiderivative = 24.88 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1389 vs. \(2 (67) = 134\).
Time = 0.24 (sec) , antiderivative size = 1389, normalized size of antiderivative = 20.13 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (67) = 134\).
Time = 0.34 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.22 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=-\frac {2 \, {\left (63 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 63 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 483 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 315 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 693 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 189 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 225 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 9 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 8 \, a^{3}\right )}}{63 \, c^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}} \]
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Time = 7.89 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.75 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=\frac {\sqrt {2}\,a^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {37\,\cos \left (3\,e+3\,f\,x\right )}{8}-\frac {63\,\sin \left (e+f\,x\right )}{2}-\frac {113\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {257\,\cos \left (e+f\,x\right )}{8}+\frac {7\,\cos \left (4\,e+4\,f\,x\right )}{16}+\frac {63\,\sin \left (2\,e+2\,f\,x\right )}{8}+\frac {9\,\sin \left (3\,e+3\,f\,x\right )}{2}-\frac {9\,\sin \left (4\,e+4\,f\,x\right )}{16}+\frac {1013}{16}\right )}{1008\,c^5\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^9} \]
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