Integrand size = 26, antiderivative size = 106 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^3} \, dx=-\frac {a^3 x}{c^3}+\frac {2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\frac {2 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}+\frac {2 a^3 \cos (e+f x)}{f \left (c^3-c^3 \sin (e+f x)\right )} \]
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Time = 0.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2759, 8} \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^3} \, dx=\frac {2 a^3 \cos (e+f x)}{f \left (c^3-c^3 \sin (e+f x)\right )}-\frac {a^3 x}{c^3}+\frac {2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\frac {2 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3} \]
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Rule 8
Rule 2759
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))^6} \, dx \\ & = \frac {2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\left (a^3 c\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^4} \, dx \\ & = \frac {2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\frac {2 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}+\frac {a^3 \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{c} \\ & = \frac {2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\frac {2 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}+\frac {2 a^3 \cos (e+f x)}{f \left (c^3-c^3 \sin (e+f x)\right )}-\frac {a^3 \int 1 \, dx}{c^3} \\ & = -\frac {a^3 x}{c^3}+\frac {2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\frac {2 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}+\frac {2 a^3 \cos (e+f x)}{f \left (c^3-c^3 \sin (e+f x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(249\) vs. \(2(106)=212\).
Time = 6.53 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.35 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (24 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-44 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-15 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+48 \sin \left (\frac {1}{2} (e+f x)\right )-88 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {1}{2} (e+f x)\right )+92 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sin \left (\frac {1}{2} (e+f x)\right )\right ) (a+a \sin (e+f x))^3}{15 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^3} \]
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Result contains complex when optimal does not.
Time = 0.95 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.92
method | result | size |
risch | \(-\frac {a^{3} x}{c^{3}}+\frac {-24 i a^{3} {\mathrm e}^{3 i \left (f x +e \right )}+12 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+\frac {56 i a^{3} {\mathrm e}^{i \left (f x +e \right )}}{3}-\frac {112 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}}{3}+\frac {92 a^{3}}{15}}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{5} f \,c^{3}}\) | \(97\) |
derivativedivides | \(\frac {2 a^{3} \left (-\frac {32}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {40}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,c^{3}}\) | \(100\) |
default | \(\frac {2 a^{3} \left (-\frac {32}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {40}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,c^{3}}\) | \(100\) |
parallelrisch | \(-\frac {\left (\left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) f x +\left (-5 f x +4\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (10 f x -8\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-10 f x +\frac {80}{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (5 f x -\frac {40}{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-f x +\frac {52}{15}\right ) a^{3}}{f \,c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(115\) |
norman | \(\frac {\frac {a^{3} x}{c}+\frac {8 a^{3} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {48 a^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {52 a^{3}}{15 c f}-\frac {5 a^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c}+\frac {13 a^{3} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {25 a^{3} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {38 a^{3} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {46 a^{3} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {46 a^{3} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {38 a^{3} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {25 a^{3} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {13 a^{3} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {5 a^{3} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {a^{3} x \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {4 a^{3} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {40 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 c f}+\frac {64 a^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {112 a^{3} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {116 a^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {472 a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}-\frac {556 a^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}-\frac {1432 a^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} c^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(489\) |
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Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (105) = 210\).
Time = 0.29 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.22 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^3} \, dx=\frac {60 \, a^{3} f x - {\left (15 \, a^{3} f x - 46 \, a^{3}\right )} \cos \left (f x + e\right )^{3} - 24 \, a^{3} - {\left (45 \, a^{3} f x + 2 \, a^{3}\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (5 \, a^{3} f x - 12 \, a^{3}\right )} \cos \left (f x + e\right ) - {\left (60 \, a^{3} f x + 24 \, a^{3} - {\left (15 \, a^{3} f x + 46 \, a^{3}\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (5 \, a^{3} f x - 8 \, a^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \, {\left (c^{3} f \cos \left (f x + e\right )^{3} + 3 \, c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) - 4 \, c^{3} f - {\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) - 4 \, c^{3} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1282 vs. \(2 (95) = 190\).
Time = 7.60 (sec) , antiderivative size = 1282, normalized size of antiderivative = 12.09 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 785 vs. \(2 (105) = 210\).
Time = 0.30 (sec) , antiderivative size = 785, normalized size of antiderivative = 7.41 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^3} \, dx=\text {Too large to display} \]
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Time = 0.31 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.99 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^3} \, dx=-\frac {\frac {15 \, {\left (f x + e\right )} a^{3}}{c^{3}} + \frac {4 \, {\left (15 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 30 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 100 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 50 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 13 \, a^{3}\right )}}{c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{5}}}{15 \, f} \]
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Time = 8.26 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.92 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^3} \, dx=-\frac {a^3\,x}{c^3}-\frac {a^3\,\left (e+f\,x\right )-\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (5\,a^3\,\left (e+f\,x\right )-\frac {a^3\,\left (75\,e+75\,f\,x-200\right )}{15}\right )-\frac {a^3\,\left (15\,e+15\,f\,x-52\right )}{15}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (5\,a^3\,\left (e+f\,x\right )-\frac {a^3\,\left (75\,e+75\,f\,x-60\right )}{15}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (10\,a^3\,\left (e+f\,x\right )-\frac {a^3\,\left (150\,e+150\,f\,x-120\right )}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (10\,a^3\,\left (e+f\,x\right )-\frac {a^3\,\left (150\,e+150\,f\,x-400\right )}{15}\right )}{c^3\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^5} \]
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