Integrand size = 32, antiderivative size = 130 \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2 c \text {arctanh}(\cos (e+f x))}{16 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot ^5(e+f x)}{5 f}+\frac {a^2 c \cot (e+f x) \csc (e+f x)}{16 f}+\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{24 f}-\frac {a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f} \]
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Time = 0.15 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3045, 3852, 3853, 3855} \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2 c \text {arctanh}(\cos (e+f x))}{16 f}-\frac {a^2 c \cot ^5(e+f x)}{5 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f}+\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{24 f}+\frac {a^2 c \cot (e+f x) \csc (e+f x)}{16 f} \]
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Rule 3045
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (-a^2 c \csc ^4(e+f x)-a^2 c \csc ^5(e+f x)+a^2 c \csc ^6(e+f x)+a^2 c \csc ^7(e+f x)\right ) \, dx \\ & = -\left (\left (a^2 c\right ) \int \csc ^4(e+f x) \, dx\right )-\left (a^2 c\right ) \int \csc ^5(e+f x) \, dx+\left (a^2 c\right ) \int \csc ^6(e+f x) \, dx+\left (a^2 c\right ) \int \csc ^7(e+f x) \, dx \\ & = \frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{4 f}-\frac {a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f}-\frac {1}{4} \left (3 a^2 c\right ) \int \csc ^3(e+f x) \, dx+\frac {1}{6} \left (5 a^2 c\right ) \int \csc ^5(e+f x) \, dx+\frac {\left (a^2 c\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (e+f x)\right )}{f}-\frac {\left (a^2 c\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (e+f x)\right )}{f} \\ & = -\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot ^5(e+f x)}{5 f}+\frac {3 a^2 c \cot (e+f x) \csc (e+f x)}{8 f}+\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{24 f}-\frac {a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f}-\frac {1}{8} \left (3 a^2 c\right ) \int \csc (e+f x) \, dx+\frac {1}{8} \left (5 a^2 c\right ) \int \csc ^3(e+f x) \, dx \\ & = \frac {3 a^2 c \text {arctanh}(\cos (e+f x))}{8 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot ^5(e+f x)}{5 f}+\frac {a^2 c \cot (e+f x) \csc (e+f x)}{16 f}+\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{24 f}-\frac {a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f}+\frac {1}{16} \left (5 a^2 c\right ) \int \csc (e+f x) \, dx \\ & = \frac {a^2 c \text {arctanh}(\cos (e+f x))}{16 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot ^5(e+f x)}{5 f}+\frac {a^2 c \cot (e+f x) \csc (e+f x)}{16 f}+\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{24 f}-\frac {a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.57 \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {2 a^2 c \cot (e+f x)}{15 f}+\frac {a^2 c \csc ^2\left (\frac {1}{2} (e+f x)\right )}{64 f}-\frac {a^2 c \csc ^6\left (\frac {1}{2} (e+f x)\right )}{384 f}+\frac {a^2 c \cot (e+f x) \csc ^2(e+f x)}{15 f}-\frac {a^2 c \cot (e+f x) \csc ^4(e+f x)}{5 f}+\frac {a^2 c \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{16 f}-\frac {a^2 c \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{16 f}-\frac {a^2 c \sec ^2\left (\frac {1}{2} (e+f x)\right )}{64 f}+\frac {a^2 c \sec ^6\left (\frac {1}{2} (e+f x)\right )}{384 f} \]
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Time = 1.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(-\frac {19 c \left (\frac {512 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{19}+\left (\sec \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\cos \left (f x +e \right )+\frac {17 \cos \left (3 f x +3 e \right )}{114}-\frac {\cos \left (5 f x +5 e \right )}{38}\right ) \csc \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {128 \cos \left (f x +e \right )}{57}+\frac {32 \cos \left (3 f x +3 e \right )}{57}-\frac {32 \cos \left (5 f x +5 e \right )}{285}\right ) \left (\sec ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\csc ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right ) a^{2}}{8192 f}\) | \(125\) |
risch | \(-\frac {a^{2} c \left (15 \,{\mathrm e}^{11 i \left (f x +e \right )}-85 \,{\mathrm e}^{9 i \left (f x +e \right )}-570 \,{\mathrm e}^{7 i \left (f x +e \right )}+480 i {\mathrm e}^{8 i \left (f x +e \right )}-570 \,{\mathrm e}^{5 i \left (f x +e \right )}-320 i {\mathrm e}^{6 i \left (f x +e \right )}-85 \,{\mathrm e}^{3 i \left (f x +e \right )}+15 \,{\mathrm e}^{i \left (f x +e \right )}-192 i {\mathrm e}^{2 i \left (f x +e \right )}+32 i\right )}{120 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{6}}-\frac {a^{2} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{16 f}+\frac {a^{2} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{16 f}\) | \(171\) |
derivativedivides | \(\frac {-a^{2} c \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )-a^{2} c \left (\left (-\frac {\left (\csc ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )+a^{2} c \left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (f x +e \right )\right )}{15}\right ) \cot \left (f x +e \right )+a^{2} c \left (\left (-\frac {\left (\csc ^{5}\left (f x +e \right )\right )}{6}-\frac {5 \left (\csc ^{3}\left (f x +e \right )\right )}{24}-\frac {5 \csc \left (f x +e \right )}{16}\right ) \cot \left (f x +e \right )+\frac {5 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{16}\right )}{f}\) | \(174\) |
default | \(\frac {-a^{2} c \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )-a^{2} c \left (\left (-\frac {\left (\csc ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )+a^{2} c \left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (f x +e \right )\right )}{15}\right ) \cot \left (f x +e \right )+a^{2} c \left (\left (-\frac {\left (\csc ^{5}\left (f x +e \right )\right )}{6}-\frac {5 \left (\csc ^{3}\left (f x +e \right )\right )}{24}-\frac {5 \csc \left (f x +e \right )}{16}\right ) \cot \left (f x +e \right )+\frac {5 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{16}\right )}{f}\) | \(174\) |
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Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (118) = 236\).
Time = 0.27 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.85 \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=-\frac {30 \, a^{2} c \cos \left (f x + e\right )^{5} - 80 \, a^{2} c \cos \left (f x + e\right )^{3} - 30 \, a^{2} c \cos \left (f x + e\right ) - 15 \, {\left (a^{2} c \cos \left (f x + e\right )^{6} - 3 \, a^{2} c \cos \left (f x + e\right )^{4} + 3 \, a^{2} c \cos \left (f x + e\right )^{2} - a^{2} c\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 15 \, {\left (a^{2} c \cos \left (f x + e\right )^{6} - 3 \, a^{2} c \cos \left (f x + e\right )^{4} + 3 \, a^{2} c \cos \left (f x + e\right )^{2} - a^{2} c\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 32 \, {\left (2 \, a^{2} c \cos \left (f x + e\right )^{5} - 5 \, a^{2} c \cos \left (f x + e\right )^{3}\right )} \sin \left (f x + e\right )}{480 \, {\left (f \cos \left (f x + e\right )^{6} - 3 \, f \cos \left (f x + e\right )^{4} + 3 \, f \cos \left (f x + e\right )^{2} - f\right )}} \]
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Timed out. \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.78 \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {5 \, a^{2} c {\left (\frac {2 \, {\left (15 \, \cos \left (f x + e\right )^{5} - 40 \, \cos \left (f x + e\right )^{3} + 33 \, \cos \left (f x + e\right )\right )}}{\cos \left (f x + e\right )^{6} - 3 \, \cos \left (f x + e\right )^{4} + 3 \, \cos \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\cos \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - 30 \, a^{2} c {\left (\frac {2 \, {\left (3 \, \cos \left (f x + e\right )^{3} - 5 \, \cos \left (f x + e\right )\right )}}{\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\cos \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\cos \left (f x + e\right ) - 1\right )\right )} + \frac {160 \, {\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a^{2} c}{\tan \left (f x + e\right )^{3}} - \frac {32 \, {\left (15 \, \tan \left (f x + e\right )^{4} + 10 \, \tan \left (f x + e\right )^{2} + 3\right )} a^{2} c}{\tan \left (f x + e\right )^{5}}}{480 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (118) = 236\).
Time = 0.38 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.86 \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {5 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 12 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 20 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 120 \, a^{2} c \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - 120 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \frac {294 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 120 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 20 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 5 \, a^{2} c}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}}}{1920 \, f} \]
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Time = 12.23 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.62 \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=-\frac {a^2\,c\,\left (5\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-12\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+12\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )-15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-20\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+120\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7-120\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+20\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+120\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\ln \left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}{1920\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6} \]
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