Integrand size = 32, antiderivative size = 105 \[ \int \csc ^6(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2 c \text {arctanh}(\cos (e+f x))}{8 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot ^5(e+f x)}{5 f}+\frac {a^2 c \cot (e+f x) \csc (e+f x)}{8 f}-\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{4 f} \]
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Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3045, 3853, 3855, 3852} \[ \int \csc ^6(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2 c \text {arctanh}(\cos (e+f x))}{8 f}-\frac {a^2 c \cot ^5(e+f x)}{5 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{4 f}+\frac {a^2 c \cot (e+f x) \csc (e+f x)}{8 f} \]
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Rule 3045
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (-a^2 c \csc ^3(e+f x)-a^2 c \csc ^4(e+f x)+a^2 c \csc ^5(e+f x)+a^2 c \csc ^6(e+f x)\right ) \, dx \\ & = -\left (\left (a^2 c\right ) \int \csc ^3(e+f x) \, dx\right )-\left (a^2 c\right ) \int \csc ^4(e+f x) \, dx+\left (a^2 c\right ) \int \csc ^5(e+f x) \, dx+\left (a^2 c\right ) \int \csc ^6(e+f x) \, dx \\ & = \frac {a^2 c \cot (e+f x) \csc (e+f x)}{2 f}-\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{4 f}-\frac {1}{2} \left (a^2 c\right ) \int \csc (e+f x) \, dx+\frac {1}{4} \left (3 a^2 c\right ) \int \csc ^3(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 \text {arctanh}(\cos (e+f x))}{2 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)}{8 f}-\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{4 f}+\frac {1}{8} \left (3 a^2 c\right ) \int \csc (e+f x) \, dx \\ & = \frac {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)}{8 f}-\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{4 f} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.94 \[ \int \csc ^6(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {2 a^2 c \cot (e+f x)}{15 f}+\frac {a^2 c \csc ^2\left (\frac {1}{2} (e+f x)\right )}{32 f}-\frac {a^2 c \csc ^4\left (\frac {1}{2} (e+f x)\right )}{64 f}+\frac {a^2 c \cot (e+f x) \csc ^2(e+f x)}{15 f}-\frac {a^2 c \cot (e+f x) \csc ^4(e+f x)}{5 f}+\frac {a^2 c \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{8 f}-\frac {a^2 c \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{8 f}-\frac {a^2 c \sec ^2\left (\frac {1}{2} (e+f x)\right )}{32 f}+\frac {a^2 c \sec ^4\left (\frac {1}{2} (e+f x)\right )}{64 f} \]
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Time = 1.24 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.17
method | result | size |
parallelrisch | \(-\frac {a^{2} c \left (-6 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+6 \left (\cot ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-15 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+15 \left (\cot ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-10 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+10 \left (\cot ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+120 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+60 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-60 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{960 f}\) | \(123\) |
risch | \(-\frac {a^{2} c \left (15 \,{\mathrm e}^{9 i \left (f x +e \right )}+240 i {\mathrm e}^{6 i \left (f x +e \right )}+90 \,{\mathrm e}^{7 i \left (f x +e \right )}+80 i {\mathrm e}^{4 i \left (f x +e \right )}+80 i {\mathrm e}^{2 i \left (f x +e \right )}-90 \,{\mathrm e}^{3 i \left (f x +e \right )}-16 i-15 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{60 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{5}}-\frac {a^{2} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{8 f}+\frac {a^{2} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{8 f}\) | \(149\) |
derivativedivides | \(\frac {-a^{2} c \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )-a^{2} c \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+a^{2} c \left (\left (-\frac {\left (\csc ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )+a^{2} c \left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (f x +e \right )\right )}{15}\right ) \cot \left (f x +e \right )}{f}\) | \(152\) |
default | \(\frac {-a^{2} c \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )-a^{2} c \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+a^{2} c \left (\left (-\frac {\left (\csc ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )+a^{2} c \left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (f x +e \right )\right )}{15}\right ) \cot \left (f x +e \right )}{f}\) | \(152\) |
norman | \(\frac {-\frac {a^{2} c}{160 f}-\frac {a^{2} c \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {3 a^{2} c \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32 f}-\frac {5 a^{2} c \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32 f}-\frac {a^{2} c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{64 f}-\frac {7 a^{2} c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{240 f}-\frac {3 a^{2} c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}+\frac {a^{2} c \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{80 f}+\frac {7 a^{2} c \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{80 f}-\frac {7 a^{2} c \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{80 f}-\frac {a^{2} c \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{80 f}+\frac {3 a^{2} c \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}+\frac {7 a^{2} c \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{240 f}+\frac {a^{2} c \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}+\frac {a^{2} c \left (\tan ^{16}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{160 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}-\frac {a^{2} c \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}\) | \(336\) |
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Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (95) = 190\).
Time = 0.27 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.91 \[ \int \csc ^6(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {32 \, a^{2} c \cos \left (f x + e\right )^{5} - 80 \, a^{2} c \cos \left (f x + e\right )^{3} + 15 \, {\left (a^{2} c \cos \left (f x + e\right )^{4} - 2 \, a^{2} c \cos \left (f x + e\right )^{2} + a^{2} c\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) - 15 \, {\left (a^{2} c \cos \left (f x + e\right )^{4} - 2 \, a^{2} c \cos \left (f x + e\right )^{2} + a^{2} c\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) - 30 \, {\left (a^{2} c \cos \left (f x + e\right )^{3} + a^{2} c \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )} \sin \left (f x + e\right )} \]
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Timed out. \[ \int \csc ^6(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.78 \[ \int \csc ^6(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {15 \, a^{2} c {\left (\frac {2 \, {\left (3 \, \cos \left (f x + e\right )^{3} - 5 \, \cos \left (f x + e\right )\right )}}{\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\cos \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - 60 \, a^{2} c {\left (\frac {2 \, \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} - 1} - \log \left (\cos \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right ) - 1\right )\right )} + \frac {80 \, {\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a^{2} c}{\tan \left (f x + e\right )^{3}} - \frac {16 \, {\left (15 \, \tan \left (f x + e\right )^{4} + 10 \, \tan \left (f x + e\right )^{2} + 3\right )} a^{2} c}{\tan \left (f x + e\right )^{5}}}{240 \, f} \]
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Time = 0.33 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.66 \[ \int \csc ^6(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {6 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 10 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 120 \, a^{2} c \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - 60 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \frac {274 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 60 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 10 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a^{2} c}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}}}{960 \, f} \]
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Time = 12.00 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.32 \[ \int \csc ^6(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=-\frac {a^2\,c\,\left (6\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-15\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )-10\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+60\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-60\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+10\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+120\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\ln \left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\right )}{960\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5} \]
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