Integrand size = 21, antiderivative size = 110 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^2 \, dx=-\frac {\left (3 a^2+4 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 f}-\frac {2 a b \cot (e+f x)}{f}-\frac {2 a b \cot ^3(e+f x)}{3 f}-\frac {\left (3 a^2+4 b^2\right ) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {a^2 \cot (e+f x) \csc ^3(e+f x)}{4 f} \]
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Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2868, 3852, 3091, 3853, 3855} \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^2 \, dx=-\frac {\left (3 a^2+4 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 f}-\frac {\left (3 a^2+4 b^2\right ) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {a^2 \cot (e+f x) \csc ^3(e+f x)}{4 f}-\frac {2 a b \cot ^3(e+f x)}{3 f}-\frac {2 a b \cot (e+f x)}{f} \]
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Rule 2868
Rule 3091
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = (2 a b) \int \csc ^4(e+f x) \, dx+\int \csc ^5(e+f x) \left (a^2+b^2 \sin ^2(e+f x)\right ) \, dx \\ & = -\frac {a^2 \cot (e+f x) \csc ^3(e+f x)}{4 f}+\frac {1}{4} \left (3 a^2+4 b^2\right ) \int \csc ^3(e+f x) \, dx-\frac {(2 a b) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (e+f x)\right )}{f} \\ & = -\frac {2 a b \cot (e+f x)}{f}-\frac {2 a b \cot ^3(e+f x)}{3 f}-\frac {\left (3 a^2+4 b^2\right ) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {a^2 \cot (e+f x) \csc ^3(e+f x)}{4 f}+\frac {1}{8} \left (3 a^2+4 b^2\right ) \int \csc (e+f x) \, dx \\ & = -\frac {\left (3 a^2+4 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 f}-\frac {2 a b \cot (e+f x)}{f}-\frac {2 a b \cot ^3(e+f x)}{3 f}-\frac {\left (3 a^2+4 b^2\right ) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {a^2 \cot (e+f x) \csc ^3(e+f x)}{4 f} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(255\) vs. \(2(110)=220\).
Time = 0.13 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.32 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^2 \, dx=-\frac {4 a b \cot (e+f x)}{3 f}-\frac {3 a^2 \csc ^2\left (\frac {1}{2} (e+f x)\right )}{32 f}-\frac {b^2 \csc ^2\left (\frac {1}{2} (e+f x)\right )}{8 f}-\frac {a^2 \csc ^4\left (\frac {1}{2} (e+f x)\right )}{64 f}-\frac {2 a b \cot (e+f x) \csc ^2(e+f x)}{3 f}-\frac {3 a^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{8 f}-\frac {b^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{2 f}+\frac {3 a^2 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{8 f}+\frac {b^2 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{2 f}+\frac {3 a^2 \sec ^2\left (\frac {1}{2} (e+f x)\right )}{32 f}+\frac {b^2 \sec ^2\left (\frac {1}{2} (e+f x)\right )}{8 f}+\frac {a^2 \sec ^4\left (\frac {1}{2} (e+f x)\right )}{64 f} \]
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Time = 1.98 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\left (-\frac {\left (\csc ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{8}\right )+2 a b \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+b^{2} \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{2}\right )}{f}\) | \(114\) |
default | \(\frac {a^{2} \left (\left (-\frac {\left (\csc ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{8}\right )+2 a b \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+b^{2} \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{2}\right )}{f}\) | \(114\) |
parallelrisch | \(\frac {-3 \left (\cot ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2}+3 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2}-16 \left (\cot ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a b +16 a b \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-24 \left (\cot ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2}-24 \left (\cot ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) b^{2}+24 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2}+24 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) b^{2}-144 \cot \left (\frac {f x}{2}+\frac {e}{2}\right ) a b +72 a^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+96 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) b^{2}+144 a b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{192 f}\) | \(189\) |
risch | \(\frac {9 a^{2} {\mathrm e}^{7 i \left (f x +e \right )}+12 b^{2} {\mathrm e}^{7 i \left (f x +e \right )}+96 i a b \,{\mathrm e}^{4 i \left (f x +e \right )}-33 a^{2} {\mathrm e}^{5 i \left (f x +e \right )}-12 b^{2} {\mathrm e}^{5 i \left (f x +e \right )}-128 i a b \,{\mathrm e}^{2 i \left (f x +e \right )}-33 a^{2} {\mathrm e}^{3 i \left (f x +e \right )}-12 b^{2} {\mathrm e}^{3 i \left (f x +e \right )}+32 i a b +9 a^{2} {\mathrm e}^{i \left (f x +e \right )}+12 b^{2} {\mathrm e}^{i \left (f x +e \right )}}{12 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{4}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{8 f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b^{2}}{2 f}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{8 f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b^{2}}{2 f}\) | \(246\) |
norman | \(\frac {-\frac {a^{2}}{64 f}+\frac {a^{2} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}-\frac {\left (5 a^{2}+4 b^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32 f}+\frac {\left (5 a^{2}+4 b^{2}\right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32 f}-\frac {\left (17 a^{2}+16 b^{2}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32 f}-\frac {\left (17 a^{2}+16 b^{2}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32 f}-\frac {a b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{12 f}-\frac {11 a b \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{12 f}-\frac {5 a b \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 f}+\frac {5 a b \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 f}+\frac {11 a b \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{12 f}+\frac {a b \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{12 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (3 a^{2}+4 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}\) | \(297\) |
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Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (102) = 204\).
Time = 0.29 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.08 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {6 \, {\left (3 \, a^{2} + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - 6 \, {\left (5 \, a^{2} + 4 \, b^{2}\right )} \cos \left (f x + e\right ) - 3 \, {\left ({\left (3 \, a^{2} + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a^{2} + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 3 \, a^{2} + 4 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (3 \, a^{2} + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a^{2} + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 3 \, a^{2} + 4 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 32 \, {\left (2 \, a b \cos \left (f x + e\right )^{3} - 3 \, a b \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )}} \]
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\[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^2 \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{2} \csc ^{5}{\left (e + f x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.34 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {3 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (f x + e\right )^{3} - 5 \, \cos \left (f x + e\right )\right )}}{\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\cos \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\cos \left (f x + e\right ) - 1\right )\right )} + 12 \, b^{2} {\left (\frac {2 \, \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} - 1} - \log \left (\cos \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - \frac {32 \, {\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a b}{\tan \left (f x + e\right )^{3}}}{48 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (102) = 204\).
Time = 0.33 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.97 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {3 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 16 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 24 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 144 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 24 \, {\left (3 \, a^{2} + 4 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - \frac {150 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 200 \, b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 144 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 24 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 16 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, a^{2}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}}}{192 \, f} \]
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Time = 6.31 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.62 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (\frac {3\,a^2}{8}+\frac {b^2}{2}\right )}{f}+\frac {a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{64\,f}-\frac {{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (2\,a^2+2\,b^2\right )+\frac {a^2}{4}+12\,a\,b\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\frac {4\,a\,b\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{3}\right )}{16\,f}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {a^2}{8}+\frac {b^2}{8}\right )}{f}+\frac {a\,b\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{12\,f}+\frac {3\,a\,b\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{4\,f} \]
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