Integrand size = 21, antiderivative size = 79 \[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=b^3 x-\frac {a \left (a^2+6 b^2\right ) \text {arctanh}(\cos (e+f x))}{2 f}-\frac {5 a^2 b \cot (e+f x)}{2 f}-\frac {a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f} \]
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Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2871, 3100, 2814, 3855} \[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=-\frac {a \left (a^2+6 b^2\right ) \text {arctanh}(\cos (e+f x))}{2 f}-\frac {5 a^2 b \cot (e+f x)}{2 f}-\frac {a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f}+b^3 x \]
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Rule 2814
Rule 2871
Rule 3100
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f}+\frac {1}{2} \int \csc ^2(e+f x) \left (5 a^2 b+a \left (a^2+6 b^2\right ) \sin (e+f x)+2 b^3 \sin ^2(e+f x)\right ) \, dx \\ & = -\frac {5 a^2 b \cot (e+f x)}{2 f}-\frac {a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f}+\frac {1}{2} \int \csc (e+f x) \left (a \left (a^2+6 b^2\right )+2 b^3 \sin (e+f x)\right ) \, dx \\ & = b^3 x-\frac {5 a^2 b \cot (e+f x)}{2 f}-\frac {a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f}+\frac {1}{2} \left (a \left (a^2+6 b^2\right )\right ) \int \csc (e+f x) \, dx \\ & = b^3 x-\frac {a \left (a^2+6 b^2\right ) \text {arctanh}(\cos (e+f x))}{2 f}-\frac {5 a^2 b \cot (e+f x)}{2 f}-\frac {a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f} \\ \end{align*}
Time = 2.22 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.92 \[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {8 b^3 e+8 b^3 f x-12 a^2 b \cot \left (\frac {1}{2} (e+f x)\right )-a^3 \csc ^2\left (\frac {1}{2} (e+f x)\right )-4 a^3 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )-24 a b^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )+4 a^3 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )+24 a b^2 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )+a^3 \sec ^2\left (\frac {1}{2} (e+f x)\right )+12 a^2 b \tan \left (\frac {1}{2} (e+f x)\right )}{8 f} \]
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Time = 1.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{2}\right )-3 a^{2} b \cot \left (f x +e \right )+3 a \,b^{2} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+b^{3} \left (f x +e \right )}{f}\) | \(86\) |
default | \(\frac {a^{3} \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{2}\right )-3 a^{2} b \cot \left (f x +e \right )+3 a \,b^{2} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+b^{3} \left (f x +e \right )}{f}\) | \(86\) |
parallelrisch | \(\frac {4 \left (a^{3}+6 a \,b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\left (\cot ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{3}+\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{3}+8 b^{3} f x -12 \cot \left (\frac {f x}{2}+\frac {e}{2}\right ) a^{2} b +12 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) a^{2} b}{8 f}\) | \(97\) |
risch | \(b^{3} x -\frac {i a^{2} \left (i a \,{\mathrm e}^{3 i \left (f x +e \right )}+i a \,{\mathrm e}^{i \left (f x +e \right )}+6 b \,{\mathrm e}^{2 i \left (f x +e \right )}-6 b \right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{2 f}+\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b^{2}}{f}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{2 f}-\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b^{2}}{f}\) | \(153\) |
norman | \(\frac {b^{3} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+b^{3} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {a^{3}}{8 f}+\frac {a^{3} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+3 b^{3} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+3 b^{3} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {3 a^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {7 a^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {11 a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {3 a^{2} b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 f}-\frac {3 a^{2} b \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {3 a^{2} b \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {3 a^{2} b \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}+\frac {a \left (a^{2}+6 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}\) | \(283\) |
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Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (73) = 146\).
Time = 0.33 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.96 \[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {4 \, b^{3} f x \cos \left (f x + e\right )^{2} - 4 \, b^{3} f x + 12 \, a^{2} b \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 2 \, a^{3} \cos \left (f x + e\right ) + {\left (a^{3} + 6 \, a b^{2} - {\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) - {\left (a^{3} + 6 \, a b^{2} - {\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right )}{4 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )}} \]
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\[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{3} \csc ^{3}{\left (e + f x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.29 \[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {4 \, {\left (f x + e\right )} b^{3} + a^{3} {\left (\frac {2 \, \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} - 1} - \log \left (\cos \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - 6 \, a b^{2} {\left (\log \left (\cos \left (f x + e\right ) + 1\right ) - \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - \frac {12 \, a^{2} b}{\tan \left (f x + e\right )}}{4 \, f} \]
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Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.70 \[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 8 \, {\left (f x + e\right )} b^{3} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, {\left (a^{3} + 6 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - \frac {6 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{3}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}}{8 \, f} \]
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Time = 6.64 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.96 \[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {2\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a^3+6\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a\,b^2+2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,b^3}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a^3+6\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a\,b^2-2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,b^3}\right )}{f}-\frac {a^3\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{8\,f}+\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{8\,f}+\frac {a^3\,\ln \left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}{2\,f}-\frac {3\,a^2\,b\,\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{2\,f}+\frac {3\,a\,b^2\,\ln \left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}{f}+\frac {3\,a^2\,b\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{2\,f} \]
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