Integrand size = 32, antiderivative size = 61 \[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2 c \text {arctanh}(\cos (e+f x))}{2 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot (e+f x) \csc (e+f x)}{2 f} \]
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Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3029, 2785, 2687, 30, 2691, 3855} \[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2 c \text {arctanh}(\cos (e+f x))}{2 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot (e+f x) \csc (e+f x)}{2 f} \]
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Rule 30
Rule 2687
Rule 2691
Rule 2785
Rule 3029
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int \frac {\cot ^4(e+f x)}{c-c \sin (e+f x)} \, dx \\ & = \left (a^2 c\right ) \int \cot ^2(e+f x) \csc (e+f x) \, dx+\left (a^2 c\right ) \int \cot ^2(e+f x) \csc ^2(e+f x) \, dx \\ & = -\frac {a^2 c \cot (e+f x) \csc (e+f x)}{2 f}-\frac {1}{2} \left (a^2 c\right ) \int \csc (e+f x) \, dx+\frac {\left (a^2 c\right ) \text {Subst}\left (\int x^2 \, 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 (e+f x) \csc (e+f x)}{2 f} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(172\) vs. \(2(61)=122\).
Time = 0.20 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.82 \[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=a^2 c \left (\frac {\cot \left (\frac {1}{2} (e+f x)\right )}{6 f}-\frac {\csc ^2\left (\frac {1}{2} (e+f x)\right )}{8 f}-\frac {\cot \left (\frac {1}{2} (e+f x)\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{24 f}+\frac {\log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{2 f}-\frac {\log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{2 f}+\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right )}{8 f}-\frac {\tan \left (\frac {1}{2} (e+f x)\right )}{6 f}+\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right ) \tan \left (\frac {1}{2} (e+f x)\right )}{24 f}\right ) \]
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Time = 0.92 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.56
method | result | size |
parallelrisch | \(\frac {a^{2} c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )-\left (\cot ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-3 \left (\cot ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-12 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+3 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}\) | \(95\) |
derivativedivides | \(\frac {-a^{2} c \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )+a^{2} c \cot \left (f x +e \right )+a^{2} c \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )+a^{2} c \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )}{f}\) | \(100\) |
default | \(\frac {-a^{2} c \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )+a^{2} c \cot \left (f x +e \right )+a^{2} c \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )+a^{2} c \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )}{f}\) | \(100\) |
risch | \(\frac {a^{2} c \left (6 i {\mathrm e}^{4 i \left (f x +e \right )}+3 \,{\mathrm e}^{5 i \left (f x +e \right )}+2 i-3 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3}}+\frac {a^{2} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{2 f}-\frac {a^{2} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{2 f}\) | \(103\) |
norman | \(\frac {-\frac {a^{2} c}{24 f}-\frac {3 a^{2} c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {7 a^{2} c \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {11 a^{2} c \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {a^{2} c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {a^{2} c \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {a^{2} c \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+\frac {a^{2} c \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+\frac {a^{2} c \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}-\frac {a^{2} c \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}\) | \(216\) |
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Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (55) = 110\).
Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.25 \[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {4 \, a^{2} c \cos \left (f x + e\right )^{3} + 6 \, a^{2} c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, {\left (a^{2} c \cos \left (f x + e\right )^{2} - a^{2} c\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) - 3 \, {\left (a^{2} c \cos \left (f x + e\right )^{2} - a^{2} c\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right )}{12 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \]
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\[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=- a^{2} c \left (\int \left (- \sin {\left (e + f x \right )} \csc ^{4}{\left (e + f x \right )}\right )\, dx + \int \sin ^{2}{\left (e + f x \right )} \csc ^{4}{\left (e + f x \right )}\, dx + \int \sin ^{3}{\left (e + f x \right )} \csc ^{4}{\left (e + f x \right )}\, dx + \int \left (- \csc ^{4}{\left (e + f x \right )}\right )\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (55) = 110\).
Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.97 \[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {3 \, a^{2} c {\left (\frac {2 \, \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} - 1} - \log \left (\cos \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right ) - 1\right )\right )} + 6 \, a^{2} c {\left (\log \left (\cos \left (f x + e\right ) + 1\right ) - \log \left (\cos \left (f x + e\right ) - 1\right )\right )} + \frac {12 \, a^{2} c}{\tan \left (f x + e\right )} - \frac {4 \, {\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a^{2} c}{\tan \left (f x + e\right )^{3}}}{12 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (55) = 110\).
Time = 0.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.28 \[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, a^{2} c \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - 3 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \frac {22 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - a^{2} c}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}}}{24 \, f} \]
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Time = 11.69 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.16 \[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{8\,f}-\frac {a^2\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8\,f}-\frac {{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (-c\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+c\,a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+\frac {c\,a^2}{3}\right )}{8\,f}+\frac {a^2\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{24\,f}-\frac {a^2\,c\,\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{2\,f} \]
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