Integrand size = 21, antiderivative size = 109 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=-\frac {b \left (3 a^2+2 b^2\right ) \text {arctanh}(\cos (e+f x))}{2 f}-\frac {a \left (2 a^2+9 b^2\right ) \cot (e+f x)}{3 f}-\frac {7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f} \]
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Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2871, 3100, 2827, 3852, 8, 3855} \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=-\frac {b \left (3 a^2+2 b^2\right ) \text {arctanh}(\cos (e+f x))}{2 f}-\frac {a \left (2 a^2+9 b^2\right ) \cot (e+f x)}{3 f}-\frac {7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f} \]
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Rule 8
Rule 2827
Rule 2871
Rule 3100
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}+\frac {1}{3} \int \csc ^3(e+f x) \left (7 a^2 b+a \left (2 a^2+9 b^2\right ) \sin (e+f x)+b \left (a^2+3 b^2\right ) \sin ^2(e+f x)\right ) \, dx \\ & = -\frac {7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}+\frac {1}{6} \int \csc ^2(e+f x) \left (2 a \left (2 a^2+9 b^2\right )+3 b \left (3 a^2+2 b^2\right ) \sin (e+f x)\right ) \, dx \\ & = -\frac {7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}+\frac {1}{2} \left (b \left (3 a^2+2 b^2\right )\right ) \int \csc (e+f x) \, dx+\frac {1}{3} \left (a \left (2 a^2+9 b^2\right )\right ) \int \csc ^2(e+f x) \, dx \\ & = -\frac {b \left (3 a^2+2 b^2\right ) \text {arctanh}(\cos (e+f x))}{2 f}-\frac {7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}-\frac {\left (a \left (2 a^2+9 b^2\right )\right ) \text {Subst}(\int 1 \, dx,x,\cot (e+f x))}{3 f} \\ & = -\frac {b \left (3 a^2+2 b^2\right ) \text {arctanh}(\cos (e+f x))}{2 f}-\frac {a \left (2 a^2+9 b^2\right ) \cot (e+f x)}{3 f}-\frac {7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac {a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(525\) vs. \(2(109)=218\).
Time = 7.68 (sec) , antiderivative size = 525, normalized size of antiderivative = 4.82 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {\left (-2 a^3 \cos \left (\frac {1}{2} (e+f x)\right )-9 a b^2 \cos \left (\frac {1}{2} (e+f x)\right )\right ) \csc \left (\frac {1}{2} (e+f x)\right ) (b+a \csc (e+f x))^3 \sin ^3(e+f x)}{6 f (a+b \sin (e+f x))^3}-\frac {3 a^2 b \csc ^2\left (\frac {1}{2} (e+f x)\right ) (b+a \csc (e+f x))^3 \sin ^3(e+f x)}{8 f (a+b \sin (e+f x))^3}-\frac {a^3 \cot \left (\frac {1}{2} (e+f x)\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right ) (b+a \csc (e+f x))^3 \sin ^3(e+f x)}{24 f (a+b \sin (e+f x))^3}+\frac {\left (-3 a^2 b-2 b^3\right ) (b+a \csc (e+f x))^3 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right ) \sin ^3(e+f x)}{2 f (a+b \sin (e+f x))^3}+\frac {\left (3 a^2 b+2 b^3\right ) (b+a \csc (e+f x))^3 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin ^3(e+f x)}{2 f (a+b \sin (e+f x))^3}+\frac {3 a^2 b (b+a \csc (e+f x))^3 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin ^3(e+f x)}{8 f (a+b \sin (e+f x))^3}+\frac {(b+a \csc (e+f x))^3 \sec \left (\frac {1}{2} (e+f x)\right ) \left (2 a^3 \sin \left (\frac {1}{2} (e+f x)\right )+9 a b^2 \sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin ^3(e+f x)}{6 f (a+b \sin (e+f x))^3}+\frac {a^3 (b+a \csc (e+f x))^3 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin ^3(e+f x) \tan \left (\frac {1}{2} (e+f x)\right )}{24 f (a+b \sin (e+f x))^3} \]
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Time = 1.85 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+3 a^{2} b \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{2}\right )-3 a \,b^{2} \cot \left (f x +e \right )+b^{3} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{f}\) | \(99\) |
default | \(\frac {a^{3} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+3 a^{2} b \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{2}\right )-3 a \,b^{2} \cot \left (f x +e \right )+b^{3} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{f}\) | \(99\) |
parallelrisch | \(\frac {\left (36 a^{2} b +24 b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\left (-\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\left (\cot ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2}+a \left (a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+9 b \right ) \cot \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2}+9 a b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+9 a^{2}+36 b^{2}\right ) a}{24 f}\) | \(134\) |
risch | \(\frac {a \left (-18 i b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+9 a b \,{\mathrm e}^{5 i \left (f x +e \right )}+12 i a^{2} {\mathrm e}^{2 i \left (f x +e \right )}+36 i b^{2} {\mathrm e}^{2 i \left (f x +e \right )}-4 i a^{2}-18 i b^{2}-9 a b \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3}}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{2 f}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f}-\frac {3 a^{2} b \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{2 f}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{f}\) | \(186\) |
norman | \(\frac {-\frac {a^{3}}{24 f}+\frac {a^{3} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}-\frac {21 a^{2} b \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {9 a^{2} b \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {33 a^{2} b \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {a \left (a^{2}+3 b^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}+\frac {a \left (a^{2}+3 b^{2}\right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}-\frac {a \left (7 a^{2}+24 b^{2}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+\frac {a \left (7 a^{2}+24 b^{2}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {3 a^{2} b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {3 a^{2} b \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}+\frac {b \left (3 a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}\) | \(290\) |
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Time = 0.31 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.75 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {18 \, a^{2} b \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 4 \, {\left (2 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3} - {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) - 3 \, {\left (3 \, a^{2} b + 2 \, b^{3} - {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) + 12 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (f x + e\right )}{12 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \]
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\[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{3} \csc ^{4}{\left (e + f x \right )}\, dx \]
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Time = 0.26 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.08 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {9 \, a^{2} b {\left (\frac {2 \, \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} - 1} - \log \left (\cos \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - 6 \, b^{3} {\left (\log \left (\cos \left (f x + e\right ) + 1\right ) - \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - \frac {36 \, a b^{2}}{\tan \left (f x + e\right )} - \frac {4 \, {\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a^{3}}{\tan \left (f x + e\right )^{3}}}{12 \, f} \]
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Time = 0.31 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.74 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 9 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 36 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - \frac {66 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 44 \, b^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 9 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{3}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}}}{24 \, f} \]
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Time = 6.53 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.38 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (\frac {3\,a^2\,b}{2}+b^3\right )}{f}+\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{24\,f}-\frac {{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (3\,a^3+12\,a\,b^2\right )+\frac {a^3}{3}+3\,a^2\,b\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{8\,f}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3\,a^3}{8}+\frac {3\,a\,b^2}{2}\right )}{f}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{8\,f} \]
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