Integrand size = 21, antiderivative size = 82 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^2 \, dx=-\frac {a b \text {arctanh}(\cos (e+f x))}{f}-\frac {\left (2 a^2+3 b^2\right ) \cot (e+f x)}{3 f}-\frac {a b \cot (e+f x) \csc (e+f x)}{f}-\frac {a^2 \cot (e+f x) \csc ^2(e+f x)}{3 f} \]
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Time = 0.06 (sec) , antiderivative size = 82, 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 = {2868, 3853, 3855, 3091, 3852, 8} \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^2 \, dx=-\frac {\left (2 a^2+3 b^2\right ) \cot (e+f x)}{3 f}-\frac {a^2 \cot (e+f x) \csc ^2(e+f x)}{3 f}-\frac {a b \text {arctanh}(\cos (e+f x))}{f}-\frac {a b \cot (e+f x) \csc (e+f x)}{f} \]
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Rule 8
Rule 2868
Rule 3091
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = (2 a b) \int \csc ^3(e+f x) \, dx+\int \csc ^4(e+f x) \left (a^2+b^2 \sin ^2(e+f x)\right ) \, dx \\ & = -\frac {a b \cot (e+f x) \csc (e+f x)}{f}-\frac {a^2 \cot (e+f x) \csc ^2(e+f x)}{3 f}+(a b) \int \csc (e+f x) \, dx+\frac {1}{3} \left (2 a^2+3 b^2\right ) \int \csc ^2(e+f x) \, dx \\ & = -\frac {a b \text {arctanh}(\cos (e+f x))}{f}-\frac {a b \cot (e+f x) \csc (e+f x)}{f}-\frac {a^2 \cot (e+f x) \csc ^2(e+f x)}{3 f}-\frac {\left (2 a^2+3 b^2\right ) \text {Subst}(\int 1 \, dx,x,\cot (e+f x))}{3 f} \\ & = -\frac {a b \text {arctanh}(\cos (e+f x))}{f}-\frac {\left (2 a^2+3 b^2\right ) \cot (e+f x)}{3 f}-\frac {a b \cot (e+f x) \csc (e+f x)}{f}-\frac {a^2 \cot (e+f x) \csc ^2(e+f x)}{3 f} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.61 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^2 \, dx=-\frac {2 a^2 \cot (e+f x)}{3 f}-\frac {b^2 \cot (e+f x)}{f}-\frac {a b \csc ^2\left (\frac {1}{2} (e+f x)\right )}{4 f}-\frac {a^2 \cot (e+f x) \csc ^2(e+f x)}{3 f}-\frac {a b \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{f}+\frac {a b \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{f}+\frac {a b \sec ^2\left (\frac {1}{2} (e+f x)\right )}{4 f} \]
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Time = 1.81 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+2 a b \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{2}\right )-b^{2} \cot \left (f x +e \right )}{f}\) | \(76\) |
default | \(\frac {a^{2} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+2 a b \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{2}\right )-b^{2} \cot \left (f x +e \right )}{f}\) | \(76\) |
parallelrisch | \(\frac {24 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a b -\left (-\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\left (\cot ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2}+a \left (a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+6 b \right ) \cot \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2}+6 a b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+9 a^{2}+12 b^{2}\right )}{24 f}\) | \(124\) |
risch | \(\frac {-2 i b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+2 a b \,{\mathrm e}^{5 i \left (f x +e \right )}+4 i a^{2} {\mathrm e}^{2 i \left (f x +e \right )}+4 i b^{2} {\mathrm e}^{2 i \left (f x +e \right )}-\frac {4 i a^{2}}{3}-2 i b^{2}-2 a b \,{\mathrm e}^{i \left (f x +e \right )}}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3}}+\frac {a b \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f}-\frac {a b \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{f}\) | \(141\) |
norman | \(\frac {\frac {a b \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {a b \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {a^{2}}{24 f}+\frac {a^{2} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}-\frac {\left (5 a^{2}+6 b^{2}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{12 f}+\frac {\left (5 a^{2}+6 b^{2}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{12 f}-\frac {\left (11 a^{2}+12 b^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}+\frac {\left (11 a^{2}+12 b^{2}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}-\frac {a b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a b \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {a b \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}\) | \(249\) |
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Time = 0.31 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.82 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^2 \, dx=-\frac {2 \, {\left (2 \, a^{2} + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - 6 \, a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, {\left (a b \cos \left (f x + e\right )^{2} - a b\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) - 3 \, {\left (a b \cos \left (f x + e\right )^{2} - a b\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) - 6 \, {\left (a^{2} + b^{2}\right )} \cos \left (f x + e\right )}{6 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \]
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\[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^2 \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{2} \csc ^{4}{\left (e + f x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.09 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {3 \, a b {\left (\frac {2 \, \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} - 1} - \log \left (\cos \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - \frac {6 \, b^{2}}{\tan \left (f x + e\right )} - \frac {2 \, {\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a^{2}}{\tan \left (f x + e\right )^{3}}}{6 \, f} \]
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Time = 0.31 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.90 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) + 9 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \frac {44 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 9 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 6 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{2}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}}}{24 \, f} \]
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Time = 6.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.66 \[ \int \csc ^4(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{24\,f}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3\,a^2}{8}+\frac {b^2}{2}\right )}{f}-\frac {{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (3\,a^2+4\,b^2\right )+\frac {a^2}{3}+2\,a\,b\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{8\,f}+\frac {a\,b\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{4\,f}+\frac {a\,b\,\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{f} \]
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