Integrand size = 21, antiderivative size = 34 \[ \int \csc ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=b^2 x-\frac {2 a b \text {arctanh}(\cos (e+f x))}{f}-\frac {a^2 \cot (e+f x)}{f} \]
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Time = 0.05 (sec) , antiderivative size = 34, 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 = {2868, 3855, 3091, 8} \[ \int \csc ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=-\frac {a^2 \cot (e+f x)}{f}-\frac {2 a b \text {arctanh}(\cos (e+f x))}{f}+b^2 x \]
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Rule 8
Rule 2868
Rule 3091
Rule 3855
Rubi steps \begin{align*} \text {integral}& = (2 a b) \int \csc (e+f x) \, dx+\int \csc ^2(e+f x) \left (a^2+b^2 \sin ^2(e+f x)\right ) \, dx \\ & = -\frac {2 a b \text {arctanh}(\cos (e+f x))}{f}-\frac {a^2 \cot (e+f x)}{f}+b^2 \int 1 \, dx \\ & = b^2 x-\frac {2 a b \text {arctanh}(\cos (e+f x))}{f}-\frac {a^2 \cot (e+f x)}{f} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(76\) vs. \(2(34)=68\).
Time = 0.48 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.24 \[ \int \csc ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {-a^2 \cot \left (\frac {1}{2} (e+f x)\right )+2 b \left (b e+b f x-2 a \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )+2 a \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )+a^2 \tan \left (\frac {1}{2} (e+f x)\right )}{2 f} \]
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Time = 1.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35
method | result | size |
derivativedivides | \(\frac {-a^{2} \cot \left (f x +e \right )+2 a b \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+b^{2} \left (f x +e \right )}{f}\) | \(46\) |
default | \(\frac {-a^{2} \cot \left (f x +e \right )+2 a b \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+b^{2} \left (f x +e \right )}{f}\) | \(46\) |
parallelrisch | \(\frac {2 b^{2} f x -\cot \left (\frac {f x}{2}+\frac {e}{2}\right ) a^{2}+4 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a b +\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) a^{2}}{2 f}\) | \(55\) |
risch | \(b^{2} x -\frac {2 i a^{2}}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}-\frac {2 a b \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{f}+\frac {2 a b \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f}\) | \(67\) |
norman | \(\frac {b^{2} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+b^{2} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {a^{2}}{2 f}-\frac {a^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}+\frac {a^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}+\frac {a^{2} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}+2 b^{2} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {2 a b \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}\) | \(159\) |
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Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (34) = 68\).
Time = 0.32 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.26 \[ \int \csc ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {b^{2} f x \sin \left (f x + e\right ) - a b \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) + a b \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) - a^{2} \cos \left (f x + e\right )}{f \sin \left (f x + e\right )} \]
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\[ \int \csc ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{2} \csc ^{2}{\left (e + f x \right )}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.53 \[ \int \csc ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {{\left (f x + e\right )} b^{2} - a b {\left (\log \left (\cos \left (f x + e\right ) + 1\right ) - \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - \frac {a^{2}}{\tan \left (f x + e\right )}}{f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (34) = 68\).
Time = 0.31 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.18 \[ \int \csc ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {2 \, {\left (f x + e\right )} b^{2} + 4 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) + a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \frac {4 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{2}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}}{2 \, f} \]
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Time = 6.25 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.09 \[ \int \csc ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {2\,b^2\,\mathrm {atan}\left (\frac {b\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+2\,a\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{2\,a\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )-b\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}{f}-\frac {a^2\,\mathrm {cot}\left (e+f\,x\right )}{f}+\frac {2\,a\,b\,\ln \left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}{f} \]
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