Integrand size = 23, antiderivative size = 82 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {3 \sqrt {b} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 a^{5/2} f}-\frac {3 \cot (e+f x)}{2 a^2 f}+\frac {\cot (e+f x)}{2 a f \left (a+b \tan ^2(e+f x)\right )} \]
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Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3744, 296, 331, 211} \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {3 \sqrt {b} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 a^{5/2} f}-\frac {3 \cot (e+f x)}{2 a^2 f}+\frac {\cot (e+f x)}{2 a f \left (a+b \tan ^2(e+f x)\right )} \]
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Rule 211
Rule 296
Rule 331
Rule 3744
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\cot (e+f x)}{2 a f \left (a+b \tan ^2(e+f x)\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{2 a f} \\ & = -\frac {3 \cot (e+f x)}{2 a^2 f}+\frac {\cot (e+f x)}{2 a f \left (a+b \tan ^2(e+f x)\right )}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{2 a^2 f} \\ & = -\frac {3 \sqrt {b} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 a^{5/2} f}-\frac {3 \cot (e+f x)}{2 a^2 f}+\frac {\cot (e+f x)}{2 a f \left (a+b \tan ^2(e+f x)\right )} \\ \end{align*}
Time = 1.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.01 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\frac {-3 \sqrt {b} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )+\sqrt {a} \left (-2 \cot (e+f x)-\frac {b \sin (2 (e+f x))}{a+b+(a-b) \cos (2 (e+f x))}\right )}{2 a^{5/2} f} \]
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Time = 0.40 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {-\frac {1}{a^{2} \tan \left (f x +e \right )}-\frac {b \left (\frac {\tan \left (f x +e \right )}{2 a +2 b \tan \left (f x +e \right )^{2}}+\frac {3 \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}}{f}\) | \(69\) |
default | \(\frac {-\frac {1}{a^{2} \tan \left (f x +e \right )}-\frac {b \left (\frac {\tan \left (f x +e \right )}{2 a +2 b \tan \left (f x +e \right )^{2}}+\frac {3 \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}}{f}\) | \(69\) |
risch | \(-\frac {i \left (2 a^{2} {\mathrm e}^{4 i \left (f x +e \right )}-3 a b \,{\mathrm e}^{4 i \left (f x +e \right )}+3 b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+4 a^{2} {\mathrm e}^{2 i \left (f x +e \right )}-6 b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+2 a^{2}-5 a b +3 b^{2}\right )}{f \left (a -b \right ) a^{2} \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}-b \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a -b \right ) \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right )}{4 a^{3} f}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right )}{4 a^{3} f}\) | \(264\) |
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Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (68) = 136\).
Time = 0.32 (sec) , antiderivative size = 373, normalized size of antiderivative = 4.55 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\left [-\frac {4 \, {\left (2 \, a - 3 \, b\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} + b\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a^{2} + a b\right )} \cos \left (f x + e\right )^{3} - a b \cos \left (f x + e\right )\right )} \sqrt {-\frac {b}{a}} \sin \left (f x + e\right ) + b^{2}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}}\right ) \sin \left (f x + e\right ) + 12 \, b \cos \left (f x + e\right )}{8 \, {\left (a^{2} b f + {\left (a^{3} - a^{2} b\right )} f \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}, -\frac {2 \, {\left (2 \, a - 3 \, b\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} + b\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {b}{a}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 6 \, b \cos \left (f x + e\right )}{4 \, {\left (a^{2} b f + {\left (a^{3} - a^{2} b\right )} f \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}\right ] \]
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\[ \int \frac {\csc ^2(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\int \frac {\csc ^{2}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{2}}\, dx \]
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Time = 0.33 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.89 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {\frac {3 \, b \tan \left (f x + e\right )^{2} + 2 \, a}{a^{2} b \tan \left (f x + e\right )^{3} + a^{3} \tan \left (f x + e\right )} + \frac {3 \, b \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}}}{2 \, f} \]
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Time = 0.56 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.07 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {\frac {3 \, {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )} b}{\sqrt {a b} a^{2}} + \frac {3 \, b \tan \left (f x + e\right )^{2} + 2 \, a}{{\left (b \tan \left (f x + e\right )^{3} + a \tan \left (f x + e\right )\right )} a^{2}}}{2 \, f} \]
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Time = 10.64 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {\frac {1}{a}+\frac {3\,b\,{\mathrm {tan}\left (e+f\,x\right )}^2}{2\,a^2}}{f\,\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^3+a\,\mathrm {tan}\left (e+f\,x\right )\right )}-\frac {3\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {tan}\left (e+f\,x\right )}{\sqrt {a}}\right )}{2\,a^{5/2}\,f} \]
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