Integrand size = 23, antiderivative size = 181 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {\cot ^2(e+f x)}{2 a^3 f}-\frac {\log (\cos (e+f x))}{(a-b)^3 f}-\frac {(a+3 b) \log (\tan (e+f x))}{a^4 f}-\frac {b^2 \left (6 a^2-8 a b+3 b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )}{2 a^4 (a-b)^3 f}+\frac {b^2}{4 a^2 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {(3 a-2 b) b^2}{2 a^3 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )} \]
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Time = 0.26 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 457, 90} \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {(a+3 b) \log (\tan (e+f x))}{a^4 f}+\frac {b^2 (3 a-2 b)}{2 a^3 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )}-\frac {\cot ^2(e+f x)}{2 a^3 f}+\frac {b^2}{4 a^2 f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b^2 \left (6 a^2-8 a b+3 b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )}{2 a^4 f (a-b)^3}-\frac {\log (\cos (e+f x))}{f (a-b)^3} \]
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Rule 90
Rule 457
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^3 \left (1+x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x^2 (1+x) (a+b x)^3} \, dx,x,\tan ^2(e+f x)\right )}{2 f} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{a^3 x^2}+\frac {-a-3 b}{a^4 x}+\frac {1}{(a-b)^3 (1+x)}-\frac {b^3}{a^2 (a-b) (a+b x)^3}-\frac {(3 a-2 b) b^3}{a^3 (a-b)^2 (a+b x)^2}-\frac {b^3 \left (6 a^2-8 a b+3 b^2\right )}{a^4 (a-b)^3 (a+b x)}\right ) \, dx,x,\tan ^2(e+f x)\right )}{2 f} \\ & = -\frac {\cot ^2(e+f x)}{2 a^3 f}-\frac {\log (\cos (e+f x))}{(a-b)^3 f}-\frac {(a+3 b) \log (\tan (e+f x))}{a^4 f}-\frac {b^2 \left (6 a^2-8 a b+3 b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )}{2 a^4 (a-b)^3 f}+\frac {b^2}{4 a^2 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {(3 a-2 b) b^2}{2 a^3 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )} \\ \end{align*}
Time = 2.23 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.80 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {\frac {\cot ^2(e+f x)}{a^3}-\frac {b^4}{2 a^4 (a-b) \left (b+a \cot ^2(e+f x)\right )^2}+\frac {(4 a-3 b) b^3}{a^4 (a-b)^2 \left (b+a \cot ^2(e+f x)\right )}+\frac {b^2 \left (6 a^2-8 a b+3 b^2\right ) \log \left (b+a \cot ^2(e+f x)\right )}{a^4 (a-b)^3}+\frac {2 \log (\sin (e+f x))}{(a-b)^3}}{2 f} \]
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Time = 0.63 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {-\frac {b^{3} \left (\frac {\left (6 a^{2}-8 a b +3 b^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )^{2}\right )}{b}-\frac {a^{2} \left (a^{2}-2 a b +b^{2}\right )}{2 b \left (a +b \tan \left (f x +e \right )^{2}\right )^{2}}-\frac {a \left (3 a^{2}-5 a b +2 b^{2}\right )}{b \left (a +b \tan \left (f x +e \right )^{2}\right )}\right )}{2 a^{4} \left (a -b \right )^{3}}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 \left (a -b \right )^{3}}-\frac {1}{2 a^{3} \tan \left (f x +e \right )^{2}}+\frac {\left (-3 b -a \right ) \ln \left (\tan \left (f x +e \right )\right )}{a^{4}}}{f}\) | \(173\) |
default | \(\frac {-\frac {b^{3} \left (\frac {\left (6 a^{2}-8 a b +3 b^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )^{2}\right )}{b}-\frac {a^{2} \left (a^{2}-2 a b +b^{2}\right )}{2 b \left (a +b \tan \left (f x +e \right )^{2}\right )^{2}}-\frac {a \left (3 a^{2}-5 a b +2 b^{2}\right )}{b \left (a +b \tan \left (f x +e \right )^{2}\right )}\right )}{2 a^{4} \left (a -b \right )^{3}}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 \left (a -b \right )^{3}}-\frac {1}{2 a^{3} \tan \left (f x +e \right )^{2}}+\frac {\left (-3 b -a \right ) \ln \left (\tan \left (f x +e \right )\right )}{a^{4}}}{f}\) | \(173\) |
norman | \(\frac {-\frac {1}{2 a f}+\frac {\left (3 a^{2} b -10 a \,b^{2}+6 b^{3}\right ) b \tan \left (f x +e \right )^{4}}{2 a^{3} f \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (4 a^{2} b -15 a \,b^{2}+9 b^{3}\right ) b^{2} \tan \left (f x +e \right )^{6}}{4 a^{4} f \left (a^{2}-2 a b +b^{2}\right )}}{\tan \left (f x +e \right )^{2} \left (a +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {\left (a +3 b \right ) \ln \left (\tan \left (f x +e \right )\right )}{a^{4} f}-\frac {b^{2} \left (6 a^{2}-8 a b +3 b^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )^{2}\right )}{2 a^{4} f \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\) | \(253\) |
parallelrisch | \(\frac {-48 \left (a^{2}-\frac {4}{3} a b +\frac {1}{2} b^{2}\right ) b^{2} \left (\frac {\left (a -b \right )^{2} \cos \left (4 f x +4 e \right )}{4}+\left (a^{2}-b^{2}\right ) \cos \left (2 f x +2 e \right )+\frac {3 a^{2}}{4}+\frac {a b}{2}+\frac {3 b^{2}}{4}\right ) \ln \left (a +b \tan \left (f x +e \right )^{2}\right )+\left (2 a^{4} \left (a -b \right )^{2} \ln \left (\sec \left (f x +e \right )^{2}\right )-4 \left (a +3 b \right ) \left (a -b \right )^{5} \ln \left (\tan \left (f x +e \right )\right )-2 a^{5} \left (a -3 b \right ) \cot \left (f x +e \right )^{2}+22 a^{3} b^{3} \csc \left (f x +e \right )^{2}-51 a^{2} b^{4}+36 a \,b^{5}-9 b^{6}\right ) \cos \left (4 f x +4 e \right )+\left (\left (8 a^{6}-8 a^{4} b^{2}\right ) \ln \left (\sec \left (f x +e \right )^{2}\right )-16 \left (a +3 b \right ) \left (a +b \right ) \left (a -b \right )^{4} \ln \left (\tan \left (f x +e \right )\right )+\left (-8 a^{6}+24 a^{5} b -48 a^{4} b^{2}\right ) \cot \left (f x +e \right )^{2}-a^{3} b^{3} \csc \left (f x +e \right )^{2}+76 a^{2} b^{4}-96 a \,b^{5}+36 b^{6}\right ) \cos \left (2 f x +2 e \right )-7 a^{3} b^{3} \csc \left (f x +e \right )^{2} \cos \left (6 f x +6 e \right )+\left (6 a^{6}+4 a^{5} b +6 a^{4} b^{2}\right ) \ln \left (\sec \left (f x +e \right )^{2}\right )-12 \left (a -b \right )^{3} \left (a^{2}+\frac {2}{3} a b +b^{2}\right ) \left (a +3 b \right ) \ln \left (\tan \left (f x +e \right )\right )-6 a^{5} \left (a -3 b \right ) \cot \left (f x +e \right )^{2}+2 a^{3} b^{3} \csc \left (f x +e \right )^{2}-25 a^{2} b^{4}+60 a \,b^{5}-27 b^{6}}{16 \left (a -b \right )^{3} a^{4} f \left (\frac {\left (a -b \right )^{2} \cos \left (4 f x +4 e \right )}{4}+\left (a^{2}-b^{2}\right ) \cos \left (2 f x +2 e \right )+\frac {3 a^{2}}{4}+\frac {a b}{2}+\frac {3 b^{2}}{4}\right )}\) | \(523\) |
risch | \(\text {Expression too large to display}\) | \(1068\) |
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Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (173) = 346\).
Time = 0.32 (sec) , antiderivative size = 545, normalized size of antiderivative = 3.01 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {{\left (2 \, a^{4} b^{2} - 6 \, a^{3} b^{3} + 13 \, a^{2} b^{4} - 6 \, a b^{5}\right )} \tan \left (f x + e\right )^{6} + 2 \, a^{6} - 6 \, a^{5} b + 6 \, a^{4} b^{2} - 2 \, a^{3} b^{3} + 2 \, {\left (2 \, a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 3 \, a b^{5}\right )} \tan \left (f x + e\right )^{4} + {\left (2 \, a^{6} - 2 \, a^{5} b - 6 \, a^{4} b^{2} + 18 \, a^{3} b^{3} - 9 \, a^{2} b^{4}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left ({\left (a^{4} b^{2} - 6 \, a^{2} b^{4} + 8 \, a b^{5} - 3 \, b^{6}\right )} \tan \left (f x + e\right )^{6} + 2 \, {\left (a^{5} b - 6 \, a^{3} b^{3} + 8 \, a^{2} b^{4} - 3 \, a b^{5}\right )} \tan \left (f x + e\right )^{4} + {\left (a^{6} - 6 \, a^{4} b^{2} + 8 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right )} \tan \left (f x + e\right )^{2}\right )} \log \left (\frac {\tan \left (f x + e\right )^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \, {\left ({\left (6 \, a^{2} b^{4} - 8 \, a b^{5} + 3 \, b^{6}\right )} \tan \left (f x + e\right )^{6} + 2 \, {\left (6 \, a^{3} b^{3} - 8 \, a^{2} b^{4} + 3 \, a b^{5}\right )} \tan \left (f x + e\right )^{4} + {\left (6 \, a^{4} b^{2} - 8 \, a^{3} b^{3} + 3 \, a^{2} b^{4}\right )} \tan \left (f x + e\right )^{2}\right )} \log \left (\frac {b \tan \left (f x + e\right )^{2} + a}{\tan \left (f x + e\right )^{2} + 1}\right )}{4 \, {\left ({\left (a^{7} b^{2} - 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} - a^{4} b^{5}\right )} f \tan \left (f x + e\right )^{6} + 2 \, {\left (a^{8} b - 3 \, a^{7} b^{2} + 3 \, a^{6} b^{3} - a^{5} b^{4}\right )} f \tan \left (f x + e\right )^{4} + {\left (a^{9} - 3 \, a^{8} b + 3 \, a^{7} b^{2} - a^{6} b^{3}\right )} f \tan \left (f x + e\right )^{2}\right )}} \]
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Timed out. \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Timed out} \]
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none
Time = 0.24 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.91 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {\frac {2 \, {\left (6 \, a^{2} b^{2} - 8 \, a b^{3} + 3 \, b^{4}\right )} \log \left (-{\left (a - b\right )} \sin \left (f x + e\right )^{2} + a\right )}{a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}} + \frac {2 \, a^{5} - 6 \, a^{4} b + 6 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + 2 \, {\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 14 \, a^{2} b^{3} + 11 \, a b^{4} - 3 \, b^{5}\right )} \sin \left (f x + e\right )^{4} - {\left (4 \, a^{5} - 16 \, a^{4} b + 24 \, a^{3} b^{2} - 24 \, a^{2} b^{3} + 9 \, a b^{4}\right )} \sin \left (f x + e\right )^{2}}{{\left (a^{8} - 5 \, a^{7} b + 10 \, a^{6} b^{2} - 10 \, a^{5} b^{3} + 5 \, a^{4} b^{4} - a^{3} b^{5}\right )} \sin \left (f x + e\right )^{6} - 2 \, {\left (a^{8} - 4 \, a^{7} b + 6 \, a^{6} b^{2} - 4 \, a^{5} b^{3} + a^{4} b^{4}\right )} \sin \left (f x + e\right )^{4} + {\left (a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right )} \sin \left (f x + e\right )^{2}} + \frac {2 \, {\left (a + 3 \, b\right )} \log \left (\sin \left (f x + e\right )^{2}\right )}{a^{4}}}{4 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 850 vs. \(2 (173) = 346\).
Time = 1.25 (sec) , antiderivative size = 850, normalized size of antiderivative = 4.70 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \]
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Time = 12.45 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.27 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}{2\,f\,{\left (a-b\right )}^3}-\frac {\frac {1}{2\,a}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (a^2\,b^2-5\,a\,b^3+3\,b^4\right )}{2\,a^3\,\left (a^2-2\,a\,b+b^2\right )}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (4\,a^2\,b-15\,a\,b^2+9\,b^3\right )}{4\,a^2\,\left (a^2-2\,a\,b+b^2\right )}}{f\,\left (a^2\,{\mathrm {tan}\left (e+f\,x\right )}^2+2\,a\,b\,{\mathrm {tan}\left (e+f\,x\right )}^4+b^2\,{\mathrm {tan}\left (e+f\,x\right )}^6\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )\right )\,\left (a+3\,b\right )}{a^4\,f}-\frac {b^2\,\ln \left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )\,\left (6\,a^2-8\,a\,b+3\,b^2\right )}{2\,a^4\,f\,{\left (a-b\right )}^3} \]
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