Optimal. Leaf size=47 \[ \frac {1}{45} \tan ^5(9 x)+\frac {4}{27} \tan ^3(9 x)+\frac {2}{3} \tan (9 x)-\frac {1}{27} \cot ^3(9 x)-\frac {4}{9} \cot (9 x) \]
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Rubi [A] time = 0.10, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3657, 2620, 270} \[ \frac {1}{45} \tan ^5(9 x)+\frac {4}{27} \tan ^3(9 x)+\frac {2}{3} \tan (9 x)-\frac {1}{27} \cot ^3(9 x)-\frac {4}{9} \cot (9 x) \]
Antiderivative was successfully verified.
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Rule 270
Rule 2620
Rule 3657
Rubi steps
\begin {align*} \int \left (1+\cot ^2(9 x)\right )^2 \left (1+\tan ^2(9 x)\right )^3 \, dx &=\int \left (1+\cot ^2(9 x)\right )^2 \sec ^6(9 x) \, dx\\ &=\int \csc ^4(9 x) \sec ^6(9 x) \, dx\\ &=\frac {1}{9} \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^4}{x^4} \, dx,x,\tan (9 x)\right )\\ &=\frac {1}{9} \operatorname {Subst}\left (\int \left (6+\frac {1}{x^4}+\frac {4}{x^2}+4 x^2+x^4\right ) \, dx,x,\tan (9 x)\right )\\ &=-\frac {4}{9} \cot (9 x)-\frac {1}{27} \cot ^3(9 x)+\frac {2}{3} \tan (9 x)+\frac {4}{27} \tan ^3(9 x)+\frac {1}{45} \tan ^5(9 x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 59, normalized size = 1.26 \[ \frac {73}{135} \tan (9 x)-\frac {11}{27} \cot (9 x)-\frac {1}{27} \cot (9 x) \csc ^2(9 x)+\frac {1}{45} \tan (9 x) \sec ^4(9 x)+\frac {14}{135} \tan (9 x) \sec ^2(9 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 42, normalized size = 0.89 \[ \frac {3 \, \tan \left (9 \, x\right )^{8} + 20 \, \tan \left (9 \, x\right )^{6} + 90 \, \tan \left (9 \, x\right )^{4} - 60 \, \tan \left (9 \, x\right )^{2} - 5}{135 \, \tan \left (9 \, x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 38, normalized size = 0.81 \[ -\frac {4 \cot \left (9 x \right )}{9}-\frac {\left (\cot ^{3}\left (9 x \right )\right )}{27}+\frac {2 \tan \left (9 x \right )}{3}+\frac {4 \left (\tan ^{3}\left (9 x \right )\right )}{27}+\frac {\left (\tan ^{5}\left (9 x \right )\right )}{45} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 41, normalized size = 0.87 \[ \frac {1}{45} \, \tan \left (9 \, x\right )^{5} + \frac {4}{27} \, \tan \left (9 \, x\right )^{3} - \frac {12 \, \tan \left (9 \, x\right )^{2} + 1}{27 \, \tan \left (9 \, x\right )^{3}} + \frac {2}{3} \, \tan \left (9 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.38, size = 42, normalized size = 0.89 \[ \frac {3\,{\mathrm {tan}\left (9\,x\right )}^8+20\,{\mathrm {tan}\left (9\,x\right )}^6+90\,{\mathrm {tan}\left (9\,x\right )}^4-60\,{\mathrm {tan}\left (9\,x\right )}^2-5}{135\,{\mathrm {tan}\left (9\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.20, size = 44, normalized size = 0.94 \[ \frac {\tan ^{5}{\left (9 x \right )}}{45} + \frac {4 \tan ^{3}{\left (9 x \right )}}{27} + \frac {2 \tan {\left (9 x \right )}}{3} - \frac {4}{9 \tan {\left (9 x \right )}} - \frac {1}{27 \tan ^{3}{\left (9 x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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