Integrand size = 21, antiderivative size = 227 \[ \int \tan ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {\sqrt {-1+\sqrt {2}} \arctan \left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{f}-\frac {\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {2 \sqrt {1+\tan (e+f x)}}{f}-\frac {22 (1+\tan (e+f x))^{5/2}}{63 f}-\frac {8 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{63 f}+\frac {2 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{9 f} \]
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Time = 0.48 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3647, 3728, 3712, 3563, 12, 3617, 3616, 209, 213} \[ \int \tan ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {\sqrt {\sqrt {2}-1} \arctan \left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{f}-\frac {\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\tan (e+f x)+1}}\right )}{f}+\frac {2 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{9 f}-\frac {8 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{63 f}-\frac {22 (\tan (e+f x)+1)^{5/2}}{63 f}+\frac {2 \sqrt {\tan (e+f x)+1}}{f} \]
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Rule 12
Rule 209
Rule 213
Rule 3563
Rule 3616
Rule 3617
Rule 3647
Rule 3712
Rule 3728
Rubi steps \begin{align*} \text {integral}& = \frac {2 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{9 f}+\frac {2}{9} \int \tan (e+f x) (1+\tan (e+f x))^{3/2} \left (-2-\frac {9}{2} \tan (e+f x)-2 \tan ^2(e+f x)\right ) \, dx \\ & = -\frac {8 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{63 f}+\frac {2 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{9 f}+\frac {4}{63} \int (1+\tan (e+f x))^{3/2} \left (2-\frac {55}{4} \tan ^2(e+f x)\right ) \, dx \\ & = -\frac {22 (1+\tan (e+f x))^{5/2}}{63 f}-\frac {8 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{63 f}+\frac {2 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{9 f}+\int (1+\tan (e+f x))^{3/2} \, dx \\ & = \frac {2 \sqrt {1+\tan (e+f x)}}{f}-\frac {22 (1+\tan (e+f x))^{5/2}}{63 f}-\frac {8 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{63 f}+\frac {2 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{9 f}+\int \frac {2 \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {2 \sqrt {1+\tan (e+f x)}}{f}-\frac {22 (1+\tan (e+f x))^{5/2}}{63 f}-\frac {8 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{63 f}+\frac {2 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{9 f}+2 \int \frac {\tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {2 \sqrt {1+\tan (e+f x)}}{f}-\frac {22 (1+\tan (e+f x))^{5/2}}{63 f}-\frac {8 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{63 f}+\frac {2 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{9 f}-\frac {\int \frac {1+\left (-1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{\sqrt {2}}+\frac {\int \frac {1+\left (-1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{\sqrt {2}} \\ & = \frac {2 \sqrt {1+\tan (e+f x)}}{f}-\frac {22 (1+\tan (e+f x))^{5/2}}{63 f}-\frac {8 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{63 f}+\frac {2 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{9 f}+\frac {\left (4-3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2 \left (-1+\sqrt {2}\right )-4 \left (-1+\sqrt {2}\right )^2+x^2} \, dx,x,\frac {1-2 \left (-1+\sqrt {2}\right )-\left (-1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {\left (4+3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2 \left (-1-\sqrt {2}\right )-4 \left (-1-\sqrt {2}\right )^2+x^2} \, dx,x,\frac {1-2 \left (-1-\sqrt {2}\right )-\left (-1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{f} \\ & = -\frac {\sqrt {-1+\sqrt {2}} \arctan \left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{f}-\frac {\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {2 \sqrt {1+\tan (e+f x)}}{f}-\frac {22 (1+\tan (e+f x))^{5/2}}{63 f}-\frac {8 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{63 f}+\frac {2 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{9 f} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.15 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.68 \[ \int \tan ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {2 \cos ^2(e+f x) \left (-63 \left (\frac {\text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )}{\sqrt {1-i}}+\frac {\text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )}{\sqrt {1+i}}\right ) (1+\tan (e+f x))^2+(1+\tan (e+f x))^{5/2} \left (71+7 \sec ^4(e+f x)-36 \tan (e+f x)+2 \sec ^2(e+f x) (-13+5 \tan (e+f x))\right )\right )}{63 f (\cos (e+f x)+\sin (e+f x))^2} \]
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Time = 0.80 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}-\frac {4 \left (1+\tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+2 \sqrt {1+\tan \left (f x +e \right )}-\frac {\sqrt {2}\, \left (-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\sqrt {2}-\sqrt {2+2 \sqrt {2}}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\tan \left (f x +e \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}-\frac {\sqrt {2}\, \left (\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\sqrt {2}+\sqrt {2+2 \sqrt {2}}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\tan \left (f x +e \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}}{f}\) | \(245\) |
default | \(\frac {\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}-\frac {4 \left (1+\tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+2 \sqrt {1+\tan \left (f x +e \right )}-\frac {\sqrt {2}\, \left (-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\sqrt {2}-\sqrt {2+2 \sqrt {2}}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\tan \left (f x +e \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}-\frac {\sqrt {2}\, \left (\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\sqrt {2}+\sqrt {2+2 \sqrt {2}}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\tan \left (f x +e \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}}{f}\) | \(245\) |
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Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (180) = 360\).
Time = 0.26 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.62 \[ \int \tan ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {63 \, \sqrt {2} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) - 63 \, \sqrt {2} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) - 63 \, \sqrt {2} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) + 63 \, \sqrt {2} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) + 4 \, {\left (7 \, \tan \left (f x + e\right )^{4} + 10 \, \tan \left (f x + e\right )^{3} - 12 \, \tan \left (f x + e\right )^{2} - 26 \, \tan \left (f x + e\right ) + 52\right )} \sqrt {\tan \left (f x + e\right ) + 1}}{126 \, f} \]
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\[ \int \tan ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\int \left (\tan {\left (e + f x \right )} + 1\right )^{\frac {3}{2}} \tan ^{4}{\left (e + f x \right )}\, dx \]
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\[ \int \tan ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\int { {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{4} \,d x } \]
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Timed out. \[ \int \tan ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\text {Timed out} \]
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Time = 7.18 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.52 \[ \int \tan ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {2\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{f}-\frac {4\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{7/2}}{7\,f}+\frac {2\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{9/2}}{9\,f}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \]
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