Optimal. Leaf size=369 \[ \frac {\sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}-\frac {\sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {2 \sqrt {1+\tan (e+f x)}}{f}+\frac {2 (1+\tan (e+f x))^{3/2}}{3 f}+\frac {20 (1+\tan (e+f x))^{5/2}}{231 f}-\frac {50 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{231 f}-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f} \]
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Rubi [A]
time = 0.38, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3647, 3728,
3729, 3711, 12, 3609, 3566, 722, 1108, 648, 632, 210, 642} \begin {gather*} \frac {\sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{f}-\frac {\sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{f}+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}-\frac {4 (\tan (e+f x)+1)^{5/2} \tan ^2(e+f x)}{33 f}-\frac {50 (\tan (e+f x)+1)^{5/2} \tan (e+f x)}{231 f}+\frac {20 (\tan (e+f x)+1)^{5/2}}{231 f}+\frac {2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac {2 \sqrt {\tan (e+f x)+1}}{f}+\frac {\log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}} f}-\frac {\log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}} f} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 632
Rule 642
Rule 648
Rule 722
Rule 1108
Rule 3566
Rule 3609
Rule 3647
Rule 3711
Rule 3728
Rule 3729
Rubi steps
\begin {align*} \int \tan ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx &=\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f}+\frac {2}{11} \int \tan ^2(e+f x) (1+\tan (e+f x))^{3/2} \left (-3-\frac {11}{2} \tan (e+f x)-3 \tan ^2(e+f x)\right ) \, dx\\ &=-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f}+\frac {4}{99} \int \tan (e+f x) (1+\tan (e+f x))^{3/2} \left (6-\frac {75}{4} \tan ^2(e+f x)\right ) \, dx\\ &=-\frac {50 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{231 f}-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f}+\frac {8}{693} \int (1+\tan (e+f x))^{3/2} \left (\frac {75}{4}+\frac {693}{8} \tan (e+f x)+\frac {75}{4} \tan ^2(e+f x)\right ) \, dx\\ &=\frac {20 (1+\tan (e+f x))^{5/2}}{231 f}-\frac {50 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{231 f}-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f}+\frac {8}{693} \int \frac {693}{8} \tan (e+f x) (1+\tan (e+f x))^{3/2} \, dx\\ &=\frac {20 (1+\tan (e+f x))^{5/2}}{231 f}-\frac {50 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{231 f}-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f}+\int \tan (e+f x) (1+\tan (e+f x))^{3/2} \, dx\\ &=\frac {2 (1+\tan (e+f x))^{3/2}}{3 f}+\frac {20 (1+\tan (e+f x))^{5/2}}{231 f}-\frac {50 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{231 f}-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f}+\int (-1+\tan (e+f x)) \sqrt {1+\tan (e+f x)} \, dx\\ &=\frac {2 \sqrt {1+\tan (e+f x)}}{f}+\frac {2 (1+\tan (e+f x))^{3/2}}{3 f}+\frac {20 (1+\tan (e+f x))^{5/2}}{231 f}-\frac {50 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{231 f}-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f}+\int -\frac {2}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {2 \sqrt {1+\tan (e+f x)}}{f}+\frac {2 (1+\tan (e+f x))^{3/2}}{3 f}+\frac {20 (1+\tan (e+f x))^{5/2}}{231 f}-\frac {50 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{231 f}-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f}-2 \int \frac {1}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {2 \sqrt {1+\tan (e+f x)}}{f}+\frac {2 (1+\tan (e+f x))^{3/2}}{3 f}+\frac {20 (1+\tan (e+f x))^{5/2}}{231 f}-\frac {50 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{231 f}-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f}-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1+x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {2 \sqrt {1+\tan (e+f x)}}{f}+\frac {2 (1+\tan (e+f x))^{3/2}}{3 f}+\frac {20 (1+\tan (e+f x))^{5/2}}{231 f}-\frac {50 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{231 f}-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f}-\frac {4 \text {Subst}\left (\int \frac {1}{2-2 x^2+x^4} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{f}\\ &=\frac {2 \sqrt {1+\tan (e+f x)}}{f}+\frac {2 (1+\tan (e+f x))^{3/2}}{3 f}+\frac {20 (1+\tan (e+f x))^{5/2}}{231 f}-\frac {50 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{231 f}-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f}-\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{\sqrt {1+\sqrt {2}} f}-\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{\sqrt {1+\sqrt {2}} f}\\ &=\frac {2 \sqrt {1+\tan (e+f x)}}{f}+\frac {2 (1+\tan (e+f x))^{3/2}}{3 f}+\frac {20 (1+\tan (e+f x))^{5/2}}{231 f}-\frac {50 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{231 f}-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{\sqrt {2} f}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{\sqrt {2} f}+\frac {\text {Subst}\left (\int \frac {-\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}-\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}\\ &=\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {2 \sqrt {1+\tan (e+f x)}}{f}+\frac {2 (1+\tan (e+f x))^{3/2}}{3 f}+\frac {20 (1+\tan (e+f x))^{5/2}}{231 f}-\frac {50 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{231 f}-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f}+\frac {\sqrt {2} \text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,-\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}\right )}{f}+\frac {\sqrt {2} \text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}\right )}{f}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{\sqrt {-1+\sqrt {2}} f}-\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{\sqrt {-1+\sqrt {2}} f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {2 \sqrt {1+\tan (e+f x)}}{f}+\frac {2 (1+\tan (e+f x))^{3/2}}{3 f}+\frac {20 (1+\tan (e+f x))^{5/2}}{231 f}-\frac {50 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{231 f}-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.90, size = 188, normalized size = 0.51 \begin {gather*} \frac {\cos (e+f x) \left ((-1+i) \left (\sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+i \sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )\right ) \cos (e+f x) (1+\tan (e+f x))^2+\frac {2}{231} \sec ^3(e+f x) (1+\tan (e+f x))^{5/2} \left (28-110 \cos ^2(e+f x)+400 \cos ^4(e+f x)+125 \cos ^3(e+f x) \sin (e+f x)-37 \sin (2 (e+f x))+21 \tan (e+f x)\right )\right )}{f (\cos (e+f x)+\sin (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 362, normalized size = 0.98
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}-\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {9}{2}}}{3}+\frac {4 \left (1+\tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 \sqrt {1+\tan \left (f x +e \right )}+\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \ln \left (1+\sqrt {2}-\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{4}+\frac {\left (-2 \sqrt {2}+\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \sqrt {2 \sqrt {2}+2}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+\tan \left (f x +e \right )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}-\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \ln \left (1+\sqrt {2}+\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{4}-\frac {\left (2 \sqrt {2}-\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \sqrt {2 \sqrt {2}+2}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {2}+2}+2 \sqrt {1+\tan \left (f x +e \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}}{f}\) | \(362\) |
default | \(\frac {\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}-\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {9}{2}}}{3}+\frac {4 \left (1+\tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 \sqrt {1+\tan \left (f x +e \right )}+\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \ln \left (1+\sqrt {2}-\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{4}+\frac {\left (-2 \sqrt {2}+\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \sqrt {2 \sqrt {2}+2}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+\tan \left (f x +e \right )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}-\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \ln \left (1+\sqrt {2}+\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{4}-\frac {\left (2 \sqrt {2}-\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \sqrt {2 \sqrt {2}+2}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {2}+2}+2 \sqrt {1+\tan \left (f x +e \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}}{f}\) | \(362\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 971 vs.
\(2 (306) = 612\).
time = 1.22, size = 971, normalized size = 2.63 \begin {gather*} \frac {924 \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (-\frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} + \frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \sqrt {\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} \cos \left (f x + e\right ) + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} - f^{2} \sqrt {\frac {1}{f^{4}}} - \sqrt {2}\right ) \cos \left (f x + e\right )^{5} + 924 \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (-\frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} + \frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \sqrt {\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} \cos \left (f x + e\right ) + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} + f^{2} \sqrt {\frac {1}{f^{4}}} + \sqrt {2}\right ) \cos \left (f x + e\right )^{5} + 231 \cdot 8^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right )^{5} - 2 \, f \cos \left (f x + e\right )^{5}\right )} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \log \left (\frac {2 \, {\left (2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} \cos \left (f x + e\right ) + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )\right )}}{\cos \left (f x + e\right )}\right ) - 231 \cdot 8^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right )^{5} - 2 \, f \cos \left (f x + e\right )^{5}\right )} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \log \left (\frac {2 \, {\left (2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} \cos \left (f x + e\right ) + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )\right )}}{\cos \left (f x + e\right )}\right ) + 16 \, {\left (400 \, \cos \left (f x + e\right )^{5} - 110 \, \cos \left (f x + e\right )^{3} + {\left (125 \, \cos \left (f x + e\right )^{4} - 74 \, \cos \left (f x + e\right )^{2} + 21\right )} \sin \left (f x + e\right ) + 28 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}}}{1848 \, f \cos \left (f x + e\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\tan {\left (e + f x \right )} + 1\right )^{\frac {3}{2}} \tan ^{5}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.84, size = 286, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {\sqrt {2} + 1} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{f} - \frac {\sqrt {\sqrt {2} + 1} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{f} - \frac {\sqrt {\sqrt {2} - 1} \log \left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{2 \, f} + \frac {\sqrt {\sqrt {2} - 1} \log \left (-2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{2 \, f} + \frac {2 \, {\left (21 \, f^{10} {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {11}{2}} - 77 \, f^{10} {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {9}{2}} + 66 \, f^{10} {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {7}{2}} + 77 \, f^{10} {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {3}{2}} + 231 \, f^{10} \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{231 \, f^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.90, size = 144, normalized size = 0.39 \begin {gather*} \frac {2\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{f}+\frac {2\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{3\,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}}{3\,f}+\frac {2\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{11/2}}{11\,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}\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}\right )\,\sqrt {\frac {-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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