Optimal. Leaf size=315 \[ -\frac {\sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}+\frac {\sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}-\frac {2 \sqrt {1+\tan (e+f x)}}{f}-\frac {2 (1+\tan (e+f x))^{3/2}}{3 f}-\frac {4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac {2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.24, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3647, 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) (\tan (e+f x)+1)^{5/2}}{7 f}-\frac {4 (\tan (e+f x)+1)^{5/2}}{35 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.
[In]
[Out]
Rule 12
Rule 210
Rule 632
Rule 642
Rule 648
Rule 722
Rule 1108
Rule 3566
Rule 3609
Rule 3647
Rule 3711
Rubi steps
\begin {align*} \int \tan ^3(e+f x) (1+\tan (e+f x))^{3/2} \, dx &=\frac {2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}+\frac {2}{7} \int (1+\tan (e+f x))^{3/2} \left (-1-\frac {7}{2} \tan (e+f x)-\tan ^2(e+f x)\right ) \, dx\\ &=-\frac {4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac {2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}+\frac {2}{7} \int -\frac {7}{2} \tan (e+f x) (1+\tan (e+f x))^{3/2} \, dx\\ &=-\frac {4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac {2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 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 {4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac {2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 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 {4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac {2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 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 {4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac {2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 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 {4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac {2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 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 {4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac {2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 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 {4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac {2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 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 {4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac {2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 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 {4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac {2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 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 {4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac {2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.64, size = 112, normalized size = 0.36 \begin {gather*} \frac {105 (1-i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+105 (1+i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )+2 \sqrt {1+\tan (e+f x)} \left (-146-32 \tan (e+f x)+24 \tan ^2(e+f x)+15 \tan ^3(e+f x)\right )}{105 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.15, size = 350, normalized size = 1.11
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {1+\tan \left (f x +e \right )}-\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \ln \left (1+\sqrt {2}-\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{4}-\frac {\left (-2 \sqrt {2}+\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \sqrt {2 \sqrt {2}+2}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+\tan \left (f x +e \right )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \ln \left (1+\sqrt {2}+\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{4}+\frac {\left (2 \sqrt {2}-\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \sqrt {2 \sqrt {2}+2}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {2}+2}+2 \sqrt {1+\tan \left (f x +e \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}}{f}\) | \(350\) |
default | \(\frac {\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {1+\tan \left (f x +e \right )}-\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \ln \left (1+\sqrt {2}-\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{4}-\frac {\left (-2 \sqrt {2}+\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \sqrt {2 \sqrt {2}+2}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+\tan \left (f x +e \right )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \ln \left (1+\sqrt {2}+\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{4}+\frac {\left (2 \sqrt {2}-\frac {\left (-\sqrt {2 \sqrt {2}+2}\, \sqrt {2}+2 \sqrt {2 \sqrt {2}+2}\right ) \sqrt {2 \sqrt {2}+2}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {2}+2}+2 \sqrt {1+\tan \left (f x +e \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}}{f}\) | \(350\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 949 vs.
\(2 (256) = 512\).
time = 1.45, size = 949, normalized size = 3.01 \begin {gather*} -\frac {420 \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 )^{3} + 420 \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 )^{3} + 105 \cdot 8^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right )^{3} - 2 \, f \cos \left (f x + e\right )^{3}\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 ) - 105 \cdot 8^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right )^{3} - 2 \, f \cos \left (f x + e\right )^{3}\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 (170 \, \cos \left (f x + e\right )^{3} + {\left (47 \, \cos \left (f x + e\right )^{2} - 15\right )} \sin \left (f x + e\right ) - 24 \, \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 )}}}{840 \, f \cos \left (f x + e\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 ^{3}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.80, size = 268, normalized size = 0.85 \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 (15 \, f^{6} {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {7}{2}} - 21 \, f^{6} {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {5}{2}} - 35 \, f^{6} {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {3}{2}} - 105 \, f^{6} \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{105 \, f^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 5.24, size = 129, normalized size = 0.41 \begin {gather*} \frac {2\,{\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 )}^{3/2}}{3\,f}-\frac {2\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{5/2}}{5\,f}-\frac {2\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{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.
[In]
[Out]
________________________________________________________________________________________