Optimal. Leaf size=166 \[ -\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {4-3 \sqrt {2}+\left (2-\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {-7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}-\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {4+3 \sqrt {2}+\left (2+\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {2 \sqrt {1+\tan (e+f x)}}{f} \]
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Rubi [A]
time = 0.11, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3609, 3617,
3616, 209, 213} \begin {gather*} -\frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \text {ArcTan}\left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{f}+\frac {2 \sqrt {\tan (e+f x)+1}}{f}-\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\tan (e+f x)+1}}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 213
Rule 3609
Rule 3616
Rule 3617
Rubi steps
\begin {align*} \int \tan (e+f x) \sqrt {1+\tan (e+f x)} \, dx &=\frac {2 \sqrt {1+\tan (e+f x)}}{f}+\int \frac {-1+\tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {2 \sqrt {1+\tan (e+f x)}}{f}-\frac {\int \frac {\sqrt {2}+\left (-2-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{2 \sqrt {2}}+\frac {\int \frac {-\sqrt {2}+\left (-2+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{2 \sqrt {2}}\\ &=\frac {2 \sqrt {1+\tan (e+f x)}}{f}+\frac {\left (4-3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-2 \sqrt {2} \left (-2+\sqrt {2}\right )-4 \left (-2+\sqrt {2}\right )^2+x^2} \, dx,x,\frac {-\sqrt {2}-2 \left (-2+\sqrt {2}\right )-\left (-2+\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 \sqrt {2} \left (-2-\sqrt {2}\right )-4 \left (-2-\sqrt {2}\right )^2+x^2} \, dx,x,\frac {\sqrt {2}-2 \left (-2-\sqrt {2}\right )-\left (-2-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{f}\\ &=-\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {4-3 \sqrt {2}+\left (2-\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {-7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}-\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {4+3 \sqrt {2}+\left (2+\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {2 \sqrt {1+\tan (e+f x)}}{f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.07, size = 78, normalized size = 0.47 \begin {gather*} -\frac {\sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+\sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )-2 \sqrt {1+\tan (e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 206, normalized size = 1.24
method | result | size |
derivativedivides | \(\frac {2 \sqrt {1+\tan \left (f x +e \right )}+\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\sqrt {2}-\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{4}-\frac {\left (\sqrt {2}-1\right ) \arctan \left (\frac {2 \sqrt {1+\tan \left (f x +e \right )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\sqrt {2}+\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{4}+\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {2}+2}+2 \sqrt {1+\tan \left (f x +e \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}}{f}\) | \(206\) |
default | \(\frac {2 \sqrt {1+\tan \left (f x +e \right )}+\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\sqrt {2}-\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{4}-\frac {\left (\sqrt {2}-1\right ) \arctan \left (\frac {2 \sqrt {1+\tan \left (f x +e \right )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\sqrt {2}+\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{4}+\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {2}+2}+2 \sqrt {1+\tan \left (f x +e \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}}{f}\) | \(206\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 981 vs.
\(2 (131) = 262\).
time = 1.01, size = 981, normalized size = 5.91 \begin {gather*} \frac {4 \cdot 2^{\frac {3}{4}} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {\frac {1}{2}} {\left (f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {2} f^{3}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 2^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 2 \, f \cos \left (f x + e\right )\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \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}} + 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}} - \frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {2} f^{3}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \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}} - f^{2} \sqrt {\frac {1}{f^{4}}} - \sqrt {2}\right ) + 4 \cdot 2^{\frac {3}{4}} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {\frac {1}{2}} {\left (f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {2} f^{3}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - 2^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 2 \, f \cos \left (f x + e\right )\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \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}} + 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}} - \frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {2} f^{3}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \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}} + f^{2} \sqrt {\frac {1}{f^{4}}} + \sqrt {2}\right ) - 2^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} + 2 \, f\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \log \left (\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 2^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 2 \, f \cos \left (f x + e\right )\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \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}} + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{2 \, \cos \left (f x + e\right )}\right ) + 2^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} + 2 \, f\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \log \left (\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - 2^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 2 \, f \cos \left (f x + e\right )\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \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}} + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{2 \, \cos \left (f x + e\right )}\right ) + 16 \, \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}}}{8 \, f} \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 \sqrt {\tan {\left (e + f x \right )} + 1} \tan {\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 = 0.64, size = 224, normalized size = 1.35 \begin {gather*} -\frac {\sqrt {2 \, \sqrt {2} - 2} \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 )}{2 \, f} - \frac {\sqrt {2 \, \sqrt {2} - 2} \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 )}{2 \, f} - \frac {\sqrt {2 \, \sqrt {2} + 2} \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 )}{4 \, f} + \frac {\sqrt {2 \, \sqrt {2} + 2} \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 )}{4 \, f} + \frac {2 \, \sqrt {\tan \left (f x + e\right ) + 1}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.10, size = 90, normalized size = 0.54 \begin {gather*} \frac {2\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{f}-\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,\left (1-\mathrm {i}\right )\right )\,\sqrt {\frac {\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{4}+\frac {1}{4}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,\left (1+1{}\mathrm {i}\right )\right )\,\sqrt {\frac {\frac {1}{4}+\frac {1}{4}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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