Optimal. Leaf size=221 \[ -\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {4-3 \sqrt {2}+\left (2-\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {-7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {9 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{4 f}-\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {4+3 \sqrt {2}+\left (2+\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}-\frac {\cot (e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{2 f} \]
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Rubi [A]
time = 0.34, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3649, 3730,
3734, 3617, 3616, 209, 213, 3715, 65} \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 {9 \tanh ^{-1}\left (\sqrt {\tan (e+f x)+1}\right )}{4 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}-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{4 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 213
Rule 3616
Rule 3617
Rule 3649
Rule 3715
Rule 3730
Rule 3734
Rubi steps
\begin {align*} \int \cot ^3(e+f x) \sqrt {1+\tan (e+f x)} \, dx &=-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{2 f}-\frac {1}{2} \int \frac {\cot ^2(e+f x) \left (-\frac {1}{2}+2 \tan (e+f x)+\frac {3}{2} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=-\frac {\cot (e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{2 f}+\frac {1}{2} \int \frac {\cot (e+f x) \left (-\frac {9}{4}-2 \tan (e+f x)-\frac {1}{4} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=-\frac {\cot (e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{2 f}+\frac {1}{2} \int \frac {-2+2 \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx-\frac {9}{8} \int \frac {\cot (e+f x) \left (1+\tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=-\frac {\cot (e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{2 f}-\frac {\int \frac {2 \sqrt {2}+\left (-4-2 \sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{4 \sqrt {2}}+\frac {\int \frac {-2 \sqrt {2}+\left (-4+2 \sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{4 \sqrt {2}}-\frac {9 \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=-\frac {\cot (e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{2 f}-\frac {9 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{4 f}+\frac {\left (2 \left (4-3 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \sqrt {2} \left (-4+2 \sqrt {2}\right )-4 \left (-4+2 \sqrt {2}\right )^2+x^2} \, dx,x,\frac {-2 \sqrt {2}-2 \left (-4+2 \sqrt {2}\right )-\left (-4+2 \sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {\left (2 \left (4+3 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{4 \sqrt {2} \left (-4-2 \sqrt {2}\right )-4 \left (-4-2 \sqrt {2}\right )^2+x^2} \, dx,x,\frac {2 \sqrt {2}-2 \left (-4-2 \sqrt {2}\right )-\left (-4-2 \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 {9 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{4 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 {\cot (e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{2 f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.49, size = 124, normalized size = 0.56 \begin {gather*} -\frac {-9 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )+4 \sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+4 \sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )+\cot (e+f x) \sqrt {1+\tan (e+f x)}+2 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.90, size = 11217, normalized size = 50.76
method | result | size |
default | \(\text {Expression too large to display}\) | \(11217\) |
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. 1206 vs.
\(2 (180) = 360\).
time = 0.78, size = 1206, normalized size = 5.46 \begin {gather*} -\frac {2^{\frac {1}{4}} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} {\left (2 \, f \cos \left (f x + e\right )^{2} + \sqrt {2} {\left (f^{3} \cos \left (f x + e\right )^{2} - f^{3}\right )} \sqrt {\frac {1}{f^{4}}} - 2 \, f\right )} \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}} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} {\left (2 \, f \cos \left (f x + e\right )^{2} + \sqrt {2} {\left (f^{3} \cos \left (f x + e\right )^{2} - f^{3}\right )} \sqrt {\frac {1}{f^{4}}} - 2 \, f\right )} \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 ) - 9 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \log \left (\sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} + 1\right ) + 9 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \log \left (\sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} - 1\right ) - 2 \, {\left (2 \, \cos \left (f x + e\right )^{2} + \cos \left (f x + e\right ) \sin \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 )}} - \frac {4 \cdot 2^{\frac {3}{4}} {\left (f^{5} \cos \left (f x + e\right )^{2} - f^{5}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \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 )}{f^{4}} - \frac {4 \cdot 2^{\frac {3}{4}} {\left (f^{5} \cos \left (f x + e\right )^{2} - f^{5}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \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 )}{f^{4}}}{8 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )}} \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} \cot ^{3}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 149, normalized size = 0.67 \begin {gather*} -\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{4\,f}-\frac {\frac {\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{4}+\frac {{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{4}}{f-2\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2}-\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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