Optimal. Leaf size=273 \[ \frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{f}-\frac {139 \tanh ^{-1}\left (\sqrt {\tan (e+f x)+1}\right )}{64 f}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\tan (e+f x)+1}}\right )}{f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{24 f}+\frac {53 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{96 f}+\frac {11 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{64 f} \]
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Rubi [A] time = 0.72, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3568, 3649, 3653, 3536, 3535, 203, 207, 3634, 63} \[ \frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{f}-\frac {139 \tanh ^{-1}\left (\sqrt {\tan (e+f x)+1}\right )}{64 f}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\tan (e+f x)+1}}\right )}{f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{24 f}+\frac {53 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{96 f}+\frac {11 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{64 f} \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 207
Rule 3535
Rule 3536
Rule 3568
Rule 3634
Rule 3649
Rule 3653
Rubi steps
\begin {align*} \int \cot ^5(e+f x) \sqrt {1+\tan (e+f x)} \, dx &=-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {1}{4} \int \frac {\cot ^4(e+f x) \left (-\frac {1}{2}+4 \tan (e+f x)+\frac {7}{2} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {1}{12} \int \frac {\cot ^3(e+f x) \left (-\frac {53}{4}-12 \tan (e+f x)-\frac {5}{4} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {53 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {1}{24} \int \frac {\cot ^2(e+f x) \left (\frac {33}{8}-24 \tan (e+f x)-\frac {159}{8} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {11 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {53 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {1}{24} \int \frac {\cot (e+f x) \left (\frac {417}{16}+24 \tan (e+f x)+\frac {33}{16} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {11 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {53 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {1}{24} \int \frac {24-24 \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx+\frac {139}{128} \int \frac {\cot (e+f x) \left (1+\tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {11 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {53 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {\int \frac {24 \sqrt {2}+\left (48-24 \sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{48 \sqrt {2}}-\frac {\int \frac {-24 \sqrt {2}+\left (48+24 \sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{48 \sqrt {2}}+\frac {139 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\tan (e+f x)\right )}{128 f}\\ &=\frac {11 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {53 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {139 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{64 f}+\frac {\left (24 \left (4-3 \sqrt {2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{48 \sqrt {2} \left (48-24 \sqrt {2}\right )-4 \left (48-24 \sqrt {2}\right )^2+x^2} \, dx,x,\frac {24 \sqrt {2}-2 \left (48-24 \sqrt {2}\right )-\left (48-24 \sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {\left (24 \left (4+3 \sqrt {2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-48 \sqrt {2} \left (48+24 \sqrt {2}\right )-4 \left (48+24 \sqrt {2}\right )^2+x^2} \, dx,x,\frac {-24 \sqrt {2}-2 \left (48+24 \sqrt {2}\right )-\left (48+24 \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 {139 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{64 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 {11 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {53 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}\\ \end {align*}
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Mathematica [C] time = 0.83, size = 169, normalized size = 0.62 \[ \frac {-417 \tanh ^{-1}\left (\sqrt {\tan (e+f x)+1}\right )+192 \sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {\tan (e+f x)+1}}{\sqrt {1-i}}\right )+192 \sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {\tan (e+f x)+1}}{\sqrt {1+i}}\right )-48 \sqrt {\tan (e+f x)+1} \cot ^4(e+f x)-8 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)+106 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)+33 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{192 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 1262, normalized size = 4.62 \[ \text {result too large to display} \]
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 \[ \int \sqrt {\tan \left (f x + e\right ) + 1} \cot \left (f x + e\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.65, size = 16815, normalized size = 61.59 \[ \text {output too large to display} \]
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 \[ \int \sqrt {\tan \left (f x + e\right ) + 1} \cot \left (f x + e\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 198, normalized size = 0.73 \[ \frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,139{}\mathrm {i}}{64\,f}+\frac {\frac {11\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{64}-\frac {121\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{192}+\frac {7\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{5/2}}{192}+\frac {11\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{7/2}}{64}}{f-4\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+6\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2-4\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^3+f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^4}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,\left (1-\mathrm {i}\right )\right )\,\sqrt {\frac {\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{4}+\frac {1}{4}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,\left (1+1{}\mathrm {i}\right )\right )\,\sqrt {\frac {\frac {1}{4}+\frac {1}{4}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \]
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 \[ \int \sqrt {\tan {\left (e + f x \right )} + 1} \cot ^{5}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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