Optimal. Leaf size=269 \[ \frac {\sqrt {-1+\sqrt {2}} \text {ArcTan}\left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{2 f}-\frac {115 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{64 f}+\frac {\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{2 f}-\frac {13 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {13 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}+\frac {7 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.43, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3650, 3730,
3731, 3735, 12, 3617, 3616, 209, 213, 3715, 65} \begin {gather*} \frac {\sqrt {\sqrt {2}-1} \text {ArcTan}\left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 f}-\frac {115 \tanh ^{-1}\left (\sqrt {\tan (e+f x)+1}\right )}{64 f}+\frac {\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}+\frac {7 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{24 f}+\frac {13 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{96 f}-\frac {13 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{64 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 65
Rule 209
Rule 213
Rule 3616
Rule 3617
Rule 3650
Rule 3715
Rule 3730
Rule 3731
Rule 3735
Rubi steps
\begin {align*} \int \frac {\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 {7}{2}+4 \tan (e+f x)+\frac {7}{2} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {7 \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 {13}{4}+\frac {35}{4} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {13 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}+\frac {7 \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 {39}{8}-24 \tan (e+f x)-\frac {39}{8} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=-\frac {13 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {13 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}+\frac {7 \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 {345}{16}-\frac {39}{16} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=-\frac {13 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {13 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}+\frac {7 \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 \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx+\frac {115}{128} \int \frac {\cot (e+f x) \left (1+\tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=-\frac {13 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {13 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}+\frac {7 \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 {115 \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\tan (e+f x)\right )}{128 f}-\int \frac {\tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=-\frac {13 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {13 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}+\frac {7 \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 {1+\left (-1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{2 \sqrt {2}}-\frac {\int \frac {1+\left (-1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{2 \sqrt {2}}+\frac {115 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{64 f}\\ &=-\frac {115 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{64 f}-\frac {13 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {13 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}+\frac {7 \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 {\left (4-3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2 \left (-1+\sqrt {2}\right )-4 \left (-1+\sqrt {2}\right )^2+x^2} \, dx,x,\frac {1-2 \left (-1+\sqrt {2}\right )-\left (-1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{2 f}-\frac {\left (4+3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2 \left (-1-\sqrt {2}\right )-4 \left (-1-\sqrt {2}\right )^2+x^2} \, dx,x,\frac {1-2 \left (-1-\sqrt {2}\right )-\left (-1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{2 f}\\ &=\frac {\sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{2 f}-\frac {115 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{64 f}+\frac {\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{2 f}-\frac {13 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {13 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}+\frac {7 \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*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 18.04, size = 440, normalized size = 1.64 \begin {gather*} \frac {338+190 \cot (e+f x)-356 \csc ^2(e+f x)-112 \cot (e+f x) \csc ^2(e+f x)+96 \csc ^4(e+f x)-678 \csc (2 (e+f x))+712 \csc ^2(2 (e+f x))-384 \csc ^3(2 (e+f x))+152 \csc ^3(e+f x) \sec (e+f x)-169 \sec ^2(e+f x)-48 \csc ^4(e+f x) \sec ^2(e+f x)+122 \csc (e+f x) \sec ^3(e+f x)+96 \sqrt {-2-2 i} \text {ArcTan}\left (\sqrt {-\frac {1}{2}-\frac {i}{2}} \sqrt {1+\sec (e+f x) \sqrt {\sin ^2(e+f x)}}\right ) \cos (2 (e+f x)) \sec ^2(e+f x) \sqrt {1+\sec (e+f x) \sqrt {\sin ^2(e+f x)}}+96 \sqrt {-2+2 i} \text {ArcTan}\left (\sqrt {-\frac {1}{2}+\frac {i}{2}} \sqrt {1+\sec (e+f x) \sqrt {\sin ^2(e+f x)}}\right ) \cos (2 (e+f x)) \sec ^2(e+f x) \sqrt {1+\sec (e+f x) \sqrt {\sin ^2(e+f x)}}+345 \tanh ^{-1}\left (\sqrt {1+\sec (e+f x) \sqrt {\sin ^2(e+f x)}}\right ) \cos (2 (e+f x)) \sec ^2(e+f x) \sqrt {1+\sec (e+f x) \sqrt {\sin ^2(e+f x)}}+148 \tan (e+f x)-74 \sec ^2(e+f x) \tan (e+f x)}{192 f \left (-2+\sec ^2(e+f x)\right ) \sqrt {1+\tan (e+f x)}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.86, size = 13527, normalized size = 50.29
method | result | size |
default | \(\text {Expression too large to display}\) | \(13527\) |
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. 1309 vs.
\(2 (225) = 450\).
time = 1.19, size = 1309, normalized size = 4.87 \begin {gather*} \text {Too large to display} \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 \frac {\cot ^{5}{\left (e + f x \right )}}{\sqrt {\tan {\left (e + f x \right )} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.22, size = 197, normalized size = 0.73 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,115{}\mathrm {i}}{64\,f}-\frac {\frac {13\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{64}+\frac {113\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{192}-\frac {143\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{5/2}}{192}+\frac {13\,{\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}{8}-\frac {1}{8}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,2{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{8}-\frac {1}{8}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{8}+\frac {1}{8}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,2{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{8}+\frac {1}{8}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________