Optimal. Leaf size=87 \[ -\frac {9 \tan ^{-1}\left (\frac {1-3 \tan (2 x)}{\sqrt {2} \sqrt {4+3 \tan (2 x)}}\right )}{250 \sqrt {2}}+\frac {13 \tanh ^{-1}\left (\frac {3+\tan (2 x)}{\sqrt {2} \sqrt {4+3 \tan (2 x)}}\right )}{250 \sqrt {2}}-\frac {3}{25 \sqrt {4+3 \tan (2 x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3564, 3617,
3616, 209, 213} \begin {gather*} -\frac {9 \text {ArcTan}\left (\frac {1-3 \tan (2 x)}{\sqrt {2} \sqrt {3 \tan (2 x)+4}}\right )}{250 \sqrt {2}}-\frac {3}{25 \sqrt {3 \tan (2 x)+4}}+\frac {13 \tanh ^{-1}\left (\frac {\tan (2 x)+3}{\sqrt {2} \sqrt {3 \tan (2 x)+4}}\right )}{250 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 213
Rule 3564
Rule 3616
Rule 3617
Rubi steps
\begin {align*} \int \frac {1}{(4+3 \tan (2 x))^{3/2}} \, dx &=-\frac {3}{25 \sqrt {4+3 \tan (2 x)}}+\frac {1}{25} \int \frac {4-3 \tan (2 x)}{\sqrt {4+3 \tan (2 x)}} \, dx\\ &=-\frac {3}{25 \sqrt {4+3 \tan (2 x)}}+\frac {1}{250} \int \frac {27+9 \tan (2 x)}{\sqrt {4+3 \tan (2 x)}} \, dx-\frac {1}{250} \int \frac {-13+39 \tan (2 x)}{\sqrt {4+3 \tan (2 x)}} \, dx\\ &=-\frac {3}{25 \sqrt {4+3 \tan (2 x)}}-\frac {81}{250} \text {Subst}\left (\int \frac {1}{162+x^2} \, dx,x,\frac {9-27 \tan (2 x)}{\sqrt {4+3 \tan (2 x)}}\right )+\frac {1521}{250} \text {Subst}\left (\int \frac {1}{-27378+x^2} \, dx,x,\frac {-351-117 \tan (2 x)}{\sqrt {4+3 \tan (2 x)}}\right )\\ &=-\frac {9 \tan ^{-1}\left (\frac {1-3 \tan (2 x)}{\sqrt {2} \sqrt {4+3 \tan (2 x)}}\right )}{250 \sqrt {2}}+\frac {13 \tanh ^{-1}\left (\frac {3+\tan (2 x)}{\sqrt {2} \sqrt {4+3 \tan (2 x)}}\right )}{250 \sqrt {2}}-\frac {3}{25 \sqrt {4+3 \tan (2 x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.07, size = 73, normalized size = 0.84 \begin {gather*} -\frac {(3+4 i) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\left (\frac {4}{25}-\frac {3 i}{25}\right ) (4+3 \tan (2 x))\right )+(3-4 i) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\left (\frac {4}{25}+\frac {3 i}{25}\right ) (4+3 \tan (2 x))\right )}{50 \sqrt {4+3 \tan (2 x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 130, normalized size = 1.49
method | result | size |
derivativedivides | \(-\frac {13 \sqrt {2}\, \ln \left (9+3 \tan \left (2 x \right )-3 \sqrt {4+3 \tan \left (2 x \right )}\, \sqrt {2}\right )}{1000}+\frac {9 \sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {4+3 \tan \left (2 x \right )}-3 \sqrt {2}\right ) \sqrt {2}}{2}\right )}{500}+\frac {13 \sqrt {2}\, \ln \left (9+3 \tan \left (2 x \right )+3 \sqrt {4+3 \tan \left (2 x \right )}\, \sqrt {2}\right )}{1000}+\frac {9 \sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {4+3 \tan \left (2 x \right )}+3 \sqrt {2}\right ) \sqrt {2}}{2}\right )}{500}-\frac {3}{25 \sqrt {4+3 \tan \left (2 x \right )}}\) | \(130\) |
default | \(-\frac {13 \sqrt {2}\, \ln \left (9+3 \tan \left (2 x \right )-3 \sqrt {4+3 \tan \left (2 x \right )}\, \sqrt {2}\right )}{1000}+\frac {9 \sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {4+3 \tan \left (2 x \right )}-3 \sqrt {2}\right ) \sqrt {2}}{2}\right )}{500}+\frac {13 \sqrt {2}\, \ln \left (9+3 \tan \left (2 x \right )+3 \sqrt {4+3 \tan \left (2 x \right )}\, \sqrt {2}\right )}{1000}+\frac {9 \sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {4+3 \tan \left (2 x \right )}+3 \sqrt {2}\right ) \sqrt {2}}{2}\right )}{500}-\frac {3}{25 \sqrt {4+3 \tan \left (2 x \right )}}\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 3213 vs.
\(2 (69) = 138\).
time = 2.76, size = 3213, normalized size = 36.93 \begin {gather*} \text {Too large to display} \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. 541 vs.
\(2 (69) = 138\).
time = 0.74, size = 541, normalized size = 6.22 \begin {gather*} -\frac {36 \, {\left (7 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right )^{2} + 24 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right ) \sin \left (2 \, x\right ) + 9 \, \sqrt {10} \sqrt {5}\right )} \arctan \left (\frac {1}{25} \, \sqrt {15} \sqrt {10} \sqrt {5} \sqrt {\frac {\sqrt {10} \sqrt {5} \sqrt {\frac {4 \, \cos \left (2 \, x\right ) + 3 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} \cos \left (2 \, x\right ) + 15 \, \cos \left (2 \, x\right ) + 5 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} - \frac {1}{5} \, \sqrt {10} \sqrt {5} \sqrt {\frac {4 \, \cos \left (2 \, x\right ) + 3 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} - 3\right ) + 36 \, {\left (7 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right )^{2} + 24 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right ) \sin \left (2 \, x\right ) + 9 \, \sqrt {10} \sqrt {5}\right )} \arctan \left (\frac {1}{25} \, \sqrt {15} \sqrt {10} \sqrt {5} \sqrt {-\frac {\sqrt {10} \sqrt {5} \sqrt {\frac {4 \, \cos \left (2 \, x\right ) + 3 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} \cos \left (2 \, x\right ) - 15 \, \cos \left (2 \, x\right ) - 5 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} - \frac {1}{5} \, \sqrt {10} \sqrt {5} \sqrt {\frac {4 \, \cos \left (2 \, x\right ) + 3 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} + 3\right ) - 13 \, {\left (7 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right )^{2} + 24 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right ) \sin \left (2 \, x\right ) + 9 \, \sqrt {10} \sqrt {5}\right )} \log \left (\frac {9375 \, {\left (\sqrt {10} \sqrt {5} \sqrt {\frac {4 \, \cos \left (2 \, x\right ) + 3 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} \cos \left (2 \, x\right ) + 15 \, \cos \left (2 \, x\right ) + 5 \, \sin \left (2 \, x\right )\right )}}{\cos \left (2 \, x\right )}\right ) + 13 \, {\left (7 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right )^{2} + 24 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right ) \sin \left (2 \, x\right ) + 9 \, \sqrt {10} \sqrt {5}\right )} \log \left (-\frac {9375 \, {\left (\sqrt {10} \sqrt {5} \sqrt {\frac {4 \, \cos \left (2 \, x\right ) + 3 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} \cos \left (2 \, x\right ) - 15 \, \cos \left (2 \, x\right ) - 5 \, \sin \left (2 \, x\right )\right )}}{\cos \left (2 \, x\right )}\right ) + 600 \, {\left (4 \, \cos \left (2 \, x\right )^{2} + 3 \, \cos \left (2 \, x\right ) \sin \left (2 \, x\right )\right )} \sqrt {\frac {4 \, \cos \left (2 \, x\right ) + 3 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}}}{5000 \, {\left (7 \, \cos \left (2 \, x\right )^{2} + 24 \, \cos \left (2 \, x\right ) \sin \left (2 \, x\right ) + 9\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 \frac {1}{\left (3 \tan {\left (2 x \right )} + 4\right )^{\frac {3}{2}}}\, 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 = 0.44, size = 63, normalized size = 0.72 \begin {gather*} -\frac {3}{25\,\sqrt {3\,\mathrm {tan}\left (2\,x\right )+4}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {3\,\mathrm {tan}\left (2\,x\right )+4}\,\left (\frac {1}{10}-\frac {3}{10}{}\mathrm {i}\right )\right )\,\left (\frac {9}{500}+\frac {13}{500}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {3\,\mathrm {tan}\left (2\,x\right )+4}\,\left (\frac {1}{10}+\frac {3}{10}{}\mathrm {i}\right )\right )\,\left (\frac {9}{500}-\frac {13}{500}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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