Optimal. Leaf size=137 \[ \sinh ^{-1}(1+\tan (x))-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {2 \sqrt {5}-\left (5+\sqrt {5}\right ) \tan (x)}{\sqrt {10 \left (1+\sqrt {5}\right )} \sqrt {2+2 \tan (x)+\tan ^2(x)}}\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2 \sqrt {5}+\left (5-\sqrt {5}\right ) \tan (x)}{\sqrt {10 \left (-1+\sqrt {5}\right )} \sqrt {2+2 \tan (x)+\tan ^2(x)}}\right ) \]
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Rubi [A]
time = 0.13, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1004, 633, 221,
1050, 1044, 213, 209} \begin {gather*} -\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {2 \sqrt {5}-\left (5+\sqrt {5}\right ) \tan (x)}{\sqrt {10 \left (1+\sqrt {5}\right )} \sqrt {\tan ^2(x)+2 \tan (x)+2}}\right )-\sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\frac {\left (5-\sqrt {5}\right ) \tan (x)+2 \sqrt {5}}{\sqrt {10 \left (\sqrt {5}-1\right )} \sqrt {\tan ^2(x)+2 \tan (x)+2}}\right )+\sinh ^{-1}(\tan (x)+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 213
Rule 221
Rule 633
Rule 1004
Rule 1044
Rule 1050
Rubi steps
\begin {align*} \int \sqrt {2+2 \tan (x)+\tan ^2(x)} \, dx &=\text {Subst}\left (\int \frac {\sqrt {2+2 x+x^2}}{1+x^2} \, dx,x,\tan (x)\right )\\ &=\text {Subst}\left (\int \frac {1}{\sqrt {2+2 x+x^2}} \, dx,x,\tan (x)\right )-\text {Subst}\left (\int \frac {-1-2 x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4}}} \, dx,x,2+2 \tan (x)\right )-\frac {\text {Subst}\left (\int \frac {5-\sqrt {5}-2 \sqrt {5} x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx,x,\tan (x)\right )}{2 \sqrt {5}}+\frac {\text {Subst}\left (\int \frac {5+\sqrt {5}+2 \sqrt {5} x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx,x,\tan (x)\right )}{2 \sqrt {5}}\\ &=\sinh ^{-1}(1+\tan (x))-\left (2 \left (5-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{20 \left (1-\sqrt {5}\right )+2 x^2} \, dx,x,\frac {-2 \sqrt {5}-\left (5-\sqrt {5}\right ) \tan (x)}{\sqrt {2+2 \tan (x)+\tan ^2(x)}}\right )-\left (2 \left (5+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{20 \left (1+\sqrt {5}\right )+2 x^2} \, dx,x,\frac {2 \sqrt {5}-\left (5+\sqrt {5}\right ) \tan (x)}{\sqrt {2+2 \tan (x)+\tan ^2(x)}}\right )\\ &=\sinh ^{-1}(1+\tan (x))-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {2 \sqrt {5}-\left (5+\sqrt {5}\right ) \tan (x)}{\sqrt {10 \left (1+\sqrt {5}\right )} \sqrt {2+2 \tan (x)+\tan ^2(x)}}\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2 \sqrt {5}+\left (5-\sqrt {5}\right ) \tan (x)}{\sqrt {10 \left (-1+\sqrt {5}\right )} \sqrt {2+2 \tan (x)+\tan ^2(x)}}\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.67, size = 99, normalized size = 0.72 \begin {gather*} \sinh ^{-1}(1+\tan (x))+\frac {1}{2} i \left (\sqrt {1+2 i} \tanh ^{-1}\left (\frac {(2+i)+(1+i) \tan (x)}{\sqrt {1+2 i} \sqrt {2+2 \tan (x)+\tan ^2(x)}}\right )-\sqrt {1-2 i} \tanh ^{-1}\left (\frac {(4-2 i)+(2-2 i) \tan (x)}{2 \sqrt {1-2 i} \sqrt {2+2 \tan (x)+\tan ^2(x)}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1603\) vs.
\(2(105)=210\).
time = 0.46, size = 1604, normalized size = 11.71
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1604\) |
default | \(\text {Expression too large to display}\) | \(1604\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 ^{2}{\left (x \right )} + 2 \tan {\left (x \right )} + 2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 495 vs.
\(2 (104) = 208\).
time = 0.51, size = 495, normalized size = 3.61 \begin {gather*} -\frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (256 \, {\left (\sqrt {5} {\left (\sqrt {\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 2} - \tan \left (x\right )\right )} + \sqrt {5} \sqrt {\sqrt {5} - 2} - \sqrt {5} - 2 \, \sqrt {\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 2} - 2 \, \sqrt {\sqrt {5} - 2} + 2 \, \tan \left (x\right ) + 2\right )}^{2} + 256 \, {\left (\sqrt {5} {\left (\sqrt {\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 2} - \tan \left (x\right )\right )} + \sqrt {5} - 2 \, \sqrt {\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 2} + \sqrt {\sqrt {5} - 2} + 2 \, \tan \left (x\right ) - 2\right )}^{2}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (256 \, {\left (\sqrt {5} {\left (\sqrt {\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 2} - \tan \left (x\right )\right )} - \sqrt {5} \sqrt {\sqrt {5} - 2} - \sqrt {5} - 2 \, \sqrt {\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 2} + 2 \, \sqrt {\sqrt {5} - 2} + 2 \, \tan \left (x\right ) + 2\right )}^{2} + 256 \, {\left (\sqrt {5} {\left (\sqrt {\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 2} - \tan \left (x\right )\right )} + \sqrt {5} - 2 \, \sqrt {\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 2} - \sqrt {\sqrt {5} - 2} + 2 \, \tan \left (x\right ) - 2\right )}^{2}\right ) + \frac {{\left (\pi + 4 \, \arctan \left (-\frac {1}{2} \, {\left (2 \, \sqrt {5} \sqrt {\sqrt {5} - 2} + \sqrt {5} + 4 \, \sqrt {\sqrt {5} - 2} + 3\right )} {\left (\sqrt {\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 2} - \tan \left (x\right )\right )} + \frac {3}{2} \, \sqrt {5} \sqrt {\sqrt {5} - 2} + \frac {1}{2} \, \sqrt {5} + \frac {7}{2} \, \sqrt {\sqrt {5} - 2} + \frac {3}{2}\right )\right )} \sqrt {2 \, \sqrt {5} - 2}}{4 \, {\left (\sqrt {5} - 1\right )}} - \frac {{\left (\pi + 4 \, \arctan \left (\frac {1}{2} \, {\left (2 \, \sqrt {5} \sqrt {\sqrt {5} - 2} - \sqrt {5} + 4 \, \sqrt {\sqrt {5} - 2} - 3\right )} {\left (\sqrt {\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 2} - \tan \left (x\right )\right )} - \frac {3}{2} \, \sqrt {5} \sqrt {\sqrt {5} - 2} + \frac {1}{2} \, \sqrt {5} - \frac {7}{2} \, \sqrt {\sqrt {5} - 2} + \frac {3}{2}\right )\right )} \sqrt {2 \, \sqrt {5} - 2}}{4 \, {\left (\sqrt {5} - 1\right )}} - \log \left (\sqrt {\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 2} - \tan \left (x\right ) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {{\mathrm {tan}\left (x\right )}^2+2\,\mathrm {tan}\left (x\right )+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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