Optimal. Leaf size=94 \[ -\frac {1}{4} \tanh ^{-1}\left (\frac {2 \tan (x)}{\sqrt {1+5 \tan ^2(x)}}\right )-\frac {\cos (x)}{4 \sqrt {1+5 \tan ^2(x)}}-\frac {5 \cot (x)}{2 \sqrt {1+5 \tan ^2(x)}}-\frac {1}{8} \cos (x) \sqrt {1+5 \tan ^2(x)}+\frac {9}{2} \cot (x) \sqrt {1+5 \tan ^2(x)} \]
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Rubi [A]
time = 0.11, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4462, 12,
3751, 483, 597, 385, 212, 3745, 277, 197} \begin {gather*} -\frac {5 \sec (x)}{8 \sqrt {5 \sec ^2(x)-4}}+\frac {\cos (x)}{4 \sqrt {5 \sec ^2(x)-4}}-\frac {1}{4} \tanh ^{-1}\left (\frac {2 \tan (x)}{\sqrt {5 \tan ^2(x)+1}}\right )+\frac {9}{2} \sqrt {5 \tan ^2(x)+1} \cot (x)-\frac {5 \cot (x)}{2 \sqrt {5 \tan ^2(x)+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 197
Rule 212
Rule 277
Rule 385
Rule 483
Rule 597
Rule 3745
Rule 3751
Rule 4462
Rubi steps
\begin {align*} \int \frac {-2 \cot ^2(x)+\sin (x)}{\left (1+5 \tan ^2(x)\right )^{3/2}} \, dx &=\int -\frac {2 \cot ^2(x)}{\left (1+5 \tan ^2(x)\right )^{3/2}} \, dx+\int \frac {\sin (x)}{\left (1+5 \tan ^2(x)\right )^{3/2}} \, dx\\ &=-\left (2 \int \frac {\cot ^2(x)}{\left (1+5 \tan ^2(x)\right )^{3/2}} \, dx\right )+\text {Subst}\left (\int \frac {1}{x^2 \left (-4+5 x^2\right )^{3/2}} \, dx,x,\sec (x)\right )\\ &=\frac {\cos (x)}{4 \sqrt {-4+5 \sec ^2(x)}}-2 \text {Subst}\left (\int \frac {1}{x^2 \left (1+x^2\right ) \left (1+5 x^2\right )^{3/2}} \, dx,x,\tan (x)\right )+\frac {5}{2} \text {Subst}\left (\int \frac {1}{\left (-4+5 x^2\right )^{3/2}} \, dx,x,\sec (x)\right )\\ &=\frac {\cos (x)}{4 \sqrt {-4+5 \sec ^2(x)}}-\frac {5 \sec (x)}{8 \sqrt {-4+5 \sec ^2(x)}}-\frac {5 \cot (x)}{2 \sqrt {1+5 \tan ^2(x)}}+\frac {1}{2} \text {Subst}\left (\int \frac {-9-10 x^2}{x^2 \left (1+x^2\right ) \sqrt {1+5 x^2}} \, dx,x,\tan (x)\right )\\ &=\frac {\cos (x)}{4 \sqrt {-4+5 \sec ^2(x)}}-\frac {5 \sec (x)}{8 \sqrt {-4+5 \sec ^2(x)}}-\frac {5 \cot (x)}{2 \sqrt {1+5 \tan ^2(x)}}+\frac {9}{2} \cot (x) \sqrt {1+5 \tan ^2(x)}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1+5 x^2}} \, dx,x,\tan (x)\right )\\ &=\frac {\cos (x)}{4 \sqrt {-4+5 \sec ^2(x)}}-\frac {5 \sec (x)}{8 \sqrt {-4+5 \sec ^2(x)}}-\frac {5 \cot (x)}{2 \sqrt {1+5 \tan ^2(x)}}+\frac {9}{2} \cot (x) \sqrt {1+5 \tan ^2(x)}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-4 x^2} \, dx,x,\frac {\tan (x)}{\sqrt {1+5 \tan ^2(x)}}\right )\\ &=-\frac {1}{4} \tanh ^{-1}\left (\frac {2 \tan (x)}{\sqrt {1+5 \tan ^2(x)}}\right )+\frac {\cos (x)}{4 \sqrt {-4+5 \sec ^2(x)}}-\frac {5 \sec (x)}{8 \sqrt {-4+5 \sec ^2(x)}}-\frac {5 \cot (x)}{2 \sqrt {1+5 \tan ^2(x)}}+\frac {9}{2} \cot (x) \sqrt {1+5 \tan ^2(x)}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 131, normalized size = 1.39 \begin {gather*} -\frac {(-3+2 \cos (2 x))^{3/2} \left (-1+2 \cot ^2(x) \csc (x)\right ) \sin ^2(x) \left (-2 \sinh ^{-1}(2 \sin (x)) \left (4+\csc ^2(x)\right )+\left (-2+164 \csc (x)-3 \csc ^2(x)+16 \csc ^3(x)\right ) \sqrt {1+4 \sin ^2(x)}\right ) \tan (x)}{2 \sqrt {-(3-2 \cos (2 x))^2} \left (5+\cot ^2(x)\right ) (4+4 \cos (2 x)-3 \sin (x)+\sin (3 x)) \sqrt {1+5 \tan ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.36, size = 975, normalized size = 10.37
method | result | size |
default | \(\text {Expression too large to display}\) | \(975\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A]
time = 0.37, size = 97, normalized size = 1.03 \begin {gather*} \frac {2 \, {\left (4 \, \cos \left (x\right )^{2} - 5\right )} \log \left (\sqrt {-\frac {4 \, \cos \left (x\right )^{2} - 5}{\cos \left (x\right )^{2}}} \cos \left (x\right ) - 2 \, \sin \left (x\right )\right ) \sin \left (x\right ) + {\left (164 \, \cos \left (x\right )^{3} - {\left (2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )\right )} \sin \left (x\right ) - 180 \, \cos \left (x\right )\right )} \sqrt {-\frac {4 \, \cos \left (x\right )^{2} - 5}{\cos \left (x\right )^{2}}}}{8 \, {\left (4 \, \cos \left (x\right )^{2} - 5\right )} \sin \left (x\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 \left (- \frac {\sin {\left (x \right )}}{5 \sqrt {5 \tan ^{2}{\left (x \right )} + 1} \tan ^{2}{\left (x \right )} + \sqrt {5 \tan ^{2}{\left (x \right )} + 1}}\right )\, dx - \int \frac {2 \cot ^{2}{\left (x \right )}}{5 \sqrt {5 \tan ^{2}{\left (x \right )} + 1} \tan ^{2}{\left (x \right )} + \sqrt {5 \tan ^{2}{\left (x \right )} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sin \left (x\right )-2\,{\mathrm {cot}\left (x\right )}^2}{{\left (5\,{\mathrm {tan}\left (x\right )}^2+1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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