Optimal. Leaf size=90 \[ \frac {1}{2} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b} \tanh ^{-1}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )-\frac {1}{2} \sqrt {a+b \cot ^4(x)} \]
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Rubi [A]
time = 0.09, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3751, 1262,
749, 858, 223, 212, 739} \begin {gather*} -\frac {1}{2} \sqrt {a+b \cot ^4(x)}+\frac {1}{2} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b} \tanh ^{-1}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 739
Rule 749
Rule 858
Rule 1262
Rule 3751
Rubi steps
\begin {align*} \int \cot (x) \sqrt {a+b \cot ^4(x)} \, dx &=-\text {Subst}\left (\int \frac {x \sqrt {a+b x^4}}{1+x^2} \, dx,x,\cot (x)\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{1+x} \, dx,x,\cot ^2(x)\right )\right )\\ &=-\frac {1}{2} \sqrt {a+b \cot ^4(x)}-\frac {1}{2} \text {Subst}\left (\int \frac {a-b x}{(1+x) \sqrt {a+b x^2}} \, dx,x,\cot ^2(x)\right )\\ &=-\frac {1}{2} \sqrt {a+b \cot ^4(x)}+\frac {1}{2} b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\cot ^2(x)\right )-\frac {1}{2} (a+b) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x^2}} \, dx,x,\cot ^2(x)\right )\\ &=-\frac {1}{2} \sqrt {a+b \cot ^4(x)}-\frac {1}{2} (-a-b) \text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {a-b \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+\frac {1}{2} b \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )\\ &=\frac {1}{2} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b} \tanh ^{-1}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )-\frac {1}{2} \sqrt {a+b \cot ^4(x)}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 86, normalized size = 0.96 \begin {gather*} \frac {1}{2} \left (\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+\sqrt {a+b} \tanh ^{-1}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )-\sqrt {a+b \cot ^4(x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 139, normalized size = 1.54
method | result | size |
derivativedivides | \(-\frac {\sqrt {b \left (1+\cot ^{2}\left (x \right )\right )^{2}-2 b \left (1+\cot ^{2}\left (x \right )\right )+a +b}}{2}+\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\cot ^{2}\left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\cot ^{2}\left (x \right )\right )^{2}-2 b \left (1+\cot ^{2}\left (x \right )\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\cot ^{2}\left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\cot ^{2}\left (x \right )\right )^{2}-2 b \left (1+\cot ^{2}\left (x \right )\right )+a +b}}{1+\cot ^{2}\left (x \right )}\right )}{2}\) | \(139\) |
default | \(-\frac {\sqrt {b \left (1+\cot ^{2}\left (x \right )\right )^{2}-2 b \left (1+\cot ^{2}\left (x \right )\right )+a +b}}{2}+\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\cot ^{2}\left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\cot ^{2}\left (x \right )\right )^{2}-2 b \left (1+\cot ^{2}\left (x \right )\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\cot ^{2}\left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\cot ^{2}\left (x \right )\right )^{2}-2 b \left (1+\cot ^{2}\left (x \right )\right )+a +b}}{1+\cot ^{2}\left (x \right )}\right )}{2}\) | \(139\) |
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 \begin {gather*} \text {Failed to integrate} \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. 252 vs.
\(2 (72) = 144\).
time = 3.96, size = 1063, normalized size = 11.81 \begin {gather*} \left [\frac {1}{4} \, \sqrt {a + b} \log \left (\frac {1}{2} \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + \frac {1}{2} \, a^{2} + \frac {1}{2} \, b^{2} + \frac {1}{2} \, {\left ({\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a - b\right )} \sqrt {a + b} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}} - {\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )\right ) + \frac {1}{4} \, \sqrt {b} \log \left (-\frac {{\left (a + 2 \, b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (\cos \left (2 \, x\right )^{2} - 1\right )} \sqrt {b} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}} - 2 \, {\left (a - 2 \, b\right )} \cos \left (2 \, x\right ) + a + 2 \, b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right ) - \frac {1}{2} \, \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}, \frac {1}{2} \, \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{b \cos \left (2 \, x\right ) + b}\right ) + \frac {1}{4} \, \sqrt {a + b} \log \left (\frac {1}{2} \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + \frac {1}{2} \, a^{2} + \frac {1}{2} \, b^{2} + \frac {1}{2} \, {\left ({\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a - b\right )} \sqrt {a + b} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}} - {\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )\right ) - \frac {1}{2} \, \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}, -\frac {1}{2} \, \sqrt {-a - b} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a - b\right )} \sqrt {-a - b} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )}\right ) + \frac {1}{4} \, \sqrt {b} \log \left (-\frac {{\left (a + 2 \, b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (\cos \left (2 \, x\right )^{2} - 1\right )} \sqrt {b} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}} - 2 \, {\left (a - 2 \, b\right )} \cos \left (2 \, x\right ) + a + 2 \, b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right ) - \frac {1}{2} \, \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}, -\frac {1}{2} \, \sqrt {-a - b} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a - b\right )} \sqrt {-a - b} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )}\right ) + \frac {1}{2} \, \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{b \cos \left (2 \, x\right ) + b}\right ) - \frac {1}{2} \, \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}\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 \sqrt {a + b \cot ^{4}{\left (x \right )}} \cot {\left (x \right )}\, 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. 204 vs.
\(2 (72) = 144\).
time = 0.47, size = 204, normalized size = 2.27 \begin {gather*} -\frac {b \arctan \left (-\frac {\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {1}{2} \, \sqrt {a + b} \log \left ({\left | -{\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )} {\left (a + b\right )} + \sqrt {a + b} b \right |}\right ) - \frac {{\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )} b - \sqrt {a + b} b}{{\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )}^{2} - b} \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 \mathrm {cot}\left (x\right )\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^4+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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