Optimal. Leaf size=33 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \]
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Rubi [A] time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3670, 444, 63, 208} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 444
Rule 3670
Rubi steps
\begin {align*} \int \frac {\cot (x)}{\sqrt {a+b \cot ^2(x)}} \, dx &=-\operatorname {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\cot ^2(x)\right )\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cot ^2(x)}\right )}{b}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 127, normalized size = 3.85 \[ \left [\frac {\log \left (-\sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )} - {\left (a - b\right )} \cos \left (2 \, x\right ) + a\right )}{2 \, \sqrt {a - b}}, \frac {\sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{a - b}\right )}{a - b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.56, size = 70, normalized size = 2.12 \[ \frac {\log \left ({\left | 2 \, {\left (\sqrt {a - b} \cos \relax (x)^{2} - \sqrt {a \cos \relax (x)^{4} - b \cos \relax (x)^{4} - 2 \, a \cos \relax (x)^{2} + b \cos \relax (x)^{2} + a}\right )} \sqrt {a - b} - 2 \, a + b \right |}\right )}{2 \, \sqrt {a - b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 29, normalized size = 0.88 \[ -\frac {\arctan \left (\frac {\sqrt {a +b \left (\cot ^{2}\relax (x )\right )}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 27, normalized size = 0.82 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {cot}\relax (x)}^2+a}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \]
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 \[ \int \frac {\cot {\relax (x )}}{\sqrt {a + b \cot ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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