Optimal. Leaf size=70 \[ \frac {\tanh ^{-1}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]
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Rubi [A] time = 0.16, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3670, 1252, 961, 725, 206, 266, 63, 208} \[ \frac {\tanh ^{-1}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 266
Rule 725
Rule 961
Rule 1252
Rule 3670
Rubi steps
\begin {align*} \int \frac {\cot (x)}{\sqrt {a+b \tan ^4(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \left (1+x^2\right ) \sqrt {a+b x^4}} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (1+x) \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{(-1-x) \sqrt {a+b x^2}}+\frac {1}{x \sqrt {a+b x^2}}\right ) \, dx,x,\tan ^2(x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(-1-x) \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan ^4(x)\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {-a+b \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan ^4(x)}\right )}{2 b}\\ &=\frac {\tanh ^{-1}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )}{2 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 70, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 475, normalized size = 6.79 \[ \left [\frac {\sqrt {a + b} a \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \relax (x)^{4} - 2 \, a b \tan \relax (x)^{2} - 2 \, \sqrt {b \tan \relax (x)^{4} + a} {\left (b \tan \relax (x)^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \relax (x)^{4} + 2 \, \tan \relax (x)^{2} + 1}\right ) + {\left (a + b\right )} \sqrt {a} \log \left (-\frac {b \tan \relax (x)^{4} - 2 \, \sqrt {b \tan \relax (x)^{4} + a} \sqrt {a} + 2 \, a}{\tan \relax (x)^{4}}\right )}{4 \, {\left (a^{2} + a b\right )}}, \frac {2 \, \sqrt {-a} {\left (a + b\right )} \arctan \left (\frac {\sqrt {b \tan \relax (x)^{4} + a} \sqrt {-a}}{a}\right ) + \sqrt {a + b} a \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \relax (x)^{4} - 2 \, a b \tan \relax (x)^{2} - 2 \, \sqrt {b \tan \relax (x)^{4} + a} {\left (b \tan \relax (x)^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \relax (x)^{4} + 2 \, \tan \relax (x)^{2} + 1}\right )}{4 \, {\left (a^{2} + a b\right )}}, \frac {2 \, a \sqrt {-a - b} \arctan \left (\frac {\sqrt {b \tan \relax (x)^{4} + a} {\left (b \tan \relax (x)^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \relax (x)^{4} + a^{2} + a b}\right ) + {\left (a + b\right )} \sqrt {a} \log \left (-\frac {b \tan \relax (x)^{4} - 2 \, \sqrt {b \tan \relax (x)^{4} + a} \sqrt {a} + 2 \, a}{\tan \relax (x)^{4}}\right )}{4 \, {\left (a^{2} + a b\right )}}, \frac {a \sqrt {-a - b} \arctan \left (\frac {\sqrt {b \tan \relax (x)^{4} + a} {\left (b \tan \relax (x)^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \relax (x)^{4} + a^{2} + a b}\right ) + \sqrt {-a} {\left (a + b\right )} \arctan \left (\frac {\sqrt {b \tan \relax (x)^{4} + a} \sqrt {-a}}{a}\right )}{2 \, {\left (a^{2} + a b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.52, size = 0, normalized size = 0.00 \[ \int \frac {\cot \relax (x )}{\sqrt {a +b \left (\tan ^{4}\relax (x )\right )}}\, dx \]
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 \[ \int \frac {\cot \relax (x)}{\sqrt {b \tan \relax (x)^{4} + a}}\,{d x} \]
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 \[ \int \frac {\mathrm {cot}\relax (x)}{\sqrt {b\,{\mathrm {tan}\relax (x)}^4+a}} \,d x \]
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 \[ \int \frac {\cot {\relax (x )}}{\sqrt {a + b \tan ^{4}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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