Optimal. Leaf size=52 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a^4+b^4 \csc ^2(x)}}{a}\right )}{a}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a^4+b^4 \csc ^2(x)}}{a}\right )}{a} \]
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Rubi [A] time = 0.09, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {4139, 266, 63, 298, 203, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a^4+b^4 \csc ^2(x)}}{a}\right )}{a}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a^4+b^4 \csc ^2(x)}}{a}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 206
Rule 266
Rule 298
Rule 4139
Rubi steps
\begin {align*} \int \frac {\cot (x)}{\sqrt [4]{a^4+b^4 \csc ^2(x)}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{a^4+b^4 x^2}} \, dx,x,\csc (x)\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{a^4+b^4 x}} \, dx,x,\csc ^2(x)\right )\right )\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {x^2}{-\frac {a^4}{b^4}+\frac {x^4}{b^4}} \, dx,x,\sqrt [4]{a^4+b^4 \csc ^2(x)}\right )}{b^4}\\ &=\operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,\sqrt [4]{a^4+b^4 \csc ^2(x)}\right )-\operatorname {Subst}\left (\int \frac {1}{a^2+x^2} \, dx,x,\sqrt [4]{a^4+b^4 \csc ^2(x)}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a^4+b^4 \csc ^2(x)}}{a}\right )}{a}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a^4+b^4 \csc ^2(x)}}{a}\right )}{a}\\ \end {align*}
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Mathematica [B] time = 0.30, size = 256, normalized size = 4.92 \[ \frac {\sqrt [4]{a^4 \cos (2 x)-a^4-2 b^4} \left (-2 \tan ^{-1}\left (1-\frac {\sqrt {2} a \sqrt {\sin (x)}}{\sqrt [4]{a^4 \left (-\sin ^2(x)\right )-b^4}}\right )+2 \tan ^{-1}\left (\frac {\sqrt {2} a \sqrt {\sin (x)}}{\sqrt [4]{a^4 \left (-\sin ^2(x)\right )-b^4}}+1\right )-\log \left (-\frac {\sqrt {2} a \sqrt {\sin (x)}}{\sqrt [4]{a^4 \left (-\sin ^2(x)\right )-b^4}}+\frac {a^2 \sin (x)}{\sqrt {a^4 \left (-\sin ^2(x)\right )-b^4}}+1\right )+\log \left (\frac {\sqrt {2} a \sqrt {\sin (x)}}{\sqrt [4]{a^4 \left (-\sin ^2(x)\right )-b^4}}+\frac {a^2 \sin (x)}{\sqrt {a^4 \left (-\sin ^2(x)\right )-b^4}}+1\right )\right )}{2\ 2^{3/4} a \sqrt {\sin (x)} \sqrt [4]{a^4+b^4 \csc ^2(x)}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot (x)}{\sqrt [4]{a^4+b^4 \csc ^2(x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.35, size = 73, normalized size = 1.40 \[ -\frac {\arctan \left (\frac {{\left (a^{4} + \frac {b^{4}}{\sin \relax (x)^{2}}\right )}^{\frac {1}{4}}}{a}\right )}{a} + \frac {\log \left ({\left | a + {\left (a^{4} + \frac {b^{4}}{\sin \relax (x)^{2}}\right )}^{\frac {1}{4}} \right |}\right )}{2 \, a} - \frac {\log \left ({\left | -a + {\left (a^{4} + \frac {b^{4}}{\sin \relax (x)^{2}}\right )}^{\frac {1}{4}} \right |}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\cot \relax (x )}{\left (a^{4}+b^{4} \left (\csc ^{2}\relax (x )\right )\right )^{\frac {1}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 71, normalized size = 1.37 \[ -\frac {\arctan \left (\frac {{\left (a^{4} + \frac {b^{4}}{\sin \relax (x)^{2}}\right )}^{\frac {1}{4}}}{a}\right )}{a} + \frac {\log \left (a + {\left (a^{4} + \frac {b^{4}}{\sin \relax (x)^{2}}\right )}^{\frac {1}{4}}\right )}{2 \, a} - \frac {\log \left (-a + {\left (a^{4} + \frac {b^{4}}{\sin \relax (x)^{2}}\right )}^{\frac {1}{4}}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 46, normalized size = 0.88 \[ -\frac {\mathrm {atan}\left (\frac {{\left (\frac {b^4}{{\sin \relax (x)}^2}+a^4\right )}^{1/4}}{a}\right )-\mathrm {atanh}\left (\frac {{\left (\frac {b^4}{{\sin \relax (x)}^2}+a^4\right )}^{1/4}}{a}\right )}{a} \]
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 \[ \int \frac {\cot {\relax (x )}}{\sqrt [4]{a^{4} + b^{4} \csc ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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