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 = 0.0893266, antiderivative size = 52, normalized size of antiderivative = 1., 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 4139
Rule 266
Rule 63
Rule 298
Rule 203
Rule 206
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*}
Mathematica [B] time = 0.407236, size = 256, normalized size = 4.92 \[ \frac{\sqrt [4]{a^4 \cos (2 x)-a^4-2 b^4} \left (-\log \left (\frac{a^2 \sin (x)}{\sqrt{a^4 \left (-\sin ^2(x)\right )-b^4}}-\frac{\sqrt{2} a \sqrt{\sin (x)}}{\sqrt [4]{a^4 \left (-\sin ^2(x)\right )-b^4}}+1\right )+\log \left (\frac{a^2 \sin (x)}{\sqrt{a^4 \left (-\sin ^2(x)\right )-b^4}}+\frac{\sqrt{2} a \sqrt{\sin (x)}}{\sqrt [4]{a^4 \left (-\sin ^2(x)\right )-b^4}}+1\right )-2 \tan ^{-1}\left (1-\frac{\sqrt{2} a \sqrt{\sin (x)}}{\sqrt [4]{a^4 \left (-\sin ^2(x)\right )-b^4}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} a \sqrt{\sin (x)}}{\sqrt [4]{a^4 \left (-\sin ^2(x)\right )-b^4}}+1\right )\right )}{2\ 2^{3/4} a \sqrt{\sin (x)} \sqrt [4]{a^4+b^4 \csc ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.112, size = 0, normalized size = 0. \begin{align*} \int{\cot \left ( x \right ){\frac{1}{\sqrt [4]{{a}^{4}+{b}^{4} \left ( \csc \left ( x \right ) \right ) ^{2}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42158, size = 96, normalized size = 1.85 \begin{align*} -\frac{\arctan \left (\frac{{\left (a^{4} + \frac{b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac{1}{4}}}{a}\right )}{a} + \frac{\log \left (a +{\left (a^{4} + \frac{b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac{1}{4}}\right )}{2 \, a} - \frac{\log \left (-a +{\left (a^{4} + \frac{b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac{1}{4}}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (x \right )}}{\sqrt [4]{a^{4} + b^{4} \csc ^{2}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (x\right )}{{\left (b^{4} \csc \left (x\right )^{2} + a^{4}\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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