Optimal. Leaf size=192 \[ \frac{\log \left (\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} \sqrt{d} f}-\frac{\log \left (\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} \sqrt{d} f}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} f}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right )}{\sqrt{2} \sqrt{d} f} \]
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Rubi [A] time = 0.111457, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{\log \left (\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} \sqrt{d} f}-\frac{\log \left (\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} \sqrt{d} f}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} f}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right )}{\sqrt{2} \sqrt{d} f} \]
Antiderivative was successfully verified.
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Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d \cot (e+f x)}} \, dx &=-\frac{d \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (d^2+x^2\right )} \, dx,x,d \cot (e+f x)\right )}{f}\\ &=-\frac{(2 d) \operatorname{Subst}\left (\int \frac{1}{d^2+x^4} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{f}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{d-x^2}{d^2+x^4} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{f}-\frac{\operatorname{Subst}\left (\int \frac{d+x^2}{d^2+x^4} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{f}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{d-\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 f}-\frac{\operatorname{Subst}\left (\int \frac{1}{d+\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 f}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}+2 x}{-d-\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} \sqrt{d} f}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}-2 x}{-d+\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} \sqrt{d} f}\\ &=\frac{\log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} \sqrt{d} f}-\frac{\log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} \sqrt{d} f}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} f}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} f}\\ &=\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} f}-\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} f}+\frac{\log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} \sqrt{d} f}-\frac{\log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} \sqrt{d} f}\\ \end{align*}
Mathematica [A] time = 0.0146639, size = 131, normalized size = 0.68 \[ \frac{\sqrt{\cot (e+f x)} \left (\log \left (\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right )-\log \left (\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right )+2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (e+f x)}\right )-2 \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (e+f x)}+1\right )\right )}{2 \sqrt{2} f \sqrt{d \cot (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 166, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{2}}{4\,fd}\sqrt [4]{{d}^{2}}\ln \left ({ \left ( d\cot \left ( fx+e \right ) +\sqrt [4]{{d}^{2}}\sqrt{d\cot \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) \left ( d\cot \left ( fx+e \right ) -\sqrt [4]{{d}^{2}}\sqrt{d\cot \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{2\,fd}\sqrt [4]{{d}^{2}}\arctan \left ({\sqrt{2}\sqrt{d\cot \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }+{\frac{\sqrt{2}}{2\,fd}\sqrt [4]{{d}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{d\cot \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{1}{\sqrt{d \cot{\left (e + f 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{1}{\sqrt{d \cot \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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