Optimal. Leaf size=74 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{f \sqrt{a-b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right )}{\sqrt{a} f} \]
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Rubi [A] time = 0.108394, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3670, 446, 86, 63, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{f \sqrt{a-b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right )}{\sqrt{a} f} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 446
Rule 86
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\cot (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (1+x) \sqrt{a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}-\frac{\operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \tan ^2(e+f x)}\right )}{b f}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \tan ^2(e+f x)}\right )}{b f}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right )}{\sqrt{a} f}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{\sqrt{a-b} f}\\ \end{align*}
Mathematica [A] time = 0.079435, size = 72, normalized size = 0.97 \[ \frac{\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{\sqrt{a-b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right )}{\sqrt{a}}}{f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.24, size = 496, normalized size = 6.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (f x + e\right )}{\sqrt{b \tan \left (f x + e\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76104, size = 1083, normalized size = 14.64 \begin{align*} \left [\frac{\sqrt{a - b} a \log \left (\frac{b \tan \left (f x + e\right )^{2} + 2 \, \sqrt{b \tan \left (f x + e\right )^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan \left (f x + e\right )^{2} + 1}\right ) +{\left (a - b\right )} \sqrt{a} \log \left (\frac{b \tan \left (f x + e\right )^{2} - 2 \, \sqrt{b \tan \left (f x + e\right )^{2} + a} \sqrt{a} + 2 \, a}{\tan \left (f x + e\right )^{2}}\right )}{2 \,{\left (a^{2} - a b\right )} f}, \frac{2 \, a \sqrt{-a + b} \arctan \left (-\frac{\sqrt{b \tan \left (f x + e\right )^{2} + a} \sqrt{-a + b}}{a - b}\right ) +{\left (a - b\right )} \sqrt{a} \log \left (\frac{b \tan \left (f x + e\right )^{2} - 2 \, \sqrt{b \tan \left (f x + e\right )^{2} + a} \sqrt{a} + 2 \, a}{\tan \left (f x + e\right )^{2}}\right )}{2 \,{\left (a^{2} - a b\right )} f}, \frac{2 \, \sqrt{-a}{\left (a - b\right )} \arctan \left (\frac{\sqrt{b \tan \left (f x + e\right )^{2} + a} \sqrt{-a}}{a}\right ) + \sqrt{a - b} a \log \left (\frac{b \tan \left (f x + e\right )^{2} + 2 \, \sqrt{b \tan \left (f x + e\right )^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan \left (f x + e\right )^{2} + 1}\right )}{2 \,{\left (a^{2} - a b\right )} f}, \frac{\sqrt{-a}{\left (a - b\right )} \arctan \left (\frac{\sqrt{b \tan \left (f x + e\right )^{2} + a} \sqrt{-a}}{a}\right ) + a \sqrt{-a + b} \arctan \left (-\frac{\sqrt{b \tan \left (f x + e\right )^{2} + a} \sqrt{-a + b}}{a - b}\right )}{{\left (a^{2} - a b\right )} f}\right ] \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 (e + f x \right )}}{\sqrt{a + b \tan ^{2}{\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{\cot \left (f x + e\right )}{\sqrt{b \tan \left (f x + e\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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