Optimal. Leaf size=66 \[ -\frac{\csc ^2(e+f x) \sqrt{a \cos ^2(e+f x)}}{2 a^2 f}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right )}{2 a^{3/2} f} \]
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Rubi [A] time = 0.126603, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3176, 3205, 16, 51, 63, 206} \[ -\frac{\csc ^2(e+f x) \sqrt{a \cos ^2(e+f x)}}{2 a^2 f}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right )}{2 a^{3/2} f} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3205
Rule 16
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\cot ^3(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac{\cot ^3(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x}{(1-x)^2 (a x)^{3/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x)^2 \sqrt{a x}} \, dx,x,\cos ^2(e+f x)\right )}{2 a f}\\ &=-\frac{\sqrt{a \cos ^2(e+f x)} \csc ^2(e+f x)}{2 a^2 f}-\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{a x}} \, dx,x,\cos ^2(e+f x)\right )}{4 a f}\\ &=-\frac{\sqrt{a \cos ^2(e+f x)} \csc ^2(e+f x)}{2 a^2 f}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-\frac{x^2}{a}} \, dx,x,\sqrt{a \cos ^2(e+f x)}\right )}{2 a^2 f}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right )}{2 a^{3/2} f}-\frac{\sqrt{a \cos ^2(e+f x)} \csc ^2(e+f x)}{2 a^2 f}\\ \end{align*}
Mathematica [A] time = 0.172236, size = 82, normalized size = 1.24 \[ -\frac{\cos ^3(e+f x) \left (\csc ^2\left (\frac{1}{2} (e+f x)\right )-\sec ^2\left (\frac{1}{2} (e+f x)\right )-4 \log \left (\sin \left (\frac{1}{2} (e+f x)\right )\right )+4 \log \left (\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )}{8 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.306, size = 69, normalized size = 1.1 \begin{align*} -{\frac{1}{2\,{a}^{2}f \left ( \sin \left ( fx+e \right ) \right ) ^{2}}\sqrt{a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}-{\frac{1}{2\,f}\ln \left ({\frac{1}{\sin \left ( fx+e \right ) } \left ( 2\,a+2\,\sqrt{a}\sqrt{a \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \right ) } \right ){a}^{-{\frac{3}{2}}}} \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 [A] time = 1.6094, size = 212, normalized size = 3.21 \begin{align*} -\frac{\sqrt{a \cos \left (f x + e\right )^{2}}{\left ({\left (\cos \left (f x + e\right )^{2} - 1\right )} \log \left (-\frac{\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1}\right ) - 2 \, \cos \left (f x + e\right )\right )}}{4 \,{\left (a^{2} f \cos \left (f x + e\right )^{3} - a^{2} f \cos \left (f x + e\right )\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 ^{3}{\left (e + f x \right )}}{\left (- a \left (\sin{\left (e + f x \right )} - 1\right ) \left (\sin{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.35796, size = 174, normalized size = 2.64 \begin{align*} -\frac{\frac{\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2}}{a^{\frac{3}{2}} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right )} + \frac{2 \, \log \left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2}\right )}{a^{\frac{3}{2}} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right )} - \frac{2 \, \sqrt{a} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + \sqrt{a}}{a^{2} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2}}}{8 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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