Optimal. Leaf size=144 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right )}{8 a^{3/2} f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a \sin (e+f x)+a}}{3 a^2 f}-\frac{\cot (e+f x)}{8 a f \sqrt{a \sin (e+f x)+a}}+\frac{11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.553107, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2717, 2772, 2773, 206, 3044, 2980} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right )}{8 a^{3/2} f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a \sin (e+f x)+a}}{3 a^2 f}-\frac{\cot (e+f x)}{8 a f \sqrt{a \sin (e+f x)+a}}+\frac{11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2717
Rule 2772
Rule 2773
Rule 206
Rule 3044
Rule 2980
Rubi steps
\begin{align*} \int \frac{\cot ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx &=\frac{\int \csc ^4(e+f x) \sqrt{a+a \sin (e+f x)} \left (1+\sin ^2(e+f x)\right ) \, dx}{a^2}-\frac{2 \int \csc ^3(e+f x) \sqrt{a+a \sin (e+f x)} \, dx}{a^2}\\ &=\frac{\cot (e+f x) \csc (e+f x)}{a f \sqrt{a+a \sin (e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{3 a^2 f}+\frac{\int \csc ^3(e+f x) \sqrt{a+a \sin (e+f x)} \left (\frac{a}{2}+\frac{9}{2} a \sin (e+f x)\right ) \, dx}{3 a^3}-\frac{3 \int \csc ^2(e+f x) \sqrt{a+a \sin (e+f x)} \, dx}{2 a^2}\\ &=\frac{3 \cot (e+f x)}{2 a f \sqrt{a+a \sin (e+f x)}}+\frac{11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt{a+a \sin (e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{3 a^2 f}-\frac{3 \int \csc (e+f x) \sqrt{a+a \sin (e+f x)} \, dx}{4 a^2}+\frac{13 \int \csc ^2(e+f x) \sqrt{a+a \sin (e+f x)} \, dx}{8 a^2}\\ &=-\frac{\cot (e+f x)}{8 a f \sqrt{a+a \sin (e+f x)}}+\frac{11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt{a+a \sin (e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{3 a^2 f}+\frac{13 \int \csc (e+f x) \sqrt{a+a \sin (e+f x)} \, dx}{16 a^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{2 a f}\\ &=\frac{3 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{2 a^{3/2} f}-\frac{\cot (e+f x)}{8 a f \sqrt{a+a \sin (e+f x)}}+\frac{11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt{a+a \sin (e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{3 a^2 f}-\frac{13 \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{8 a f}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{8 a^{3/2} f}-\frac{\cot (e+f x)}{8 a f \sqrt{a+a \sin (e+f x)}}+\frac{11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt{a+a \sin (e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{3 a^2 f}\\ \end{align*}
Mathematica [B] time = 0.760534, size = 294, normalized size = 2.04 \[ \frac{\csc ^9\left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3 \left (132 \sin \left (\frac{1}{2} (e+f x)\right )+62 \sin \left (\frac{3}{2} (e+f x)\right )-6 \sin \left (\frac{5}{2} (e+f x)\right )-132 \cos \left (\frac{1}{2} (e+f x)\right )+62 \cos \left (\frac{3}{2} (e+f x)\right )+6 \cos \left (\frac{5}{2} (e+f x)\right )-9 \sin (e+f x) \log \left (-\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )+1\right )+9 \sin (e+f x) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )-\cos \left (\frac{1}{2} (e+f x)\right )+1\right )+3 \sin (3 (e+f x)) \log \left (-\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )+1\right )-3 \sin (3 (e+f x)) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )-\cos \left (\frac{1}{2} (e+f x)\right )+1\right )\right )}{24 f (a (\sin (e+f x)+1))^{3/2} \left (\csc ^2\left (\frac{1}{4} (e+f x)\right )-\sec ^2\left (\frac{1}{4} (e+f x)\right )\right )^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.621, size = 144, normalized size = 1. \begin{align*}{\frac{1+\sin \left ( fx+e \right ) }{24\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}\cos \left ( fx+e \right ) f}\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( 3\,\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }{a}^{7/2}-8\, \left ( -a \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{3/2}{a}^{5/2}-3\, \left ( -a \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{5/2}{a}^{3/2}-3\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }}{\sqrt{a}}} \right ){a}^{4} \left ( \sin \left ( fx+e \right ) \right ) ^{3} \right ){a}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [B] time = 1.57176, size = 1003, normalized size = 6.97 \begin{align*} \frac{3 \,{\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} -{\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right ) + 1\right )} \sqrt{a} \log \left (\frac{a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 4 \,{\left (\cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{a} - 9 \, a \cos \left (f x + e\right ) +{\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) + 4 \,{\left (3 \, \cos \left (f x + e\right )^{3} + 17 \, \cos \left (f x + e\right )^{2} -{\left (3 \, \cos \left (f x + e\right )^{2} - 14 \, \cos \left (f x + e\right ) - 25\right )} \sin \left (f x + e\right ) - 11 \, \cos \left (f x + e\right ) - 25\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{96 \,{\left (a^{2} f \cos \left (f x + e\right )^{4} - 2 \, a^{2} f \cos \left (f x + e\right )^{2} + a^{2} f -{\left (a^{2} f \cos \left (f x + e\right )^{3} + a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - a^{2} f\right )} \sin \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 ^{4}{\left (e + f x \right )}}{\left (a \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 = 3.13146, size = 828, normalized size = 5.75 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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