Optimal. Leaf size=80 \[ \frac{12 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{8/3} F_1\left (\frac{13}{6};-\frac{3}{2},4;\frac{19}{6};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )}{13 a^3 f} \]
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Rubi [A] time = 0.0935749, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2719, 137, 136} \[ \frac{12 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{8/3} F_1\left (\frac{13}{6};-\frac{3}{2},4;\frac{19}{6};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )}{13 a^3 f} \]
Antiderivative was successfully verified.
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Rule 2719
Rule 137
Rule 136
Rubi steps
\begin{align*} \int \frac{\cot ^4(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx &=\frac{\left (\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{(a-x)^{3/2} (a+x)^{7/6}}{x^4} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{\left (2 \sqrt{2} \sec (e+f x) (a-a \sin (e+f x)) \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{(a+x)^{7/6} \left (\frac{1}{2}-\frac{x}{2 a}\right )^{3/2}}{x^4} \, dx,x,a \sin (e+f x)\right )}{f \sqrt{\frac{a-a \sin (e+f x)}{a}}}\\ &=\frac{12 \sqrt{2} F_1\left (\frac{13}{6};-\frac{3}{2},4;\frac{19}{6};\frac{1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sec (e+f x) \sqrt{1-\sin (e+f x)} (a+a \sin (e+f x))^{8/3}}{13 a^3 f}\\ \end{align*}
Mathematica [F] time = 9.12866, size = 0, normalized size = 0. \[ \int \frac{\cot ^4(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.105, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cot \left ( fx+e \right ) \right ) ^{4}{\frac{1}{\sqrt [3]{a+a\sin \left ( fx+e \right ) }}}}\, dx \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 [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 ^{4}{\left (e + f x \right )}}{\sqrt [3]{a \left (\sin{\left (e + f x \right )} + 1\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 )^{4}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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