Optimal. Leaf size=80 \[ \frac {6 \sqrt {2} \sqrt {1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{5/3} F_1\left (\frac {7}{6};-\frac {1}{2},2;\frac {13}{6};\frac {1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )}{7 a^2 f} \]
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Rubi [A] time = 0.09, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2719, 137, 136} \[ \frac {6 \sqrt {2} \sqrt {1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{5/3} F_1\left (\frac {7}{6};-\frac {1}{2},2;\frac {13}{6};\frac {1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )}{7 a^2 f} \]
Antiderivative was successfully verified.
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Rule 136
Rule 137
Rule 2719
Rubi steps
\begin {align*} \int \frac {\cot ^2(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 {\sqrt {a-x} \sqrt [6]{a+x}}{x^2} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac {\left (\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 {\sqrt [6]{a+x} \sqrt {\frac {1}{2}-\frac {x}{2 a}}}{x^2} \, dx,x,a \sin (e+f x)\right )}{a f \sqrt {\frac {a-a \sin (e+f x)}{a}}}\\ &=\frac {6 \sqrt {2} F_1\left (\frac {7}{6};-\frac {1}{2},2;\frac {13}{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))^{5/3}}{7 a^2 f}\\ \end {align*}
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Mathematica [F] time = 12.57, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^2(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{2}\left (f x +e \right )}{\left (a +a \sin \left (f x +e \right )\right )^{\frac {1}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cot}\left (e+f\,x\right )}^2}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{2}{\left (e + f x \right )}}{\sqrt [3]{a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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