Optimal. Leaf size=80 \[ \frac {6 \sqrt {2} F_1\left (\frac {7}{6};-\frac {1}{2},2;\frac {13}{6};\frac {1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sec (e+f x) \sqrt {1-\sin (e+f x)} (a+a \sin (e+f x))^{5/3}}{7 a^2 f} \]
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Rubi [A]
time = 0.07, 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 = {2798, 142, 141}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 141
Rule 142
Rule 2798
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 ) \text {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 ) \text {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 = 7.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^2(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.12, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {\cot ^{2}{\left (e + f x \right )}}{\sqrt [3]{a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {cot}\left (e+f\,x\right )}^2}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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