Optimal. Leaf size=88 \[ \frac{\sqrt [3]{a+b \sec (e+f x)} \sqrt [3]{c \cos (e+f x)+d} \text{Unintegrable}\left (\frac{\sqrt [3]{a \cos (e+f x)+b}}{\sqrt [3]{c \cos (e+f x)+d}},x\right )}{\sqrt [3]{a \cos (e+f x)+b} \sqrt [3]{c+d \sec (e+f x)}} \]
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Rubi [A] time = 0.190879, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt [3]{a+b \sec (e+f x)}}{\sqrt [3]{c+d \sec (e+f x)}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b \sec (e+f x)}}{\sqrt [3]{c+d \sec (e+f x)}} \, dx &=\frac{\left (\sqrt [3]{d+c \cos (e+f x)} \sqrt [3]{a+b \sec (e+f x)}\right ) \int \frac{\sqrt [3]{b+a \cos (e+f x)}}{\sqrt [3]{d+c \cos (e+f x)}} \, dx}{\sqrt [3]{b+a \cos (e+f x)} \sqrt [3]{c+d \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 2.15962, size = 0, normalized size = 0. \[ \int \frac{\sqrt [3]{a+b \sec (e+f x)}}{\sqrt [3]{c+d \sec (e+f x)}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.257, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [3]{a+b\sec \left ( fx+e \right ) }{\frac{1}{\sqrt [3]{c+d\sec \left ( fx+e \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac{1}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac{1}{3}}}\,{d x} \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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{a + b \sec{\left (e + f x \right )}}}{\sqrt [3]{c + d \sec{\left (e + f x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac{1}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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