Optimal. Leaf size=89 \[ \frac {(d+c \cos (e+f x))^{2/3} (a+b \sec (e+f x))^{2/3} \text {Int}\left (\frac {(b+a \cos (e+f x))^{2/3}}{(d+c \cos (e+f x))^{2/3}},x\right )}{(b+a \cos (e+f x))^{2/3} (c+d \sec (e+f x))^{2/3}} \]
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Rubi [A]
time = 0.14, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{2/3}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{2/3}} \, dx &=\frac {\left ((d+c \cos (e+f x))^{2/3} (a+b \sec (e+f x))^{2/3}\right ) \int \frac {(b+a \cos (e+f x))^{2/3}}{(d+c \cos (e+f x))^{2/3}} \, dx}{(b+a \cos (e+f x))^{2/3} (c+d \sec (e+f x))^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 2.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{2/3}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \sec \left (f x +e \right )\right )^{\frac {2}{3}}}{\left (c +d \sec \left (f x +e \right )\right )^{\frac {2}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \sec {\left (e + f x \right )}\right )^{\frac {2}{3}}}{\left (c + d \sec {\left (e + f x \right )}\right )^{\frac {2}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{2/3}}{{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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