Optimal. Leaf size=28 \[ \text {Int}\left (\frac {1}{\sec ^{\frac {4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}},x\right ) \]
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Rubi [A] time = 0.06, 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 = {} \[ \int \frac {1}{\sec ^{\frac {4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\sec ^{\frac {4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx &=\int \frac {1}{\sec ^{\frac {4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx\\ \end {align*}
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Mathematica [A] time = 58.91, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sec ^{\frac {4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 2.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {2}{3}}}{b^{3} \sec \left (d x + c\right )^{5} + 3 \, a b^{2} \sec \left (d x + c\right )^{4} + 3 \, a^{2} b \sec \left (d x + c\right )^{3} + a^{3} \sec \left (d x + c\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 1.58, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sec \left (d x +c \right )^{\frac {4}{3}} \left (a +b \sec \left (d x +c \right )\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{4/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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