Optimal. Leaf size=28 \[ \text {Int}\left (\frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}},x\right ) \]
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Rubi [A] time = 0.07, 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 = {} \[ \int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx &=\int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx\\ \end {align*}
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Mathematica [A] time = 2.74, size = 0, normalized size = 0.00 \[ \int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (f x + e\right ) + a} \left (d \sec \left (f x + e\right )\right )^{n}}{b^{2} \sec \left (f x + e\right )^{2} + 2 \, a b \sec \left (f x + e\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {\left (d \sec \left (f x + e\right )\right )^{n}}{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sec \left (f x +e \right )\right )^{n}}{\left (a +b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\left (d \sec \left (f x + e\right )\right )^{n}}{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^n}{{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (d \sec {\left (e + f x \right )}\right )^{n}}{\left (a + b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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