Optimal. Leaf size=64 \[ \frac {F_1\left (\frac {1}{2};1-n,2;\frac {3}{2};1+\sec (e+f x),\frac {1}{2} (1+\sec (e+f x))\right ) \tan (e+f x)}{2 a f \sqrt {a-a \sec (e+f x)}} \]
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Rubi [A]
time = 0.12, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3913, 3910,
129, 440} \begin {gather*} \frac {\tan (e+f x) F_1\left (\frac {1}{2};1-n,2;\frac {3}{2};\sec (e+f x)+1,\frac {1}{2} (\sec (e+f x)+1)\right )}{2 a f \sqrt {a-a \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 129
Rule 440
Rule 3910
Rule 3913
Rubi steps
\begin {align*} \int \frac {(-\sec (e+f x))^n}{(a-a \sec (e+f x))^{3/2}} \, dx &=\frac {\sqrt {1-\sec (e+f x)} \int \frac {(-\sec (e+f x))^n}{(1-\sec (e+f x))^{3/2}} \, dx}{a \sqrt {a-a \sec (e+f x)}}\\ &=\frac {\tan (e+f x) \text {Subst}\left (\int \frac {(1-x)^{-1+n}}{(2-x)^2 \sqrt {x}} \, dx,x,1+\sec (e+f x)\right )}{a f \sqrt {1+\sec (e+f x)} \sqrt {a-a \sec (e+f x)}}\\ &=\frac {(2 \tan (e+f x)) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{-1+n}}{\left (2-x^2\right )^2} \, dx,x,\sqrt {1+\sec (e+f x)}\right )}{a f \sqrt {1+\sec (e+f x)} \sqrt {a-a \sec (e+f x)}}\\ &=\frac {F_1\left (\frac {1}{2};1-n,2;\frac {3}{2};1+\sec (e+f x),\frac {1}{2} (1+\sec (e+f x))\right ) \tan (e+f x)}{2 a f \sqrt {a-a \sec (e+f x)}}\\ \end {align*}
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Mathematica [F]
time = 74.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-\sec (e+f x))^n}{(a-a \sec (e+f x))^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (-\sec \left (f x +e \right )\right )^{n}}{\left (a -a \sec \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \sec {\left (e + f x \right )}\right )^{n}}{\left (- a \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.02 \begin {gather*} \int \frac {{\left (-\frac {1}{\cos \left (e+f\,x\right )}\right )}^n}{{\left (a-\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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