Optimal. Leaf size=62 \[ \frac {2 \, _2F_1\left (1,\frac {2-n}{4};\frac {6-n}{4};-\tan ^2(e+f x)\right ) \tan (e+f x)}{f (2-n) \sqrt {b \tan ^n(e+f x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3740, 3557,
371} \begin {gather*} \frac {2 \tan (e+f x) \, _2F_1\left (1,\frac {2-n}{4};\frac {6-n}{4};-\tan ^2(e+f x)\right )}{f (2-n) \sqrt {b \tan ^n(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 3557
Rule 3740
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b \tan ^n(e+f x)}} \, dx &=\frac {\tan ^{\frac {n}{2}}(e+f x) \int \tan ^{-\frac {n}{2}}(e+f x) \, dx}{\sqrt {b \tan ^n(e+f x)}}\\ &=\frac {\tan ^{\frac {n}{2}}(e+f x) \text {Subst}\left (\int \frac {x^{-n/2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f \sqrt {b \tan ^n(e+f x)}}\\ &=\frac {2 \, _2F_1\left (1,\frac {2-n}{4};\frac {6-n}{4};-\tan ^2(e+f x)\right ) \tan (e+f x)}{f (2-n) \sqrt {b \tan ^n(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 60, normalized size = 0.97 \begin {gather*} -\frac {2 \, _2F_1\left (1,\frac {2-n}{4};\frac {6-n}{4};-\tan ^2(e+f x)\right ) \tan (e+f x)}{f (-2+n) \sqrt {b \tan ^n(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {b \left (\tan ^{n}\left (f x +e \right )\right )}}\, dx\]
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(-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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {b \tan ^{n}{\left (e + f x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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