Optimal. Leaf size=71 \[ \frac {2 \tan ^{1-n}(e+f x) \, _2F_1\left (1,\frac {1}{4} (2-3 n);\frac {3 (2-n)}{4};-\tan ^2(e+f x)\right )}{b f (2-3 n) \sqrt {b \tan ^n(e+f x)}} \]
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Rubi [A] time = 0.05, antiderivative size = 71, 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 = {3659, 3476, 364} \[ \frac {2 \tan ^{1-n}(e+f x) \, _2F_1\left (1,\frac {1}{4} (2-3 n);\frac {3 (2-n)}{4};-\tan ^2(e+f x)\right )}{b f (2-3 n) \sqrt {b \tan ^n(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 3476
Rule 3659
Rubi steps
\begin {align*} \int \frac {1}{\left (b \tan ^n(e+f x)\right )^{3/2}} \, dx &=\frac {\tan ^{\frac {n}{2}}(e+f x) \int \tan ^{-\frac {3 n}{2}}(e+f x) \, dx}{b \sqrt {b \tan ^n(e+f x)}}\\ &=\frac {\tan ^{\frac {n}{2}}(e+f x) \operatorname {Subst}\left (\int \frac {x^{-3 n/2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{b f \sqrt {b \tan ^n(e+f x)}}\\ &=\frac {2 \, _2F_1\left (1,\frac {1}{4} (2-3 n);\frac {3 (2-n)}{4};-\tan ^2(e+f x)\right ) \tan ^{1-n}(e+f x)}{b f (2-3 n) \sqrt {b \tan ^n(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 60, normalized size = 0.85 \[ -\frac {2 \tan (e+f x) \, _2F_1\left (1,\frac {1}{4} (2-3 n);-\frac {3}{4} (n-2);-\tan ^2(e+f x)\right )}{f (3 n-2) \left (b \tan ^n(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {1}{\left (b \tan \left (f x + e\right )^{n}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.23, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \left (\tan ^{n}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {1}{\left (b \tan \left (f x + e\right )^{n}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^n\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (b \tan ^{n}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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