Optimal. Leaf size=71 \[ \frac{2 \tan ^{1-p}(c+d x) \, _2F_1\left (1,\frac{1}{4} (2-3 p);\frac{3 (2-p)}{4};-\tan ^2(c+d x)\right )}{b d (2-3 p) \sqrt{b \tan ^p(c+d x)}} \]
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Rubi [A] time = 0.0481324, antiderivative size = 71, normalized size of antiderivative = 1., 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} \[ \frac{2 \tan ^{1-p}(c+d x) \, _2F_1\left (1,\frac{1}{4} (2-3 p);\frac{3 (2-p)}{4};-\tan ^2(c+d x)\right )}{b d (2-3 p) \sqrt{b \tan ^p(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{\left (b \tan ^p(c+d x)\right )^{3/2}} \, dx &=\frac{\tan ^{\frac{p}{2}}(c+d x) \int \tan ^{-\frac{3 p}{2}}(c+d x) \, dx}{b \sqrt{b \tan ^p(c+d x)}}\\ &=\frac{\tan ^{\frac{p}{2}}(c+d x) \operatorname{Subst}\left (\int \frac{x^{-3 p/2}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b d \sqrt{b \tan ^p(c+d x)}}\\ &=\frac{2 \, _2F_1\left (1,\frac{1}{4} (2-3 p);\frac{3 (2-p)}{4};-\tan ^2(c+d x)\right ) \tan ^{1-p}(c+d x)}{b d (2-3 p) \sqrt{b \tan ^p(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0678254, size = 60, normalized size = 0.85 \[ -\frac{2 \tan (c+d x) \, _2F_1\left (1,\frac{1}{4} (2-3 p);-\frac{3}{4} (p-2);-\tan ^2(c+d x)\right )}{d (3 p-2) \left (b \tan ^p(c+d x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.129, size = 0, normalized size = 0. \begin{align*} \int \left ( b \left ( \tan \left ( dx+c \right ) \right ) ^{p} \right ) ^{-{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \tan \left (d x + c\right )^{p}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \tan ^{p}{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \tan \left (d x + c\right )^{p}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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