Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\tan ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^n,x\right ) \]
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Rubi [A] time = 0.0459208, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (a+b \sec (c+d x))^n \tan ^{\frac{3}{2}}(c+d x) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (a+b \sec (c+d x))^n \tan ^{\frac{3}{2}}(c+d x) \, dx &=\int (a+b \sec (c+d x))^n \tan ^{\frac{3}{2}}(c+d x) \, dx\\ \end{align*}
Mathematica [A] time = 4.16643, size = 0, normalized size = 0. \[ \int (a+b \sec (c+d x))^n \tan ^{\frac{3}{2}}(c+d x) \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.303, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\sec \left ( dx+c \right ) \right ) ^{n} \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{\frac{3}{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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