Optimal. Leaf size=26 \[ \frac{(a+b \tan (c+d x))^{n+1}}{b d (n+1)} \]
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Rubi [A] time = 0.0435077, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3506, 32} \[ \frac{(a+b \tan (c+d x))^{n+1}}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 32
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+b \tan (c+d x))^n \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^n \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{(a+b \tan (c+d x))^{1+n}}{b d (1+n)}\\ \end{align*}
Mathematica [A] time = 0.178291, size = 26, normalized size = 1. \[ \frac{(a+b \tan (c+d x))^{n+1}}{b d (n+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 27, normalized size = 1. \begin{align*}{\frac{ \left ( a+b\tan \left ( dx+c \right ) \right ) ^{1+n}}{bd \left ( 1+n \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.55939, size = 155, normalized size = 5.96 \begin{align*} \frac{{\left (a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )\right )} \left (\frac{a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{n}}{{\left (b d n + b d\right )} \cos \left (d x + c\right )} \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 \left (a + b \tan{\left (c + d x \right )}\right )^{n} \sec ^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.96348, size = 35, normalized size = 1.35 \begin{align*} \frac{{\left (b \tan \left (d x + c\right ) + a\right )}^{n + 1}}{b d{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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