Optimal. Leaf size=24 \[ \frac{(d \tan (a+b x))^{n+1}}{b d (n+1)} \]
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Rubi [A] time = 0.0385258, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2607, 32} \[ \frac{(d \tan (a+b x))^{n+1}}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 32
Rubi steps
\begin{align*} \int \sec ^2(a+b x) (d \tan (a+b x))^n \, dx &=\frac{\operatorname{Subst}\left (\int (d x)^n \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{(d \tan (a+b x))^{1+n}}{b d (1+n)}\\ \end{align*}
Mathematica [A] time = 0.020364, size = 25, normalized size = 1.04 \[ \frac{\tan (a+b x) (d \tan (a+b x))^n}{b (n+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 25, normalized size = 1. \begin{align*}{\frac{ \left ( d\tan \left ( bx+a \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 [A] time = 1.65202, size = 96, normalized size = 4. \begin{align*} \frac{\left (\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}\right )^{n} \sin \left (b x + a\right )}{{\left (b n + b\right )} \cos \left (b x + a\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 (d \tan{\left (a + b x \right )}\right )^{n} \sec ^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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