Optimal. Leaf size=15 \[ \frac{\tan ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.0295692, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2607, 30} \[ \frac{\tan ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \sec ^2(a+b x) \tan ^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\tan ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0068882, size = 15, normalized size = 1. \[ \frac{\tan ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 22, normalized size = 1.5 \begin{align*}{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{3\,b \left ( \cos \left ( bx+a \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02254, size = 18, normalized size = 1.2 \begin{align*} \frac{\tan \left (b x + a\right )^{3}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73518, size = 80, normalized size = 5.33 \begin{align*} -\frac{{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )}{3 \, b \cos \left (b x + a\right )^{3}} \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 = 1.16347, size = 18, normalized size = 1.2 \begin{align*} \frac{\tan \left (b x + a\right )^{3}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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