Optimal. Leaf size=22 \[ \frac{(a+b \tan (c+d x))^3}{3 b d} \]
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Rubi [A] time = 0.0365484, antiderivative size = 22, 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} \[ \frac{(a+b \tan (c+d x))^3}{3 b d} \]
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))^2 \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^2 \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{(a+b \tan (c+d x))^3}{3 b d}\\ \end{align*}
Mathematica [B] time = 0.0419874, size = 46, normalized size = 2.09 \[ \frac{a^2 \tan (c+d x)}{d}+\frac{a b \tan ^2(c+d x)}{d}+\frac{b^2 \tan ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 48, normalized size = 2.2 \begin{align*}{\frac{1}{d} \left ({\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{ab}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{a}^{2}\tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.36252, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (b \tan \left (d x + c\right ) + a\right )}^{3}}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71534, size = 131, normalized size = 5.95 \begin{align*} \frac{3 \, a b \cos \left (d x + c\right ) +{\left ({\left (3 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \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 )^{2} \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 [B] time = 1.36354, size = 55, normalized size = 2.5 \begin{align*} \frac{b^{2} \tan \left (d x + c\right )^{3} + 3 \, a b \tan \left (d x + c\right )^{2} + 3 \, a^{2} \tan \left (d x + c\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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