Optimal. Leaf size=22 \[ \frac{(a+b \tan (c+d x))^4}{4 b d} \]
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Rubi [A] time = 0.0363167, 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} \[ \frac{(a+b \tan (c+d x))^4}{4 b d} \]
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))^3 \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^3 \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{(a+b \tan (c+d x))^4}{4 b d}\\ \end{align*}
Mathematica [B] time = 0.154683, size = 57, normalized size = 2.59 \[ \frac{\tan (c+d x) \left (6 a^2 b \tan (c+d x)+4 a^3+4 a b^2 \tan ^2(c+d x)+b^3 \tan ^3(c+d x)\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.061, size = 72, normalized size = 3.3 \begin{align*}{\frac{1}{d} \left ({\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{3\,b{a}^{2}}{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{a}^{3}\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 = 1.18808, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (b \tan \left (d x + c\right ) + a\right )}^{4}}{4 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.83864, size = 181, normalized size = 8.23 \begin{align*} \frac{b^{3} + 2 \,{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (a b^{2} \cos \left (d x + c\right ) +{\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{4}} \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 )^{3} \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 = 1.82681, size = 77, normalized size = 3.5 \begin{align*} \frac{b^{3} \tan \left (d x + c\right )^{4} + 4 \, a b^{2} \tan \left (d x + c\right )^{3} + 6 \, a^{2} b \tan \left (d x + c\right )^{2} + 4 \, a^{3} \tan \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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