Optimal. Leaf size=30 \[ \frac{\tan ^3(c+d x) (a \cot (c+d x)+b)^3}{3 b d} \]
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Rubi [A] time = 0.0464568, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3088, 37} \[ \frac{\tan ^3(c+d x) (a \cot (c+d x)+b)^3}{3 b d} \]
Antiderivative was successfully verified.
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Rule 3088
Rule 37
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(b+a x)^2}{x^4} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{(b+a \cot (c+d x))^3 \tan ^3(c+d x)}{3 b d}\\ \end{align*}
Mathematica [A] time = 0.0404834, size = 46, normalized size = 1.53 \[ \frac{a^2 \tan (c+d x)}{d}+\frac{a b \tan ^2(c+d x)}{d}+\frac{b^2 \tan ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.103, size = 48, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ({a}^{2}\tan \left ( dx+c \right ) +{\frac{ab}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0484, size = 61, normalized size = 2.03 \begin{align*} \frac{b^{2} \tan \left (d x + c\right )^{3} + 3 \, a^{2} \tan \left (d x + c\right ) - \frac{3 \, a b}{\sin \left (d x + c\right )^{2} - 1}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.463329, size = 131, normalized size = 4.37 \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(-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 = 1.16747, size = 55, normalized size = 1.83 \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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