Optimal. Leaf size=56 \[ -\frac{a^3 \tan ^5(c+d x)}{5 d}+\frac{a^3 \tan ^3(c+d x)}{3 d}-\frac{a^3 \tan (c+d x)}{d}+a^3 x \]
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Rubi [A] time = 0.0385128, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4120, 3473, 8} \[ -\frac{a^3 \tan ^5(c+d x)}{5 d}+\frac{a^3 \tan ^3(c+d x)}{3 d}-\frac{a^3 \tan (c+d x)}{d}+a^3 x \]
Antiderivative was successfully verified.
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Rule 4120
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \left (a-a \sec ^2(c+d x)\right )^3 \, dx &=-\left (a^3 \int \tan ^6(c+d x) \, dx\right )\\ &=-\frac{a^3 \tan ^5(c+d x)}{5 d}+a^3 \int \tan ^4(c+d x) \, dx\\ &=\frac{a^3 \tan ^3(c+d x)}{3 d}-\frac{a^3 \tan ^5(c+d x)}{5 d}-a^3 \int \tan ^2(c+d x) \, dx\\ &=-\frac{a^3 \tan (c+d x)}{d}+\frac{a^3 \tan ^3(c+d x)}{3 d}-\frac{a^3 \tan ^5(c+d x)}{5 d}+a^3 \int 1 \, dx\\ &=a^3 x-\frac{a^3 \tan (c+d x)}{d}+\frac{a^3 \tan ^3(c+d x)}{3 d}-\frac{a^3 \tan ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.03132, size = 58, normalized size = 1.04 \[ -a^3 \left (\frac{\tan ^5(c+d x)}{5 d}-\frac{\tan ^3(c+d x)}{3 d}-\frac{\tan ^{-1}(\tan (c+d x))}{d}+\frac{\tan (c+d x)}{d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 81, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( dx+c \right ) -3\,{a}^{3}\tan \left ( dx+c \right ) -3\,{a}^{3} \left ( -2/3-1/3\, \left ( \sec \left ( dx+c \right ) \right ) ^{2} \right ) \tan \left ( dx+c \right ) +{a}^{3} \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \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 = 1.04516, size = 109, normalized size = 1.95 \begin{align*} a^{3} x - \frac{{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{3}}{15 \, d} + \frac{{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3}}{d} - \frac{3 \, a^{3} \tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.48466, size = 167, normalized size = 2.98 \begin{align*} \frac{15 \, a^{3} d x \cos \left (d x + c\right )^{5} -{\left (23 \, a^{3} \cos \left (d x + c\right )^{4} - 11 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3}\right )} \sin \left (d x + c\right )}{15 \, d \cos \left (d x + c\right )^{5}} \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*} - a^{3} \left (\int \left (-1\right )\, dx + \int 3 \sec ^{2}{\left (c + d x \right )}\, dx + \int - 3 \sec ^{4}{\left (c + d x \right )}\, dx + \int \sec ^{6}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17522, size = 72, normalized size = 1.29 \begin{align*} -\frac{3 \, a^{3} \tan \left (d x + c\right )^{5} - 5 \, a^{3} \tan \left (d x + c\right )^{3} - 15 \,{\left (d x + c\right )} a^{3} + 15 \, a^{3} \tan \left (d x + c\right )}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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