Optimal. Leaf size=44 \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.036244, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2669, 3767} \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 3767
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{a \sec ^3(c+d x)}{3 d}+a \int \sec ^4(c+d x) \, dx\\ &=\frac{a \sec ^3(c+d x)}{3 d}-\frac{a \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{a \sec ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0600332, size = 41, normalized size = 0.93 \[ \frac{a \left (\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac{a \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.084, size = 38, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{a}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-a \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \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 = 0.952829, size = 47, normalized size = 1.07 \begin{align*} \frac{{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a + \frac{a}{\cos \left (d x + c\right )^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62443, size = 131, normalized size = 2.98 \begin{align*} -\frac{2 \, a \cos \left (d x + c\right )^{2} + 2 \, a \sin \left (d x + c\right ) - a}{3 \,{\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\right )}} \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 = 1.17314, size = 89, normalized size = 2.02 \begin{align*} -\frac{\frac{3 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1} + \frac{9 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 7 \, a}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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