Optimal. Leaf size=59 \[ \frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^3 x}{2} \]
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Rubi [A] time = 0.066875, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3791, 2637, 2635, 8, 3770} \[ \frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^3 x}{2} \]
Antiderivative was successfully verified.
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Rule 3791
Rule 2637
Rule 2635
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+a \sec (c+d x))^3 \, dx &=\int \left (3 a^3+3 a^3 \cos (c+d x)+a^3 \cos ^2(c+d x)+a^3 \sec (c+d x)\right ) \, dx\\ &=3 a^3 x+a^3 \int \cos ^2(c+d x) \, dx+a^3 \int \sec (c+d x) \, dx+\left (3 a^3\right ) \int \cos (c+d x) \, dx\\ &=3 a^3 x+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} a^3 \int 1 \, dx\\ &=\frac{7 a^3 x}{2}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0700981, size = 81, normalized size = 1.37 \[ \frac{a^3 \left (12 \sin (c+d x)+\sin (2 (c+d x))-4 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+4 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+14 d x\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 72, normalized size = 1.2 \begin{align*}{\frac{{a}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{7\,{a}^{3}x}{2}}+{\frac{7\,{a}^{3}c}{2\,d}}+3\,{\frac{{a}^{3}\sin \left ( dx+c \right ) }{d}}+{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10061, size = 100, normalized size = 1.69 \begin{align*} \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} + 12 \,{\left (d x + c\right )} a^{3} + 2 \, a^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{3} \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75841, size = 159, normalized size = 2.69 \begin{align*} \frac{7 \, a^{3} d x + a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (a^{3} \cos \left (d x + c\right ) + 6 \, a^{3}\right )} \sin \left (d x + c\right )}{2 \, d} \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 = 1.39002, size = 135, normalized size = 2.29 \begin{align*} \frac{7 \,{\left (d x + c\right )} a^{3} + 2 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (5 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 7 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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