Optimal. Leaf size=59 \[ \frac{3 a^3 \tan (c+d x)}{d}+\frac{7 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}+a^3 x \]
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Rubi [A] time = 0.0826457, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2757, 3770, 3767, 8, 3768} \[ \frac{3 a^3 \tan (c+d x)}{d}+\frac{7 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}+a^3 x \]
Antiderivative was successfully verified.
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Rule 2757
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^3 \sec ^3(c+d x) \, dx &=\int \left (a^3+3 a^3 \sec (c+d x)+3 a^3 \sec ^2(c+d x)+a^3 \sec ^3(c+d x)\right ) \, dx\\ &=a^3 x+a^3 \int \sec ^3(c+d x) \, dx+\left (3 a^3\right ) \int \sec (c+d x) \, dx+\left (3 a^3\right ) \int \sec ^2(c+d x) \, dx\\ &=a^3 x+\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} a^3 \int \sec (c+d x) \, dx-\frac{\left (3 a^3\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=a^3 x+\frac{7 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{3 a^3 \tan (c+d x)}{d}+\frac{a^3 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0244912, size = 50, normalized size = 0.85 \[ a^3 \left (\frac{3 \tan (c+d x)}{d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 d}+x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 71, normalized size = 1.2 \begin{align*}{a}^{3}x+{\frac{{a}^{3}c}{d}}+{\frac{7\,{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+3\,{\frac{{a}^{3}\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21225, size = 134, normalized size = 2.27 \begin{align*} \frac{4 \,{\left (d x + c\right )} a^{3} - a^{3}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, 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} \tan \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 = 1.80309, size = 251, normalized size = 4.25 \begin{align*} \frac{4 \, a^{3} d x \cos \left (d x + c\right )^{2} + 7 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 7 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (6 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \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 = 1.42431, size = 135, normalized size = 2.29 \begin{align*} \frac{2 \,{\left (d x + c\right )} a^{3} + 7 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 7 \, 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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