Optimal. Leaf size=66 \[ \frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{2 a^2 \tan (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{d} \]
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Rubi [A] time = 0.0877679, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2757, 3767, 8, 3768, 3770} \[ \frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{2 a^2 \tan (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2757
Rule 3767
Rule 8
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^2 \sec ^4(c+d x) \, dx &=\int \left (a^2 \sec ^2(c+d x)+2 a^2 \sec ^3(c+d x)+a^2 \sec ^4(c+d x)\right ) \, dx\\ &=a^2 \int \sec ^2(c+d x) \, dx+a^2 \int \sec ^4(c+d x) \, dx+\left (2 a^2\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{a^2 \sec (c+d x) \tan (c+d x)}{d}+a^2 \int \sec (c+d x) \, dx-\frac{a^2 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}-\frac{a^2 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a^2 \tan (c+d x)}{d}+\frac{a^2 \sec (c+d x) \tan (c+d x)}{d}+\frac{a^2 \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [B] time = 5.54138, size = 162, normalized size = 2.45 \[ -\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (-2 \tan (c) \cos (c+d x)-\sec (c) (-4 \sin (2 c+d x)+3 \sin (c+2 d x)+3 \sin (3 c+2 d x)+5 \sin (2 c+3 d x)+13 \sin (d x))+12 \cos ^3(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{48 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 78, normalized size = 1.2 \begin{align*}{\frac{5\,{a}^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{{a}^{2}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2} \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13642, size = 115, normalized size = 1.74 \begin{align*} \frac{2 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{2} - 3 \, a^{2}{\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^{2} \tan \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67818, size = 246, normalized size = 3.73 \begin{align*} \frac{3 \, a^{2} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a^{2} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (5 \, a^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sin \left (d x + c\right )}{6 \, 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 = 1.40979, size = 143, normalized size = 2.17 \begin{align*} \frac{3 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 8 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, a^{2} \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 )}^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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