Optimal. Leaf size=48 \[ \frac{a^2 \sec ^2(c+d x)}{2 d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0356408, antiderivative size = 48, 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 = {3879, 43} \[ \frac{a^2 \sec ^2(c+d x)}{2 d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 43
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^2 \tan (c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a+a x)^2}{x^3} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{x^3}+\frac{2 a^2}{x^2}+\frac{a^2}{x}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a^2 \sec (c+d x)}{d}+\frac{a^2 \sec ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0972948, size = 51, normalized size = 1.06 \[ -\frac{a^2 \sec ^2(c+d x) (-4 \cos (c+d x)+\cos (2 (c+d x)) \log (\cos (c+d x))+\log (\cos (c+d x))-1)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 46, normalized size = 1. \begin{align*}{\frac{{a}^{2} \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+2\,{\frac{{a}^{2}\sec \left ( dx+c \right ) }{d}}+{\frac{{a}^{2}\ln \left ( \sec \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 = 1.19007, size = 58, normalized size = 1.21 \begin{align*} -\frac{2 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac{4 \, a^{2} \cos \left (d x + c\right ) + a^{2}}{\cos \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.13372, size = 127, normalized size = 2.65 \begin{align*} -\frac{2 \, a^{2} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 4 \, a^{2} \cos \left (d x + c\right ) - a^{2}}{2 \, 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 [A] time = 0.93767, size = 60, normalized size = 1.25 \begin{align*} \begin{cases} \frac{a^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a^{2} \sec ^{2}{\left (c + d x \right )}}{2 d} + \frac{2 a^{2} \sec{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sec{\left (c \right )} + a\right )^{2} \tan{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.41181, size = 192, normalized size = 4. \begin{align*} \frac{2 \, a^{2} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2 \, a^{2} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{11 \, a^{2} + \frac{10 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{3 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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