Optimal. Leaf size=56 \[ -\frac{2}{a^3 d (\cos (c+d x)+1)}+\frac{1}{2 a^3 d (\cos (c+d x)+1)^2}-\frac{\log (\cos (c+d x)+1)}{a^3 d} \]
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Rubi [A] time = 0.0413221, antiderivative size = 56, 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} \[ -\frac{2}{a^3 d (\cos (c+d x)+1)}+\frac{1}{2 a^3 d (\cos (c+d x)+1)^2}-\frac{\log (\cos (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 43
Rubi steps
\begin{align*} \int \frac{\tan (c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^2}{(a+a x)^3} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^3 (1+x)^3}-\frac{2}{a^3 (1+x)^2}+\frac{1}{a^3 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{1}{2 a^3 d (1+\cos (c+d x))^2}-\frac{2}{a^3 d (1+\cos (c+d x))}-\frac{\log (1+\cos (c+d x))}{a^3 d}\\ \end{align*}
Mathematica [A] time = 0.129941, size = 79, normalized size = 1.41 \[ -\frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (8 \cos ^2\left (\frac{1}{2} (c+d x)\right )+16 \cos ^4\left (\frac{1}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-1\right )}{a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 68, normalized size = 1.2 \begin{align*}{\frac{1}{2\,d{a}^{3} \left ( 1+\sec \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{d{a}^{3} \left ( 1+\sec \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{d{a}^{3}}}+{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11954, size = 81, normalized size = 1.45 \begin{align*} -\frac{\frac{4 \, \cos \left (d x + c\right ) + 3}{a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} \cos \left (d x + c\right ) + a^{3}} + \frac{2 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.10605, size = 204, normalized size = 3.64 \begin{align*} -\frac{2 \,{\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 4 \, \cos \left (d x + c\right ) + 3}{2 \,{\left (a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 27.8445, size = 411, normalized size = 7.34 \begin{align*} \begin{cases} \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \sec ^{2}{\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} + \frac{2 \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \sec{\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} + \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} - \frac{2 \log{\left (\sec{\left (c + d x \right )} + 1 \right )} \sec ^{2}{\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} - \frac{4 \log{\left (\sec{\left (c + d x \right )} + 1 \right )} \sec{\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} - \frac{2 \log{\left (\sec{\left (c + d x \right )} + 1 \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} + \frac{2 \sec{\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} + \frac{3}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} & \text{for}\: d \neq 0 \\\frac{x \tan{\left (c \right )}}{\left (a \sec{\left (c \right )} + a\right )^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3264, size = 117, normalized size = 2.09 \begin{align*} \frac{\frac{8 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} + \frac{\frac{6 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{6}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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