Optimal. Leaf size=48 \[ \frac{a^3 \sec (c+d x)}{d}+\frac{4 a^3 \log (1-\cos (c+d x))}{d}-\frac{3 a^3 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0460482, 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, 88} \[ \frac{a^3 \sec (c+d x)}{d}+\frac{4 a^3 \log (1-\cos (c+d x))}{d}-\frac{3 a^3 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 88
Rubi steps
\begin{align*} \int \cot (c+d x) (a+a \sec (c+d x))^3 \, dx &=-\frac{a^2 \operatorname{Subst}\left (\int \frac{(a+a x)^2}{x^2 (a-a x)} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^2 \operatorname{Subst}\left (\int \left (-\frac{4 a}{-1+x}+\frac{a}{x^2}+\frac{3 a}{x}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{4 a^3 \log (1-\cos (c+d x))}{d}-\frac{3 a^3 \log (\cos (c+d x))}{d}+\frac{a^3 \sec (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0876936, size = 36, normalized size = 0.75 \[ \frac{a^3 \left (\sec (c+d x)+8 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-3 \log (\cos (c+d x))\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 47, normalized size = 1. \begin{align*}{\frac{{a}^{3}\sec \left ( dx+c \right ) }{d}}+4\,{\frac{{a}^{3}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3}\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 = 1.50157, size = 58, normalized size = 1.21 \begin{align*} \frac{4 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 3 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac{a^{3}}{\cos \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19775, size = 155, normalized size = 3.23 \begin{align*} -\frac{3 \, a^{3} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - 4 \, a^{3} \cos \left (d x + c\right ) \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - a^{3}}{d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int 3 \cot{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int 3 \cot{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cot{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \cot{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26674, size = 196, normalized size = 4.08 \begin{align*} \frac{4 \, a^{3} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 3 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{5 \, a^{3} + \frac{3 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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