Optimal. Leaf size=73 \[ -\frac{a^2}{2 d (a-a \cos (c+d x))}+\frac{3 a \log (1-\cos (c+d x))}{4 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{a \log (\cos (c+d x)+1)}{4 d} \]
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Rubi [A] time = 0.0954302, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3872, 2836, 12, 72} \[ -\frac{a^2}{2 d (a-a \cos (c+d x))}+\frac{3 a \log (1-\cos (c+d x))}{4 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{a \log (\cos (c+d x)+1)}{4 d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2836
Rule 12
Rule 72
Rubi steps
\begin{align*} \int \csc ^3(c+d x) (a+a \sec (c+d x)) \, dx &=-\int (-a-a \cos (c+d x)) \csc ^3(c+d x) \sec (c+d x) \, dx\\ &=\frac{a^3 \operatorname{Subst}\left (\int \frac{a}{(-a-x)^2 x (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^4 \operatorname{Subst}\left (\int \frac{1}{(-a-x)^2 x (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^4 \operatorname{Subst}\left (\int \left (-\frac{1}{4 a^3 (a-x)}-\frac{1}{a^3 x}+\frac{1}{2 a^2 (a+x)^2}+\frac{3}{4 a^3 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac{a^2}{2 d (a-a \cos (c+d x))}+\frac{3 a \log (1-\cos (c+d x))}{4 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{a \log (1+\cos (c+d x))}{4 d}\\ \end{align*}
Mathematica [A] time = 0.820608, size = 114, normalized size = 1.56 \[ -\frac{a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}-\frac{a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}-\frac{a \left (\csc ^2(c+d x)-2 \log (\sin (c+d x))+2 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 48, normalized size = 0.7 \begin{align*}{\frac{a\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{4\,d}}-{\frac{a}{2\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}+{\frac{3\,a\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{4\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09293, size = 70, normalized size = 0.96 \begin{align*} \frac{a \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - 4 \, a \log \left (\cos \left (d x + c\right )\right ) + \frac{2 \, a}{\cos \left (d x + c\right ) - 1}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80028, size = 246, normalized size = 3.37 \begin{align*} -\frac{4 \,{\left (a \cos \left (d x + c\right ) - a\right )} \log \left (-\cos \left (d x + c\right )\right ) -{\left (a \cos \left (d x + c\right ) - a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 3 \,{\left (a \cos \left (d x + c\right ) - a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 2 \, a}{4 \,{\left (d \cos \left (d x + c\right ) - d\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 \left (\int \csc ^{3}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \csc ^{3}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.47429, size = 138, normalized size = 1.89 \begin{align*} \frac{3 \, a \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 4 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{{\left (a - \frac{3 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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