Optimal. Leaf size=71 \[ \frac{a^2}{2 d (a-a \sin (c+d x))}+\frac{a \sin (c+d x)}{d}+\frac{5 a \log (1-\sin (c+d x))}{4 d}-\frac{a \log (\sin (c+d x)+1)}{4 d} \]
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Rubi [A] time = 0.048039, antiderivative size = 71, 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 = {2707, 88} \[ \frac{a^2}{2 d (a-a \sin (c+d x))}+\frac{a \sin (c+d x)}{d}+\frac{5 a \log (1-\sin (c+d x))}{4 d}-\frac{a \log (\sin (c+d x)+1)}{4 d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 88
Rubi steps
\begin{align*} \int (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{(a-x)^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{a^2}{2 (a-x)^2}-\frac{5 a}{4 (a-x)}-\frac{a}{4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{5 a \log (1-\sin (c+d x))}{4 d}-\frac{a \log (1+\sin (c+d x))}{4 d}+\frac{a \sin (c+d x)}{d}+\frac{a^2}{2 d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.108376, size = 77, normalized size = 1.08 \[ -\frac{a \sin (c+d x) \tan ^2(c+d x)}{d}+\frac{a \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d}-\frac{3 a \left (\tanh ^{-1}(\sin (c+d x))-\tan (c+d x) \sec (c+d x)\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 96, normalized size = 1.4 \begin{align*}{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{2\,d}}+{\frac{3\,a\sin \left ( dx+c \right ) }{2\,d}}-{\frac{3\,a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{a\ln \left ( \cos \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 = 0.99153, size = 69, normalized size = 0.97 \begin{align*} -\frac{a \log \left (\sin \left (d x + c\right ) + 1\right ) - 5 \, a \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, a \sin \left (d x + c\right ) + \frac{2 \, a}{\sin \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 = 1.53161, size = 224, normalized size = 3.15 \begin{align*} -\frac{4 \, a \cos \left (d x + c\right )^{2} +{\left (a \sin \left (d x + c\right ) - a\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 5 \,{\left (a \sin \left (d x + c\right ) - a\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 4 \, a \sin \left (d x + c\right ) - 2 \, a}{4 \,{\left (d \sin \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 \sin{\left (c + d x \right )} \tan ^{3}{\left (c + d x \right )}\, dx + \int \tan ^{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 [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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