Optimal. Leaf size=70 \[ -\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{3 a^3 \sin ^2(c+d x)}{2 d}-\frac{4 a^3 \sin (c+d x)}{d}-\frac{4 a^3 \log (1-\sin (c+d x))}{d} \]
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Rubi [A] time = 0.0449772, antiderivative size = 70, 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, 77} \[ -\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{3 a^3 \sin ^2(c+d x)}{2 d}-\frac{4 a^3 \sin (c+d x)}{d}-\frac{4 a^3 \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 77
Rubi steps
\begin{align*} \int (a+a \sin (c+d x))^3 \tan (c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x (a+x)^2}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-4 a^2+\frac{4 a^3}{a-x}-3 a x-x^2\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{4 a^3 \log (1-\sin (c+d x))}{d}-\frac{4 a^3 \sin (c+d x)}{d}-\frac{3 a^3 \sin ^2(c+d x)}{2 d}-\frac{a^3 \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.045258, size = 52, normalized size = 0.74 \[ -\frac{a^3 \left (2 \sin ^3(c+d x)+9 \sin ^2(c+d x)+24 \sin (c+d x)+24 \log (1-\sin (c+d x))\right )}{6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 85, normalized size = 1.2 \begin{align*} -{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-4\,{\frac{{a}^{3}\sin \left ( dx+c \right ) }{d}}+4\,{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-4\,{\frac{{a}^{3}\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 = 1.07111, size = 77, normalized size = 1.1 \begin{align*} -\frac{2 \, a^{3} \sin \left (d x + c\right )^{3} + 9 \, a^{3} \sin \left (d x + c\right )^{2} + 24 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) + 24 \, a^{3} \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51562, size = 147, normalized size = 2.1 \begin{align*} \frac{9 \, a^{3} \cos \left (d x + c\right )^{2} - 24 \, a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - 13 \, a^{3}\right )} \sin \left (d x + c\right )}{6 \, d} \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 \sin{\left (c + d x \right )} \tan{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \tan{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \tan{\left (c + d x \right )}\, dx + \int \tan{\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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