Optimal. Leaf size=91 \[ \frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{2 a^4}{d (a-a \sin (c+d x))}+\frac{5 a^3 \sin (c+d x)}{d}+\frac{7 a^3 \log (1-\sin (c+d x))}{d} \]
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Rubi [A] time = 0.0707841, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2707, 77} \[ \frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{2 a^4}{d (a-a \sin (c+d x))}+\frac{5 a^3 \sin (c+d x)}{d}+\frac{7 a^3 \log (1-\sin (c+d x))}{d} \]
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 ^3(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3 (a+x)}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (5 a^2+\frac{2 a^4}{(a-x)^2}-\frac{7 a^3}{a-x}+3 a x+x^2\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{7 a^3 \log (1-\sin (c+d x))}{d}+\frac{5 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}+\frac{2 a^4}{d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.156734, size = 66, normalized size = 0.73 \[ \frac{a^3 \left (2 \sin ^3(c+d x)+9 \sin ^2(c+d x)+30 \sin (c+d x)+\frac{12}{1-\sin (c+d x)}+42 \log (1-\sin (c+d x))\right )}{6 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.078, size = 205, normalized size = 2.3 \begin{align*}{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{2\,d}}+{\frac{7\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+7\,{\frac{{a}^{3}\sin \left ( dx+c \right ) }{d}}-7\,{\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 ) ^{6}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{2\,d}}+3\,{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}+7\,{\frac{{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07083, size = 97, normalized size = 1.07 \begin{align*} \frac{2 \, a^{3} \sin \left (d x + c\right )^{3} + 9 \, a^{3} \sin \left (d x + c\right )^{2} + 42 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) + 30 \, a^{3} \sin \left (d x + c\right ) - \frac{12 \, a^{3}}{\sin \left (d x + c\right ) - 1}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59931, size = 248, normalized size = 2.73 \begin{align*} \frac{4 \, a^{3} \cos \left (d x + c\right )^{4} - 50 \, a^{3} \cos \left (d x + c\right )^{2} + 31 \, a^{3} + 84 \,{\left (a^{3} \sin \left (d x + c\right ) - a^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) -{\left (14 \, a^{3} \cos \left (d x + c\right )^{2} + 55 \, a^{3}\right )} \sin \left (d x + c\right )}{12 \,{\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(-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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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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