Optimal. Leaf size=64 \[ -\frac{2 a^5 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}+a^3 x+\frac{\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d} \]
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Rubi [A] time = 0.138811, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2855, 2670, 2680, 8} \[ -\frac{2 a^5 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}+a^3 x+\frac{\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 2855
Rule 2670
Rule 2680
Rule 8
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx &=\frac{\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-a \int \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=\frac{\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-a^5 \int \frac{\cos ^2(c+d x)}{(a-a \sin (c+d x))^2} \, dx\\ &=\frac{\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-\frac{2 a^5 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}+a^3 \int 1 \, dx\\ &=a^3 x+\frac{\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-\frac{2 a^5 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.719077, size = 107, normalized size = 1.67 \[ -\frac{a^3 \left (-9 (c+d x+2) \cos \left (\frac{1}{2} (c+d x)\right )+(3 c+3 d x+14) \cos \left (\frac{3}{2} (c+d x)\right )+6 \sin \left (\frac{1}{2} (c+d x)\right ) (2 (c+d x+2)+(c+d x) \cos (c+d x))\right )}{6 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.071, size = 126, normalized size = 2. \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ({\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3}}-\tan \left ( dx+c \right ) +dx+c \right ) +3\,{a}^{3} \left ( 1/3\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-1/3\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{\cos \left ( dx+c \right ) }}-1/3\, \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) \right ) +{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{a}^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.69052, size = 113, normalized size = 1.77 \begin{align*} \frac{3 \, a^{3} \tan \left (d x + c\right )^{3} +{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{3} - \frac{3 \,{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{3}}{\cos \left (d x + c\right )^{3}} + \frac{a^{3}}{\cos \left (d x + c\right )^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.35476, size = 335, normalized size = 5.23 \begin{align*} -\frac{6 \, a^{3} d x + 2 \, a^{3} -{\left (3 \, a^{3} d x + 7 \, a^{3}\right )} \cos \left (d x + c\right )^{2} +{\left (3 \, a^{3} d x - 5 \, a^{3}\right )} \cos \left (d x + c\right ) -{\left (6 \, a^{3} d x - 2 \, a^{3} +{\left (3 \, a^{3} d x - 7 \, a^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) +{\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, 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 [A] time = 1.25368, size = 90, normalized size = 1.41 \begin{align*} \frac{3 \,{\left (d x + c\right )} a^{3} + \frac{2 \,{\left (3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5 \, a^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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