Optimal. Leaf size=62 \[ -\frac{a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{3 a^3 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0914808, antiderivative size = 62, 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, 2833, 12, 43} \[ -\frac{a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{3 a^3 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^3 \sin (c+d x) \, dx &=-\int (-a-a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{a^3 (-a+x)^3}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \frac{(-a+x)^3}{x^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \left (1-\frac{a^3}{x^3}+\frac{3 a^2}{x^2}-\frac{3 a}{x}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac{a^3 \cos (c+d x)}{d}-\frac{3 a^3 \log (\cos (c+d x))}{d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.228813, size = 65, normalized size = 1.05 \[ -\frac{a^3 \sec ^2(c+d x) (-9 \cos (c+d x)+\cos (3 (c+d x))+6 \log (\cos (c+d x))+\cos (2 (c+d x)) (6 \log (\cos (c+d x))-2)-4)}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 63, normalized size = 1. \begin{align*}{\frac{{a}^{3} \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+3\,{\frac{{a}^{3}\sec \left ( dx+c \right ) }{d}}+3\,{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3}}{d\sec \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986018, size = 74, normalized size = 1.19 \begin{align*} -\frac{2 \, a^{3} \cos \left (d x + c\right ) + 6 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac{6 \, a^{3}}{\cos \left (d x + c\right )} - \frac{a^{3}}{\cos \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84729, size = 158, normalized size = 2.55 \begin{align*} -\frac{2 \, a^{3} \cos \left (d x + c\right )^{3} + 6 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 6 \, a^{3} \cos \left (d x + c\right ) - a^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \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 )} \sec{\left (c + d x \right )}\, dx + \int 3 \sin{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sin{\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 = 1.29489, size = 86, normalized size = 1.39 \begin{align*} -\frac{a^{3} \cos \left (d x + c\right )}{d} - \frac{3 \, a^{3} \log \left (\frac{{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac{6 \, a^{3} \cos \left (d x + c\right ) + a^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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