Optimal. Leaf size=40 \[ \frac{2 a^4}{d (a-a \sin (c+d x))}+\frac{a^3 \log (1-\sin (c+d x))}{d} \]
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Rubi [A] time = 0.0522718, antiderivative size = 40, 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 = {2667, 43} \[ \frac{2 a^4}{d (a-a \sin (c+d x))}+\frac{a^3 \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{a^3 \operatorname{Subst}\left (\int \frac{a+x}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{2 a}{(a-x)^2}+\frac{1}{-a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \log (1-\sin (c+d x))}{d}+\frac{2 a^4}{d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0410921, size = 59, normalized size = 1.48 \[ \frac{a^3 (1-\sin (c+d x)) (\sin (c+d x)+1) \sec ^2(c+d x) \left (\frac{2}{1-\sin (c+d x)}+\log (1-\sin (c+d x))\right )}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 128, normalized size = 3.2 \begin{align*}{\frac{{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{a}^{3}\sin \left ( dx+c \right ) }{2\,d}}-{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{3\,{a}^{3}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.95084, size = 45, normalized size = 1.12 \begin{align*} \frac{a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac{2 \, a^{3}}{\sin \left (d x + c\right ) - 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6929, size = 109, normalized size = 2.72 \begin{align*} -\frac{2 \, a^{3} -{\left (a^{3} \sin \left (d x + c\right ) - a^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{d \sin \left (d x + c\right ) - d} \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 [B] time = 1.20148, size = 124, normalized size = 3.1 \begin{align*} -\frac{a^{3} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 2 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 10 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{3}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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