Optimal. Leaf size=20 \[ \frac{a^3}{d (a-a \sin (c+d x))} \]
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Rubi [A] time = 0.0376204, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 32} \[ \frac{a^3}{d (a-a \sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3}{d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.138102, size = 32, normalized size = 1.6 \[ \frac{a^2}{d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 75, normalized size = 3.8 \begin{align*}{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{2}\sin \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{2}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{2}\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 = 0.932716, size = 24, normalized size = 1.2 \begin{align*} -\frac{a^{2}}{d{\left (\sin \left (d x + c\right ) - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54091, size = 36, normalized size = 1.8 \begin{align*} -\frac{a^{2}}{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 [A] time = 1.17787, size = 41, normalized size = 2.05 \begin{align*} \frac{2 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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