Optimal. Leaf size=31 \[ \frac{a^5 \sin ^2(c+d x)}{2 d (a-a \sin (c+d x))^2} \]
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Rubi [A] time = 0.059388, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 37} \[ \frac{a^5 \sin ^2(c+d x)}{2 d (a-a \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 37
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{x}{a (a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^4 \operatorname{Subst}\left (\int \frac{x}{(a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^5 \sin ^2(c+d x)}{2 d (a-a \sin (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 0.0304367, size = 30, normalized size = 0.97 \[ \frac{a^3 \sin ^2(c+d x)}{2 d (1-\sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.073, size = 154, normalized size = 5. \begin{align*}{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10199, size = 57, normalized size = 1.84 \begin{align*} \frac{2 \, a^{3} \sin \left (d x + c\right ) - a^{3}}{2 \,{\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32256, size = 104, normalized size = 3.35 \begin{align*} -\frac{2 \, a^{3} \sin \left (d x + c\right ) - a^{3}}{2 \,{\left (d \cos \left (d x + c\right )^{2} + 2 \, d \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 = 1.19195, size = 43, normalized size = 1.39 \begin{align*} \frac{2 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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