Optimal. Leaf size=23 \[ \frac{a^5}{2 d (a-a \sin (c+d x))^2} \]
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Rubi [A] time = 0.0399265, antiderivative size = 23, 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} \[ \frac{a^5}{2 d (a-a \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{1}{(a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^5}{2 d (a-a \sin (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 0.233885, size = 35, normalized size = 1.52 \[ \frac{a^3}{2 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.074, size = 146, normalized size = 6.4 \begin{align*}{\frac{{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{3\,{a}^{3}\sin \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969195, size = 38, normalized size = 1.65 \begin{align*} \frac{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 = 1.61176, size = 73, normalized size = 3.17 \begin{align*} -\frac{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 [B] time = 1.19545, size = 85, normalized size = 3.7 \begin{align*} \frac{2 \,{\left (a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{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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