Optimal. Leaf size=61 \[ \frac{2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{d}-\frac{8 a^2 \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d} \]
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Rubi [A] time = 0.113317, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2674, 2673} \[ \frac{2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{d}-\frac{8 a^2 \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 2674
Rule 2673
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=\frac{2 a \sec ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{d}-(4 a) \int \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx\\ &=-\frac{8 a^2 \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}+\frac{2 a \sec ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{d}\\ \end{align*}
Mathematica [A] time = 5.29493, size = 82, normalized size = 1.34 \[ \frac{2 a^3 (3 \sin (c+d x)-1) \sqrt{a (\sin (c+d x)+1)}}{3 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.102, size = 57, normalized size = 0.9 \begin{align*} -{\frac{2\,{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) \left ( 3\,\sin \left ( dx+c \right ) -1 \right ) }{ \left ( 3\,\sin \left ( dx+c \right ) -3 \right ) \cos \left ( dx+c \right ) d}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.47511, size = 432, normalized size = 7.08 \begin{align*} \frac{2 \,{\left (a^{\frac{7}{2}} - \frac{6 \, a^{\frac{7}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{5 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{24 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{10 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{36 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{10 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{24 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{5 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{6 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac{a^{\frac{7}{2}} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )}}{3 \, d{\left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62355, size = 142, normalized size = 2.33 \begin{align*} -\frac{2 \,{\left (3 \, a^{3} \sin \left (d x + c\right ) - a^{3}\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{3 \,{\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\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 [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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