Optimal. Leaf size=119 \[ -\frac{256 a^4 \cos (c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{35 d}-\frac{24 a^2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}-\frac{2 a \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d} \]
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Rubi [A] time = 0.0706675, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ -\frac{256 a^4 \cos (c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{35 d}-\frac{24 a^2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}-\frac{2 a \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \sin (c+d x))^{7/2} \, dx &=-\frac{2 a \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 d}+\frac{1}{7} (12 a) \int (a+a \sin (c+d x))^{5/2} \, dx\\ &=-\frac{24 a^2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac{2 a \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 d}+\frac{1}{35} \left (96 a^2\right ) \int (a+a \sin (c+d x))^{3/2} \, dx\\ &=-\frac{64 a^3 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{35 d}-\frac{24 a^2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac{2 a \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 d}+\frac{1}{35} \left (128 a^3\right ) \int \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{256 a^4 \cos (c+d x)}{35 d \sqrt{a+a \sin (c+d x)}}-\frac{64 a^3 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{35 d}-\frac{24 a^2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac{2 a \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 d}\\ \end{align*}
Mathematica [A] time = 0.654412, size = 154, normalized size = 1.29 \[ -\frac{a^3 (\sin (c+d x)+1)^3 \sqrt{a (\sin (c+d x)+1)} \left (-1225 \sin \left (\frac{1}{2} (c+d x)\right )+245 \sin \left (\frac{3}{2} (c+d x)\right )+49 \sin \left (\frac{5}{2} (c+d x)\right )-5 \sin \left (\frac{7}{2} (c+d x)\right )+1225 \cos \left (\frac{1}{2} (c+d x)\right )+245 \cos \left (\frac{3}{2} (c+d x)\right )-49 \cos \left (\frac{5}{2} (c+d x)\right )-5 \cos \left (\frac{7}{2} (c+d x)\right )\right )}{140 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 75, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ){a}^{4} \left ( \sin \left ( dx+c \right ) -1 \right ) \left ( 5\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}+27\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}+71\,\sin \left ( dx+c \right ) +177 \right ) }{35\,d\cos \left ( dx+c \right ) }{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63007, size = 360, normalized size = 3.03 \begin{align*} \frac{2 \,{\left (5 \, a^{3} \cos \left (d x + c\right )^{4} + 27 \, a^{3} \cos \left (d x + c\right )^{3} - 54 \, a^{3} \cos \left (d x + c\right )^{2} - 204 \, a^{3} \cos \left (d x + c\right ) - 128 \, a^{3} +{\left (5 \, a^{3} \cos \left (d x + c\right )^{3} - 22 \, a^{3} \cos \left (d x + c\right )^{2} - 76 \, a^{3} \cos \left (d x + c\right ) + 128 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{35 \,{\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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