Optimal. Leaf size=169 \[ -\frac{7 a^3 \cos (c+d x)}{16 d (a \sin (c+d x)+a)^{3/2}}+\frac{7 a^2 \sec (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}-\frac{7 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{16 \sqrt{2} d}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}+\frac{7 a \sec ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{30 d} \]
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Rubi [A] time = 0.221389, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2675, 2687, 2650, 2649, 206} \[ -\frac{7 a^3 \cos (c+d x)}{16 d (a \sin (c+d x)+a)^{3/2}}+\frac{7 a^2 \sec (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}-\frac{7 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{16 \sqrt{2} d}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}+\frac{7 a \sec ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{30 d} \]
Antiderivative was successfully verified.
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Rule 2675
Rule 2687
Rule 2650
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \sec ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac{\sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{1}{10} (7 a) \int \sec ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{7 a \sec ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{30 d}+\frac{\sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{1}{12} \left (7 a^2\right ) \int \frac{\sec ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=\frac{7 a^2 \sec (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}+\frac{7 a \sec ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{30 d}+\frac{\sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{1}{8} \left (7 a^3\right ) \int \frac{1}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac{7 a^3 \cos (c+d x)}{16 d (a+a \sin (c+d x))^{3/2}}+\frac{7 a^2 \sec (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}+\frac{7 a \sec ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{30 d}+\frac{\sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{1}{32} \left (7 a^2\right ) \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{7 a^3 \cos (c+d x)}{16 d (a+a \sin (c+d x))^{3/2}}+\frac{7 a^2 \sec (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}+\frac{7 a \sec ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{30 d}+\frac{\sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac{\left (7 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{16 d}\\ &=-\frac{7 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{16 \sqrt{2} d}-\frac{7 a^3 \cos (c+d x)}{16 d (a+a \sin (c+d x))^{3/2}}+\frac{7 a^2 \sec (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}+\frac{7 a \sec ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{30 d}+\frac{\sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}\\ \end{align*}
Mathematica [C] time = 0.421191, size = 288, normalized size = 1.7 \[ \frac{(a (\sin (c+d x)+1))^{3/2} \left (30 \sin \left (\frac{1}{2} (c+d x)\right )+\frac{90 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{40 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{24 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^5}-15 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+(105+105 i) (-1)^{3/4} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2 \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )\right )}{240 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.132, size = 172, normalized size = 1. \begin{align*} -{\frac{1}{480\, \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}\cos \left ( dx+c \right ) d} \left ( 210\,{a}^{7/2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+ \left ( 105\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) a-168\,{a}^{7/2} \right ) \sin \left ( dx+c \right ) -350\,{a}^{7/2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}+105\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) a+72\,{a}^{7/2} \right ){a}^{-{\frac{3}{2}}}{\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(-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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Fricas [A] time = 1.95671, size = 667, normalized size = 3.95 \begin{align*} \frac{105 \,{\left (\sqrt{2} a \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - \sqrt{2} a \cos \left (d x + c\right )^{3}\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{a \sin \left (d x + c\right ) + a}{\left (\sqrt{2} \cos \left (d x + c\right ) - \sqrt{2} \sin \left (d x + c\right ) + \sqrt{2}\right )} \sqrt{a} + 3 \, a \cos \left (d x + c\right ) -{\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) - 4 \,{\left (175 \, a \cos \left (d x + c\right )^{2} - 21 \,{\left (5 \, a \cos \left (d x + c\right )^{2} - 4 \, a\right )} \sin \left (d x + c\right ) - 36 \, a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{960 \,{\left (d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{3}\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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