Optimal. Leaf size=159 \[ -\frac{35 a^3}{128 d \sqrt{a \sin (c+d x)+a}}+\frac{35 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{128 \sqrt{2} d}+\frac{35 a^2 \sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{192 d}+\frac{\sec ^6(c+d x) (a \sin (c+d x)+a)^{5/2}}{6 d}+\frac{7 a \sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{48 d} \]
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Rubi [A] time = 0.241828, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2675, 2667, 51, 63, 206} \[ -\frac{35 a^3}{128 d \sqrt{a \sin (c+d x)+a}}+\frac{35 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{128 \sqrt{2} d}+\frac{35 a^2 \sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{192 d}+\frac{\sec ^6(c+d x) (a \sin (c+d x)+a)^{5/2}}{6 d}+\frac{7 a \sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{48 d} \]
Antiderivative was successfully verified.
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Rule 2675
Rule 2667
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=\frac{\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac{1}{12} (7 a) \int \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=\frac{7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac{\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac{1}{96} \left (35 a^2\right ) \int \sec ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{35 a^2 \sec ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{192 d}+\frac{7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac{\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac{1}{128} \left (35 a^3\right ) \int \frac{\sec (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=\frac{35 a^2 \sec ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{192 d}+\frac{7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac{\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac{\left (35 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{128 d}\\ &=-\frac{35 a^3}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{35 a^2 \sec ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{192 d}+\frac{7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac{\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac{\left (35 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{a+x}} \, dx,x,a \sin (c+d x)\right )}{256 d}\\ &=-\frac{35 a^3}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{35 a^2 \sec ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{192 d}+\frac{7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac{\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac{\left (35 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+a \sin (c+d x)}\right )}{128 d}\\ &=\frac{35 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{128 \sqrt{2} d}-\frac{35 a^3}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{35 a^2 \sec ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{192 d}+\frac{7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac{\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}\\ \end{align*}
Mathematica [C] time = 0.0922075, size = 44, normalized size = 0.28 \[ -\frac{a^3 \, _2F_1\left (-\frac{1}{2},4;\frac{1}{2};\frac{1}{2} (\sin (c+d x)+1)\right )}{8 d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.234, size = 113, normalized size = 0.7 \begin{align*} 2\,{\frac{{a}^{7}}{d} \left ( -1/16\,{\frac{1}{{a}^{4}} \left ( -1/48\,{\frac{{a}^{2}\sqrt{a+a\sin \left ( dx+c \right ) } \left ( 57\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+158\,\sin \left ( dx+c \right ) -190 \right ) }{ \left ( a\sin \left ( dx+c \right ) -a \right ) ^{3}}}-{\frac{35\,\sqrt{2}}{32\,\sqrt{a}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) } \right ) }-1/16\,{\frac{1}{{a}^{4}\sqrt{a+a\sin \left ( dx+c \right ) }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75733, size = 549, normalized size = 3.45 \begin{align*} \frac{105 \,{\left (\sqrt{2} a^{2} \cos \left (d x + c\right )^{4} + 2 \, \sqrt{2} a^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, \sqrt{2} a^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt{a} \log \left (-\frac{a \sin \left (d x + c\right ) + 2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \,{\left (245 \, a^{2} \cos \left (d x + c\right )^{2} - 160 \, a^{2} - 7 \,{\left (15 \, a^{2} \cos \left (d x + c\right )^{2} - 32 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{1536 \,{\left (d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )^{2}\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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