Optimal. Leaf size=103 \[ \frac{3 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{16 \sqrt{2} d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{4 d}+\frac{3 a \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{16 d} \]
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Rubi [A] time = 0.172129, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2675, 2667, 63, 206} \[ \frac{3 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{16 \sqrt{2} d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{4 d}+\frac{3 a \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{16 d} \]
Antiderivative was successfully verified.
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Rule 2675
Rule 2667
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \sec ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=\frac{\sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{4 d}+\frac{1}{8} (3 a) \int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=\frac{3 a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{16 d}+\frac{\sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{4 d}+\frac{1}{32} \left (3 a^2\right ) \int \sec (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{3 a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{16 d}+\frac{\sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{4 d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{a+x}} \, dx,x,a \sin (c+d x)\right )}{32 d}\\ &=\frac{3 a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{16 d}+\frac{\sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{4 d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+a \sin (c+d x)}\right )}{16 d}\\ &=\frac{3 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{16 \sqrt{2} d}+\frac{3 a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{16 d}+\frac{\sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{4 d}\\ \end{align*}
Mathematica [A] time = 0.277936, size = 110, normalized size = 1.07 \[ \frac{2 a^2 (7-3 \sin (c+d x)) \sqrt{a (\sin (c+d x)+1)}+3 \sqrt{2} a^{5/2} \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^4 \tanh ^{-1}\left (\frac{\sqrt{a (\sin (c+d x)+1)}}{\sqrt{2} \sqrt{a}}\right )}{32 d (\sin (c+d x)-1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.164, size = 107, normalized size = 1. \begin{align*} -2\,{\frac{{a}^{5}}{d} \left ( -1/8\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }}{a \left ( a\sin \left ( dx+c \right ) -a \right ) ^{2}}}-3/8\,{\frac{1}{a} \left ( -1/4\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }}{a \left ( a\sin \left ( dx+c \right ) -a \right ) }}+1/8\,{\frac{\sqrt{2}}{{a}^{3/2}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) } \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 = 1.74697, size = 387, normalized size = 3.76 \begin{align*} \frac{3 \,{\left (\sqrt{2} a^{2} \cos \left (d x + c\right )^{2} + 2 \, \sqrt{2} a^{2} \sin \left (d x + c\right ) - 2 \, \sqrt{2} a^{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 (3 \, a^{2} \sin \left (d x + c\right ) - 7 \, a^{2}\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{64 \,{\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 [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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