Optimal. Leaf size=113 \[ -\frac{1}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}-\frac{1}{6 a d (a \sin (c+d x)+a)^{3/2}}-\frac{1}{5 d (a \sin (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.0886034, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2667, 51, 63, 206} \[ -\frac{1}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}-\frac{1}{6 a d (a \sin (c+d x)+a)^{3/2}}-\frac{1}{5 d (a \sin (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac{a \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{7/2}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{1}{5 d (a+a \sin (c+d x))^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{5/2}} \, dx,x,a \sin (c+d x)\right )}{2 d}\\ &=-\frac{1}{5 d (a+a \sin (c+d x))^{5/2}}-\frac{1}{6 a d (a+a \sin (c+d x))^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{4 a d}\\ &=-\frac{1}{5 d (a+a \sin (c+d x))^{5/2}}-\frac{1}{6 a d (a+a \sin (c+d x))^{3/2}}-\frac{1}{4 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{a+x}} \, dx,x,a \sin (c+d x)\right )}{8 a^2 d}\\ &=-\frac{1}{5 d (a+a \sin (c+d x))^{5/2}}-\frac{1}{6 a d (a+a \sin (c+d x))^{3/2}}-\frac{1}{4 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+a \sin (c+d x)}\right )}{4 a^2 d}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}-\frac{1}{5 d (a+a \sin (c+d x))^{5/2}}-\frac{1}{6 a d (a+a \sin (c+d x))^{3/2}}-\frac{1}{4 a^2 d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.0842239, size = 41, normalized size = 0.36 \[ -\frac{\, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right )}{5 d (a \sin (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.097, size = 88, normalized size = 0.8 \begin{align*} -2\,{\frac{a}{d} \left ( -1/16\,{\frac{\sqrt{2}}{{a}^{7/2}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) }+1/8\,{\frac{1}{{a}^{3}\sqrt{a+a\sin \left ( dx+c \right ) }}}+1/12\,{\frac{1}{{a}^{2} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{3/2}}}+1/10\,{\frac{1}{a \left ( a+a\sin \left ( dx+c \right ) \right ) ^{5/2}}} \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 = 2.47456, size = 452, normalized size = 4. \begin{align*} \frac{15 \, \sqrt{2}{\left (3 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 4\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 (15 \, \cos \left (d x + c\right )^{2} - 40 \, \sin \left (d x + c\right ) - 52\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{240 \,{\left (3 \, a^{3} d \cos \left (d x + c\right )^{2} - 4 \, a^{3} d +{\left (a^{3} d \cos \left (d x + c\right )^{2} - 4 \, a^{3} d\right )} \sin \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 [A] time = 1.12648, size = 130, normalized size = 1.15 \begin{align*} -\frac{1}{120} \, a{\left (\frac{15 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a} a^{3} d} + \frac{2 \,{\left (15 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2} + 10 \,{\left (a \sin \left (d x + c\right ) + a\right )} a + 12 \, a^{2}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{3} d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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