Optimal. Leaf size=167 \[ \frac{35 \sec (c+d x)}{96 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{35 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{128 \sqrt{2} a^{5/2} d}-\frac{35 \cos (c+d x)}{128 a d (a \sin (c+d x)+a)^{3/2}}-\frac{7 \sec (c+d x)}{48 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec (c+d x)}{6 d (a \sin (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.230418, antiderivative size = 167, 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 = {2681, 2687, 2650, 2649, 206} \[ \frac{35 \sec (c+d x)}{96 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{35 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{128 \sqrt{2} a^{5/2} d}-\frac{35 \cos (c+d x)}{128 a d (a \sin (c+d x)+a)^{3/2}}-\frac{7 \sec (c+d x)}{48 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec (c+d x)}{6 d (a \sin (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2681
Rule 2687
Rule 2650
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=-\frac{\sec (c+d x)}{6 d (a+a \sin (c+d x))^{5/2}}+\frac{7 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{12 a}\\ &=-\frac{\sec (c+d x)}{6 d (a+a \sin (c+d x))^{5/2}}-\frac{7 \sec (c+d x)}{48 a d (a+a \sin (c+d x))^{3/2}}+\frac{35 \int \frac{\sec ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{96 a^2}\\ &=-\frac{\sec (c+d x)}{6 d (a+a \sin (c+d x))^{5/2}}-\frac{7 \sec (c+d x)}{48 a d (a+a \sin (c+d x))^{3/2}}+\frac{35 \sec (c+d x)}{96 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{35 \int \frac{1}{(a+a \sin (c+d x))^{3/2}} \, dx}{64 a}\\ &=-\frac{\sec (c+d x)}{6 d (a+a \sin (c+d x))^{5/2}}-\frac{35 \cos (c+d x)}{128 a d (a+a \sin (c+d x))^{3/2}}-\frac{7 \sec (c+d x)}{48 a d (a+a \sin (c+d x))^{3/2}}+\frac{35 \sec (c+d x)}{96 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{35 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{256 a^2}\\ &=-\frac{\sec (c+d x)}{6 d (a+a \sin (c+d x))^{5/2}}-\frac{35 \cos (c+d x)}{128 a d (a+a \sin (c+d x))^{3/2}}-\frac{7 \sec (c+d x)}{48 a d (a+a \sin (c+d x))^{3/2}}+\frac{35 \sec (c+d x)}{96 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{35 \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 )}{128 a^2 d}\\ &=-\frac{35 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{128 \sqrt{2} a^{5/2} d}-\frac{\sec (c+d x)}{6 d (a+a \sin (c+d x))^{5/2}}-\frac{35 \cos (c+d x)}{128 a d (a+a \sin (c+d x))^{3/2}}-\frac{7 \sec (c+d x)}{48 a d (a+a \sin (c+d x))^{3/2}}+\frac{35 \sec (c+d x)}{96 a^2 d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.395521, size = 284, normalized size = 1.7 \[ \frac{\frac{48 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}-57 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4+114 \sin \left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3-44 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2+88 \sin \left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{64 \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}+(105+105 i) (-1)^{3/4} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5 \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )-32}{384 d (a (\sin (c+d x)+1))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.187, size = 266, normalized size = 1.6 \begin{align*} -{\frac{1}{768\, \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) d} \left ( \left ( 210\,{a}^{7/2}-105\,\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \right ) \sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+ \left ( -448\,{a}^{7/2}+420\,\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \right ) \sin \left ( dx+c \right ) + \left ( 490\,{a}^{7/2}-315\,\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}-320\,{a}^{7/2}+420\,\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \right ){a}^{-{\frac{11}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.44893, size = 755, normalized size = 4.52 \begin{align*} \frac{105 \, \sqrt{2}{\left (3 \, \cos \left (d x + c\right )^{3} +{\left (\cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 4 \, \cos \left (d x + c\right )\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a}{\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 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 (245 \, \cos \left (d x + c\right )^{2} + 7 \,{\left (15 \, \cos \left (d x + c\right )^{2} - 32\right )} \sin \left (d x + c\right ) - 160\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{1536 \,{\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right ) +{\left (a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right )\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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