Optimal. Leaf size=237 \[ \frac{95 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{299 \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{163 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{17 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.640021, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {4222, 2766, 2978, 2984, 12, 2782, 205} \[ \frac{95 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{299 \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{163 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{17 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 4222
Rule 2766
Rule 2978
Rule 2984
Rule 12
Rule 2782
Rule 205
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx\\ &=-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{11 a}{2}-3 a \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{17 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{95 a^2}{4}-17 a^2 \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{8 a^4}\\ &=-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{17 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{95 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{48 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{299 a^3}{8}+\frac{95}{4} a^3 \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{12 a^5}\\ &=-\frac{299 \sqrt{\sec (c+d x)} \sin (c+d x)}{48 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{17 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{95 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{48 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{489 a^4}{16 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{6 a^6}\\ &=-\frac{299 \sqrt{\sec (c+d x)} \sin (c+d x)}{48 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{17 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{95 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{48 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (163 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{32 a^2}\\ &=-\frac{299 \sqrt{\sec (c+d x)} \sin (c+d x)}{48 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{17 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{95 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{48 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{\left (163 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{2 a^2+a x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\right )}{16 a d}\\ &=\frac{163 \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{16 \sqrt{2} a^{5/2} d}-\frac{299 \sqrt{\sec (c+d x)} \sin (c+d x)}{48 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{17 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{95 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{48 a^2 d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 7.98176, size = 641, normalized size = 2.7 \[ -\frac{\left (\frac{1}{1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}\right )^{7/2} \cot ^5\left (\frac{c}{2}+\frac{d x}{2}\right ) \csc ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right ) \left (640 \sin ^{12}\left (\frac{c}{2}+\frac{d x}{2}\right ) \cos ^8\left (\frac{1}{2} (c+d x)\right ) \, _5F_4\left (2,2,2,2,\frac{7}{2};1,1,1,\frac{13}{2};\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )-1}\right )-1280 \left (5 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )-6\right ) \sin ^{12}\left (\frac{c}{2}+\frac{d x}{2}\right ) \cos ^6\left (\frac{1}{2} (c+d x)\right ) \, _4F_3\left (2,2,2,\frac{7}{2};1,1,\frac{13}{2};\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )-1}\right )+33 \left (1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )\right )^3 \sqrt{\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )-1}} \left (\sqrt{\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )-1}} \left (4344400 \sin ^{14}\left (\frac{c}{2}+\frac{d x}{2}\right )-26448512 \sin ^{12}\left (\frac{c}{2}+\frac{d x}{2}\right )+68243596 \sin ^{10}\left (\frac{c}{2}+\frac{d x}{2}\right )-96281836 \sin ^8\left (\frac{c}{2}+\frac{d x}{2}\right )+79946462 \sin ^6\left (\frac{c}{2}+\frac{d x}{2}\right )-38990350 \sin ^4\left (\frac{c}{2}+\frac{d x}{2}\right )+10333785 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )-1148175\right )-105 \left (33208 \sin ^{10}\left (\frac{c}{2}+\frac{d x}{2}\right )-140732 \sin ^8\left (\frac{c}{2}+\frac{d x}{2}\right )+234156 \sin ^6\left (\frac{c}{2}+\frac{d x}{2}\right )-188110 \sin ^4\left (\frac{c}{2}+\frac{d x}{2}\right )+72902 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )-10935\right ) \cos ^4\left (\frac{1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sqrt{\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )-1}}\right )\right )\right )}{41580 d (a (\cos (c+d x)+1))^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.446, size = 316, normalized size = 1.3 \begin{align*}{\frac{\cos \left ( dx+c \right ) \sqrt{2}}{96\,d{a}^{3}\sin \left ( dx+c \right ) \left ( 1+\cos \left ( dx+c \right ) \right ) ^{2}} \left ( -489\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) \arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{3/2}-1467\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) \arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{3/2}-1467\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{3/2}+299\,\sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}-489\,\sin \left ( dx+c \right ) \arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{3/2}+204\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{2}-343\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{2}-192\,\cos \left ( dx+c \right ) \sqrt{2}+32\,\sqrt{2} \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{5}{2}}}\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \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 = 1.90572, size = 544, normalized size = 2.3 \begin{align*} -\frac{489 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt{a} \arctan \left (\frac{\sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )}}{\sqrt{a} \sin \left (d x + c\right )}\right ) + \frac{2 \,{\left (299 \, \cos \left (d x + c\right )^{3} + 503 \, \cos \left (d x + c\right )^{2} + 160 \, \cos \left (d x + c\right ) - 32\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{96 \,{\left (a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d \cos \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*} \int \frac{\sec \left (d x + c\right )^{\frac{5}{2}}}{{\left (a \cos \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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