Optimal. Leaf size=294 \[ -\frac{259 \sin (c+d x)}{192 a^2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{7 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{a^{7/2} d}+\frac{189 \sin (c+d x)}{64 a^3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{637 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{64 \sqrt{2} a^{7/2} d}-\frac{7 \sin (c+d x)}{16 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}} \]
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Rubi [A] time = 0.816935, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {4222, 2765, 2977, 2983, 2982, 2782, 205, 2774, 216} \[ -\frac{259 \sin (c+d x)}{192 a^2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{7 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{a^{7/2} d}+\frac{189 \sin (c+d x)}{64 a^3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{637 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{64 \sqrt{2} a^{7/2} d}-\frac{7 \sin (c+d x)}{16 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 4222
Rule 2765
Rule 2977
Rule 2983
Rule 2982
Rule 2782
Rule 205
Rule 2774
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{9}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx\\ &=-\frac{\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{7}{2}}(c+d x)}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (\frac{7 a}{2}-7 a \cos (c+d x)\right )}{(a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2}\\ &=-\frac{\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{7}{2}}(c+d x)}-\frac{7 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (\frac{105 a^2}{4}-\frac{77}{2} a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4}\\ &=-\frac{\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{7}{2}}(c+d x)}-\frac{7 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)}-\frac{259 \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\cos (c+d x)} \left (\frac{777 a^3}{8}-\frac{567}{4} a^3 \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{48 a^6}\\ &=-\frac{\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{7}{2}}(c+d x)}-\frac{7 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)}-\frac{259 \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{189 \sin (c+d x)}{64 a^3 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{567 a^4}{8}+168 a^4 \cos (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{48 a^7}\\ &=-\frac{\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{7}{2}}(c+d x)}-\frac{7 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)}-\frac{259 \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{189 \sin (c+d x)}{64 a^3 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{\left (7 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx}{2 a^4}+\frac{\left (637 \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}{128 a^3}\\ &=-\frac{\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{7}{2}}(c+d x)}-\frac{7 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)}-\frac{259 \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{189 \sin (c+d x)}{64 a^3 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{\left (7 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a}}} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{a^4 d}-\frac{\left (637 \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 )}{64 a^2 d}\\ &=-\frac{7 \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{a^{7/2} d}+\frac{637 \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)}}{64 \sqrt{2} a^{7/2} d}-\frac{\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{7}{2}}(c+d x)}-\frac{7 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)}-\frac{259 \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{189 \sin (c+d x)}{64 a^3 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 3.515, size = 460, normalized size = 1.56 \[ \frac{e^{-\frac{1}{2} i (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left (672 i \sqrt{2} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^7\left (\frac{1}{2} (c+d x)\right ) \sinh ^{-1}\left (e^{i (c+d x)}\right )-\frac{1}{64} i e^{-4 i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \cos \left (\frac{1}{2} (c+d x)\right ) \left (\sqrt{2} \left (-1003 e^{i (c+d x)}-2169 e^{2 i (c+d x)}-2297 e^{3 i (c+d x)}-779 e^{4 i (c+d x)}+779 e^{5 i (c+d x)}+2297 e^{6 i (c+d x)}+2169 e^{7 i (c+d x)}+1003 e^{8 i (c+d x)}+96 e^{9 i (c+d x)}+672 e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left (1+e^{i (c+d x)}\right )^6 \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )-96\right )-1911 e^{i (c+d x)} \left (1+e^{i (c+d x)}\right )^6 \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )\right )\right )}{24 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.458, size = 472, normalized size = 1.6 \begin{align*} \text{result too large to display} \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{1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{7}{2}} \sec \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 5.24897, size = 817, normalized size = 2.78 \begin{align*} -\frac{1911 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\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 ) - 2688 \,{\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \sqrt{a} \arctan \left (\frac{\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 (192 \, \cos \left (d x + c\right )^{4} + 1099 \, \cos \left (d x + c\right )^{3} + 1442 \, \cos \left (d x + c\right )^{2} + 567 \, \cos \left (d x + c\right )\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{384 \,{\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{7}{2}} \sec \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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