Optimal. Leaf size=254 \[ -\frac{259 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{7 \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{a^{7/2} d}+\frac{189 \sin (c+d x) \sqrt{\cos (c+d x)}}{64 a^3 d \sqrt{a \cos (c+d x)+a}}+\frac{637 \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{64 \sqrt{2} a^{7/2} d}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac{7 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.748577, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {2765, 2977, 2983, 2982, 2782, 205, 2774, 216} \[ -\frac{259 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{7 \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{a^{7/2} d}+\frac{189 \sin (c+d x) \sqrt{\cos (c+d x)}}{64 a^3 d \sqrt{a \cos (c+d x)+a}}+\frac{637 \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{64 \sqrt{2} a^{7/2} d}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac{7 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2765
Rule 2977
Rule 2983
Rule 2982
Rule 2782
Rule 205
Rule 2774
Rule 216
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx &=-\frac{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac{\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{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac{7 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac{\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{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac{7 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac{259 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac{\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{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac{7 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac{259 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac{189 \sqrt{\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt{a+a \cos (c+d x)}}-\frac{\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{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac{7 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac{259 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac{189 \sqrt{\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt{a+a \cos (c+d x)}}-\frac{7 \int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx}{2 a^4}+\frac{637 \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{128 a^3}\\ &=-\frac{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac{7 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac{259 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac{189 \sqrt{\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt{a+a \cos (c+d x)}}+\frac{7 \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{637 \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 )}{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 )}{64 \sqrt{2} a^{7/2} d}-\frac{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac{7 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac{259 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac{189 \sqrt{\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.72162, size = 448, normalized size = 1.76 \[ \frac{\sqrt{\cos (c+d x)} \cos ^7\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (\frac{16 \sin \left (\frac{c}{2}\right ) \cos \left (\frac{d x}{2}\right )}{d}+\frac{16 \cos \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )}{d}+\frac{\sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d}-\frac{15 \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{4 d}+\frac{523 \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{24 d}+\frac{\tan \left (\frac{c}{2}\right ) \sec ^5\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d}-\frac{15 \tan \left (\frac{c}{2}\right ) \sec ^3\left (\frac{c}{2}+\frac{d x}{2}\right )}{4 d}+\frac{523 \tan \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right )}{24 d}\right )}{(a (\cos (c+d x)+1))^{7/2}}+\frac{7 i e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \cos ^7\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (64 \sinh ^{-1}\left (e^{i (c+d x)}\right )+91 \sqrt{2} \tanh ^{-1}\left (\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )-64 \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )\right )}{8 \sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} (a (\cos (c+d x)+1))^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.408, size = 464, normalized size = 1.8 \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{\cos \left (d x + c\right )^{\frac{9}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 6.25466, size = 797, normalized size = 3.14 \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 ) - 2 \,{\left (192 \, \cos \left (d x + c\right )^{3} + 1099 \, \cos \left (d x + c\right )^{2} + 1442 \, \cos \left (d x + c\right ) + 567\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\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 )}{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{\cos \left (d x + c\right )^{\frac{9}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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