Optimal. Leaf size=181 \[ \frac{11 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac{119 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{119 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.322797, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2765, 2977, 2748, 2639, 2635, 2641} \[ \frac{11 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac{119 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{119 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2765
Rule 2977
Rule 2748
Rule 2639
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx &=-\frac{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (\frac{7 a}{2}-\frac{13}{2} a \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{2 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2}-\frac{\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (25 a^2-\frac{69}{2} a^2 \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{2 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2}-\frac{119 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac{\int \sqrt{\cos (c+d x)} \left (\frac{357 a^3}{4}-\frac{495}{4} a^3 \cos (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{2 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2}-\frac{119 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac{119 \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}+\frac{33 \int \cos ^{\frac{3}{2}}(c+d x) \, dx}{4 a^3}\\ &=-\frac{119 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{11 \sqrt{\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{2 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2}-\frac{119 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{11 \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{4 a^3}\\ &=-\frac{119 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{11 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac{11 \sqrt{\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{2 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2}-\frac{119 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 1.86659, size = 369, normalized size = 2.04 \[ \frac{\cos ^6\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\csc (c) \sqrt{\cos (c+d x)} \left (1961 \cos \left (\frac{1}{2} (c-d x)\right )+1609 \cos \left (\frac{1}{2} (3 c+d x)\right )+1165 \cos \left (\frac{1}{2} (c+3 d x)\right )+620 \cos \left (\frac{1}{2} (5 c+3 d x)\right )+292 \cos \left (\frac{1}{2} (3 c+5 d x)\right )+65 \cos \left (\frac{1}{2} (7 c+5 d x)\right )+5 \cos \left (\frac{1}{2} (5 c+7 d x)\right )-5 \cos \left (\frac{1}{2} (9 c+7 d x)\right )\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right )}{12 d}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \left (119 \left (-1+e^{2 i c}\right ) \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )+55 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )+119 \left (1+e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) d \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{5 a^3 (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.313, size = 283, normalized size = 1.6 \begin{align*} -{\frac{1}{60\,{a}^{3}d}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 160\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+468\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+330\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+714\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -1058\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+474\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-47\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+3 \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (d x + c\right )^{\frac{9}{2}}}{a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}}, x\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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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