Optimal. Leaf size=181 \[ \frac{2 a^3 (10 A+11 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (10 A+7 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 (230 A+301 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.55189, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {2975, 2980, 2771} \[ \frac{2 a^3 (10 A+11 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (10 A+7 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 (230 A+301 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2975
Rule 2980
Rule 2771
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{2 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2}{7} \int \frac{(a+a \cos (c+d x))^{3/2} \left (\frac{1}{2} a (10 A+7 B)+\frac{1}{2} a (2 A+7 B) \cos (c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (10 A+7 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4}{35} \int \frac{\sqrt{a+a \cos (c+d x)} \left (\frac{7}{4} a^2 (10 A+11 B)+\frac{1}{4} a^2 (30 A+49 B) \cos (c+d x)\right )}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a^3 (10 A+11 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (10 A+7 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{1}{105} \left (a^2 (230 A+301 B)\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^3 (10 A+11 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a^3 (230 A+301 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (10 A+7 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.600312, size = 104, normalized size = 0.57 \[ \frac{a^2 \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} ((930 A+987 B) \cos (c+d x)+2 (115 A+98 B) \cos (2 (c+d x))+230 A \cos (3 (c+d x))+290 A+301 B \cos (3 (c+d x))+196 B)}{210 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.601, size = 111, normalized size = 0.6 \begin{align*} -{\frac{2\,{a}^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 230\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+301\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+115\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+98\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+60\,A\cos \left ( dx+c \right ) +21\,B\cos \left ( dx+c \right ) +15\,A \right ) }{105\,d\sin \left ( dx+c \right ) }\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) } \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.00511, size = 535, normalized size = 2.96 \begin{align*} \frac{8 \,{\left (\frac{7 \,{\left (\frac{15 \, \sqrt{2} a^{\frac{5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{35 \, \sqrt{2} a^{\frac{5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{28 \, \sqrt{2} a^{\frac{5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{8 \, \sqrt{2} a^{\frac{5}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )} B}{{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{7}{2}}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{7}{2}}} + \frac{5 \,{\left (\frac{21 \, \sqrt{2} a^{\frac{5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{56 \, \sqrt{2} a^{\frac{5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{63 \, \sqrt{2} a^{\frac{5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{36 \, \sqrt{2} a^{\frac{5}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{8 \, \sqrt{2} a^{\frac{5}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}\right )} A{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{9}{2}}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{9}{2}}{\left (\frac{2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )}}\right )}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26924, size = 297, normalized size = 1.64 \begin{align*} \frac{2 \,{\left ({\left (230 \, A + 301 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} +{\left (115 \, A + 98 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 3 \,{\left (20 \, A + 7 \, B\right )} a^{2} \cos \left (d x + c\right ) + 15 \, A a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\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{{\left (B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\cos \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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