Optimal. Leaf size=184 \[ \frac{2 a^2 (4 A+6 B+5 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+126 B+175 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+7 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.579203, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.089, Rules used = {3043, 2975, 2980, 2771} \[ \frac{2 a^2 (4 A+6 B+5 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+126 B+175 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+7 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3043
Rule 2975
Rule 2980
Rule 2771
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{2 A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \int \frac{(a+a \cos (c+d x))^{3/2} \left (\frac{1}{2} a (3 A+7 B)+\frac{1}{2} a (2 A+7 C) \cos (c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx}{7 a}\\ &=\frac{2 a (3 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 \cos (c+d x))^{3/2} \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 \int \frac{\sqrt{a+a \cos (c+d x)} \left (\frac{7}{4} a^2 (4 A+6 B+5 C)+\frac{1}{4} a^2 (16 A+14 B+35 C) \cos (c+d x)\right )}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{35 a}\\ &=\frac{2 a^2 (4 A+6 B+5 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a (3 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 \cos (c+d x))^{3/2} \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{1}{105} (a (104 A+126 B+175 C)) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (4 A+6 B+5 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (104 A+126 B+175 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}+\frac{2 a (3 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 \cos (c+d x))^{3/2} \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.703005, size = 122, normalized size = 0.66 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} ((468 A+462 B+525 C) \cos (c+d x)+2 (52 A+63 B+35 C) \cos (2 (c+d x))+104 A \cos (3 (c+d x))+164 A+126 B \cos (3 (c+d x))+126 B+175 C \cos (3 (c+d x))+70 C)}{210 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.114, size = 131, normalized size = 0.7 \begin{align*} -{\frac{2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 104\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+126\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+175\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+52\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+63\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+35\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+39\,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 = 1.75735, size = 815, normalized size = 4.43 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04511, size = 305, normalized size = 1.66 \begin{align*} \frac{2 \,{\left ({\left (104 \, A + 126 \, B + 175 \, C\right )} a \cos \left (d x + c\right )^{3} +{\left (52 \, A + 63 \, B + 35 \, C\right )} a \cos \left (d x + c\right )^{2} + 3 \,{\left (13 \, A + 7 \, B\right )} a \cos \left (d x + c\right ) + 15 \, A a\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 (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{3}{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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