Optimal. Leaf size=233 \[ \frac{a^{5/2} (304 A+200 B+163 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{64 d}+\frac{a^3 (432 A+392 B+299 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (16 A+24 B+17 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{32 d}+\frac{a (8 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{24 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}{4 d} \]
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Rubi [A] time = 0.78535, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3045, 2976, 2981, 2774, 216} \[ \frac{a^{5/2} (304 A+200 B+163 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{64 d}+\frac{a^3 (432 A+392 B+299 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (16 A+24 B+17 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{32 d}+\frac{a (8 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{24 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}{4 d} \]
Antiderivative was successfully verified.
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Rule 3045
Rule 2976
Rule 2981
Rule 2774
Rule 216
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx &=\frac{C \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac{\int \frac{(a+a \cos (c+d x))^{5/2} \left (\frac{1}{2} a (8 A+C)+\frac{1}{2} a (8 B+5 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{4 a}\\ &=\frac{a (8 B+5 C) \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac{C \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac{\int \frac{(a+a \cos (c+d x))^{3/2} \left (\frac{1}{4} a^2 (48 A+8 B+11 C)+\frac{3}{4} a^2 (16 A+24 B+17 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{12 a}\\ &=\frac{a^2 (16 A+24 B+17 C) \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{32 d}+\frac{a (8 B+5 C) \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac{C \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac{\int \frac{\sqrt{a+a \cos (c+d x)} \left (\frac{1}{8} a^3 (240 A+104 B+95 C)+\frac{1}{8} a^3 (432 A+392 B+299 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{24 a}\\ &=\frac{a^3 (432 A+392 B+299 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (16 A+24 B+17 C) \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{32 d}+\frac{a (8 B+5 C) \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac{C \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac{1}{128} \left (a^2 (304 A+200 B+163 C)\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{a^3 (432 A+392 B+299 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (16 A+24 B+17 C) \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{32 d}+\frac{a (8 B+5 C) \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac{C \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{4 d}-\frac{\left (a^2 (304 A+200 B+163 C)\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 )}{64 d}\\ &=\frac{a^{5/2} (304 A+200 B+163 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{64 d}+\frac{a^3 (432 A+392 B+299 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (16 A+24 B+17 C) \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{32 d}+\frac{a (8 B+5 C) \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac{C \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 1.06692, size = 146, normalized size = 0.63 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} \left (3 \sqrt{2} (304 A+200 B+163 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )+2 \sin \left (\frac{1}{2} (c+d x)\right ) \sqrt{\cos (c+d x)} ((96 A+272 B+362 C) \cos (c+d x)+528 A+4 (8 B+23 C) \cos (2 (c+d x))+632 B+12 C \cos (3 (c+d x))+581 C)\right )}{384 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.124, size = 625, normalized size = 2.7 \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(-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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Fricas [A] time = 5.81653, size = 520, normalized size = 2.23 \begin{align*} \frac{{\left (48 \, C a^{2} \cos \left (d x + c\right )^{3} + 8 \,{\left (8 \, B + 23 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \,{\left (48 \, A + 136 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right ) + 3 \,{\left (176 \, A + 200 \, B + 163 \, C\right )} 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 ) - 3 \,{\left ({\left (304 \, A + 200 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right ) +{\left (304 \, A + 200 \, B + 163 \, C\right )} a^{2}\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 )}{192 \,{\left (d \cos \left (d x + c\right ) + 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{{\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{5}{2}}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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