Optimal. Leaf size=303 \[ \frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{128 d}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (10 B+3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{5}{2}}(c+d x)} \]
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Rubi [A] time = 0.90757, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {4221, 3045, 2976, 2981, 2770, 2774, 216} \[ \frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{128 d}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (10 B+3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3045
Rule 2976
Rule 2981
Rule 2770
Rule 2774
Rule 216
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 )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (\frac{5}{2} a (2 A+C)+\frac{1}{2} a (10 B+3 C) \cos (c+d x)\right ) \, dx}{5 a}\\ &=\frac{a (10 B+3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \left (\frac{5}{4} a^2 (16 A+10 B+11 C)+\frac{1}{4} a^2 (80 A+90 B+67 C) \cos (c+d x)\right ) \, dx}{20 a}\\ &=\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x)}+\frac{a (10 B+3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{1}{96} \left (a (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x)}+\frac{a (10 B+3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{128} \left (a (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x)}+\frac{a (10 B+3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{1}{256} \left (a (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x)}+\frac{a (10 B+3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{\left (a (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\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 )}{128 d}\\ &=\frac{a^{3/2} (176 A+150 B+133 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{128 d}+\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x)}+\frac{a (10 B+3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.95439, size = 190, normalized size = 0.63 \[ \frac{a \sqrt{\cos (c+d x)} \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left (15 \sqrt{2} (176 A+150 B+133 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )+2 \sin \left (\frac{1}{2} (c+d x)\right ) \sqrt{\cos (c+d x)} (2 (880 A+930 B+1007 C) \cos (c+d x)+4 (80 A+150 B+181 C) \cos (2 (c+d x))+2960 A+120 B \cos (3 (c+d x))+2850 B+228 C \cos (3 (c+d x))+48 C \cos (4 (c+d x))+2671 C)\right )}{3840 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.222, size = 589, normalized size = 1.9 \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 = 4.92131, size = 595, normalized size = 1.96 \begin{align*} -\frac{15 \,{\left ({\left (176 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right ) +{\left (176 \, A + 150 \, B + 133 \, C\right )} a\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 ) - \frac{{\left (384 \, C a \cos \left (d x + c\right )^{5} + 48 \,{\left (10 \, B + 19 \, C\right )} a \cos \left (d x + c\right )^{4} + 8 \,{\left (80 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right )^{3} + 10 \,{\left (176 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right )^{2} + 15 \,{\left (176 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right )\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{1920 \,{\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{3}{2}}}{\sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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