Optimal. Leaf size=279 \[ \frac{4 a^3 (121 A+95 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{4 a^3 (221 A+175 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{195 d}+\frac{40 a^3 (143 A+118 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d}+\frac{4 a^3 (221 A+175 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{585 d}+\frac{2 (143 A+145 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{1287 d}+\frac{4 a^3 (121 A+95 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{12 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{143 a d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d} \]
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Rubi [A] time = 0.656285, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {3046, 2976, 2968, 3023, 2748, 2635, 2641, 2639} \[ \frac{4 a^3 (121 A+95 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{4 a^3 (221 A+175 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{195 d}+\frac{40 a^3 (143 A+118 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d}+\frac{4 a^3 (221 A+175 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{585 d}+\frac{2 (143 A+145 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{1287 d}+\frac{4 a^3 (121 A+95 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{12 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{143 a d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d} \]
Antiderivative was successfully verified.
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Rule 3046
Rule 2976
Rule 2968
Rule 3023
Rule 2748
Rule 2635
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac{2 \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (\frac{1}{2} a (13 A+5 C)+3 a C \cos (c+d x)\right ) \, dx}{13 a}\\ &=\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac{12 C \cos ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac{4 \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \left (\frac{1}{4} a^2 (143 A+85 C)+\frac{1}{4} a^2 (143 A+145 C) \cos (c+d x)\right ) \, dx}{143 a}\\ &=\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac{12 C \cos ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac{2 (143 A+145 C) \cos ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d}+\frac{8 \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x)) \left (\frac{1}{4} a^3 (1001 A+745 C)+\frac{5}{2} a^3 (143 A+118 C) \cos (c+d x)\right ) \, dx}{1287 a}\\ &=\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac{12 C \cos ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac{2 (143 A+145 C) \cos ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d}+\frac{8 \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{1}{4} a^4 (1001 A+745 C)+\left (\frac{5}{2} a^4 (143 A+118 C)+\frac{1}{4} a^4 (1001 A+745 C)\right ) \cos (c+d x)+\frac{5}{2} a^4 (143 A+118 C) \cos ^2(c+d x)\right ) \, dx}{1287 a}\\ &=\frac{40 a^3 (143 A+118 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac{12 C \cos ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac{2 (143 A+145 C) \cos ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d}+\frac{16 \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{117}{8} a^4 (121 A+95 C)+\frac{77}{8} a^4 (221 A+175 C) \cos (c+d x)\right ) \, dx}{9009 a}\\ &=\frac{40 a^3 (143 A+118 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac{12 C \cos ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac{2 (143 A+145 C) \cos ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d}+\frac{1}{77} \left (2 a^3 (121 A+95 C)\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{117} \left (2 a^3 (221 A+175 C)\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx\\ &=\frac{4 a^3 (121 A+95 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{4 a^3 (221 A+175 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{585 d}+\frac{40 a^3 (143 A+118 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac{12 C \cos ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac{2 (143 A+145 C) \cos ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d}+\frac{1}{231} \left (2 a^3 (121 A+95 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{195} \left (2 a^3 (221 A+175 C)\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{4 a^3 (221 A+175 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{195 d}+\frac{4 a^3 (121 A+95 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{4 a^3 (121 A+95 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{4 a^3 (221 A+175 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{585 d}+\frac{40 a^3 (143 A+118 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac{12 C \cos ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac{2 (143 A+145 C) \cos ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d}\\ \end{align*}
Mathematica [C] time = 6.33281, size = 1028, normalized size = 3.68 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.099, size = 464, normalized size = 1.7 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \cos \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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C a^{3} \cos \left (d x + c\right )^{6} + 3 \, C a^{3} \cos \left (d x + c\right )^{5} +{\left (A + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} +{\left (3 \, A + C\right )} a^{3} \cos \left (d x + c\right )^{3} + 3 \, A a^{3} \cos \left (d x + c\right )^{2} + A a^{3} \cos \left (d x + c\right )\right )} \sqrt{\cos \left (d x + c\right )}, 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{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \cos \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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