Optimal. Leaf size=194 \[ \frac{2 a \left (7 a^2+15 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 b \left (27 a^2+7 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b \left (27 a^2+7 b^2\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a \left (7 a^2+15 b^2\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}+\frac{40 a b^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d} \]
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Rubi [A] time = 0.22039, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2793, 3023, 2748, 2635, 2641, 2639} \[ \frac{2 a \left (7 a^2+15 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 b \left (27 a^2+7 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b \left (27 a^2+7 b^2\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a \left (7 a^2+15 b^2\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}+\frac{40 a b^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d} \]
Antiderivative was successfully verified.
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Rule 2793
Rule 3023
Rule 2748
Rule 2635
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \, dx &=\frac{2 b^2 \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac{2}{9} \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{1}{2} a \left (9 a^2+5 b^2\right )+\frac{1}{2} b \left (27 a^2+7 b^2\right ) \cos (c+d x)+10 a b^2 \cos ^2(c+d x)\right ) \, dx\\ &=\frac{40 a b^2 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b^2 \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac{4}{63} \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{9}{4} a \left (7 a^2+15 b^2\right )+\frac{7}{4} b \left (27 a^2+7 b^2\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{40 a b^2 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b^2 \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac{1}{9} \left (b \left (27 a^2+7 b^2\right )\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{7} \left (a \left (7 a^2+15 b^2\right )\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 a \left (7 a^2+15 b^2\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 b \left (27 a^2+7 b^2\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{40 a b^2 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b^2 \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac{1}{15} \left (b \left (27 a^2+7 b^2\right )\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (a \left (7 a^2+15 b^2\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b \left (27 a^2+7 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a \left (7 a^2+15 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 a \left (7 a^2+15 b^2\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 b \left (27 a^2+7 b^2\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{40 a b^2 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b^2 \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 0.91535, size = 137, normalized size = 0.71 \[ \frac{60 \left (7 a^3+15 a b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+84 \left (27 a^2 b+7 b^3\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt{\cos (c+d x)} \left (7 b \left (108 a^2+43 b^2\right ) \cos (c+d x)+5 \left (84 a^3+54 a b^2 \cos (2 (c+d x))+234 a b^2+7 b^3 \cos (3 (c+d x))\right )\right )}{630 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.132, size = 470, normalized size = 2.4 \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 (b \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 (b^{3} \cos \left (d x + c\right )^{4} + 3 \, a b^{2} \cos \left (d x + c\right )^{3} + 3 \, a^{2} b \cos \left (d x + c\right )^{2} + 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 (b \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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