Optimal. Leaf size=187 \[ \frac{2 a^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{6 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{52 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{28 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{52 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{28 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d} \]
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Rubi [A] time = 0.233406, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3238, 3791, 3768, 3771, 2639, 2641} \[ \frac{2 a^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{6 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{52 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{28 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{52 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{28 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d} \]
Antiderivative was successfully verified.
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Rule 3238
Rule 3791
Rule 3768
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x) \, dx &=\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \, dx\\ &=\int \left (a^3 \sec ^{\frac{3}{2}}(c+d x)+3 a^3 \sec ^{\frac{5}{2}}(c+d x)+3 a^3 \sec ^{\frac{7}{2}}(c+d x)+a^3 \sec ^{\frac{9}{2}}(c+d x)\right ) \, dx\\ &=a^3 \int \sec ^{\frac{3}{2}}(c+d x) \, dx+a^3 \int \sec ^{\frac{9}{2}}(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^{\frac{5}{2}}(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^{\frac{7}{2}}(c+d x) \, dx\\ &=\frac{2 a^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 a^3 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{d}+\frac{6 a^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a^3 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{7} \left (5 a^3\right ) \int \sec ^{\frac{5}{2}}(c+d x) \, dx-a^3 \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+a^3 \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{5} \left (9 a^3\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{28 a^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{52 a^3 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{6 a^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a^3 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{21} \left (5 a^3\right ) \int \sqrt{\sec (c+d x)} \, dx-\frac{1}{5} \left (9 a^3\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\left (a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\left (a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 a^3 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{d}+\frac{2 a^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{d}+\frac{28 a^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{52 a^3 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{6 a^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a^3 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{21} \left (5 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\frac{1}{5} \left (9 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{28 a^3 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 d}+\frac{52 a^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 d}+\frac{28 a^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{52 a^3 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{6 a^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a^3 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [C] time = 2.71249, size = 279, normalized size = 1.49 \[ \frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \left (\sqrt{\sec (c+d x)} \left (294 \csc (c) \cos (d x)+(63 \cos (c+d x)+65 \cos (2 (c+d x))+80) \tan (c+d x) \sec ^2(c+d x)\right )-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left (147 \left (-1+e^{2 i c}\right ) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right )+65 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right )+147 \left (1+e^{2 i (c+d x)}\right )\right )}{-1+e^{2 i c}}\right )}{420 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.434, size = 439, 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 (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \sec \left (d x + c\right )^{\frac{9}{2}}, 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 (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \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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