Optimal. Leaf size=160 \[ \frac{2 a \left (a^2+9 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 b \left (3 a^2-b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}{3 d}+\frac{16 a^2 b \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d} \]
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Rubi [A] time = 0.23123, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3238, 3842, 4047, 3771, 2641, 4046, 2639} \[ \frac{2 a \left (a^2+9 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 b \left (3 a^2-b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}{3 d}+\frac{16 a^2 b \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3238
Rule 3842
Rule 4047
Rule 3771
Rule 2641
Rule 4046
Rule 2639
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \, dx &=\int \frac{(b+a \sec (c+d x))^3}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a^2 \sqrt{\sec (c+d x)} (b+a \sec (c+d x)) \sin (c+d x)}{3 d}+\frac{2}{3} \int \frac{-\frac{1}{2} b \left (a^2-3 b^2\right )+\frac{1}{2} a \left (a^2+9 b^2\right ) \sec (c+d x)+4 a^2 b \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a^2 \sqrt{\sec (c+d x)} (b+a \sec (c+d x)) \sin (c+d x)}{3 d}+\frac{2}{3} \int \frac{-\frac{1}{2} b \left (a^2-3 b^2\right )+4 a^2 b \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (a \left (a^2+9 b^2\right )\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{16 a^2 b \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 a^2 \sqrt{\sec (c+d x)} (b+a \sec (c+d x)) \sin (c+d x)}{3 d}-\left (b \left (3 a^2-b^2\right )\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (a \left (a^2+9 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 a \left (a^2+9 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 d}+\frac{16 a^2 b \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 a^2 \sqrt{\sec (c+d x)} (b+a \sec (c+d x)) \sin (c+d x)}{3 d}-\left (b \left (3 a^2-b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 b \left (3 a^2-b^2\right ) \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 \left (a^2+9 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 d}+\frac{16 a^2 b \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 a^2 \sqrt{\sec (c+d x)} (b+a \sec (c+d x)) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.44409, size = 106, normalized size = 0.66 \[ \frac{\sec ^{\frac{3}{2}}(c+d x) \left (6 b \left (b^2-3 a^2\right ) \cos ^{\frac{3}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+a \left (2 \left (a^2+9 b^2\right ) \cos ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 a \sin (c+d x) (a+9 b \cos (c+d x))\right )\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 6.875, size = 631, normalized size = 3.9 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac{5}{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 )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}\right )} \sec \left (d x + c\right )^{\frac{5}{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 (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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