Optimal. Leaf size=234 \[ \frac{2 a \left (5 a^2+21 b^2\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 b \left (9 a^2+5 b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a \left (5 a^2+21 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 b \left (9 a^2+5 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}+\frac{32 a^2 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d} \]
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Rubi [A] time = 0.268907, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {3238, 3842, 4047, 3768, 3771, 2641, 4046, 2639} \[ \frac{2 a \left (5 a^2+21 b^2\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 b \left (9 a^2+5 b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a \left (5 a^2+21 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 b \left (9 a^2+5 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}+\frac{32 a^2 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d} \]
Antiderivative was successfully verified.
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Rule 3238
Rule 3842
Rule 4047
Rule 3768
Rule 3771
Rule 2641
Rule 4046
Rule 2639
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x) \, dx &=\int \sec ^{\frac{3}{2}}(c+d x) (b+a \sec (c+d x))^3 \, dx\\ &=\frac{2 a^2 \sec ^{\frac{5}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{7 d}+\frac{2}{7} \int \sec ^{\frac{3}{2}}(c+d x) \left (\frac{1}{2} b \left (3 a^2+7 b^2\right )+\frac{1}{2} a \left (5 a^2+21 b^2\right ) \sec (c+d x)+8 a^2 b \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 a^2 \sec ^{\frac{5}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{7 d}+\frac{2}{7} \int \sec ^{\frac{3}{2}}(c+d x) \left (\frac{1}{2} b \left (3 a^2+7 b^2\right )+8 a^2 b \sec ^2(c+d x)\right ) \, dx+\frac{1}{7} \left (a \left (5 a^2+21 b^2\right )\right ) \int \sec ^{\frac{5}{2}}(c+d x) \, dx\\ &=\frac{2 a \left (5 a^2+21 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{32 a^2 b \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 a^2 \sec ^{\frac{5}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{7 d}+\frac{1}{5} \left (b \left (9 a^2+5 b^2\right )\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{21} \left (a \left (5 a^2+21 b^2\right )\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 b \left (9 a^2+5 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a \left (5 a^2+21 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{32 a^2 b \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 a^2 \sec ^{\frac{5}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{7 d}-\frac{1}{5} \left (b \left (9 a^2+5 b^2\right )\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{21} \left (a \left (5 a^2+21 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 (5 a^2+21 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)}}{21 d}+\frac{2 b \left (9 a^2+5 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a \left (5 a^2+21 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{32 a^2 b \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 a^2 \sec ^{\frac{5}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{7 d}-\frac{1}{5} \left (b \left (9 a^2+5 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 (9 a^2+5 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)}}{5 d}+\frac{2 a \left (5 a^2+21 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)}}{21 d}+\frac{2 b \left (9 a^2+5 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a \left (5 a^2+21 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{32 a^2 b \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 a^2 \sec ^{\frac{5}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 1.079, size = 191, normalized size = 0.82 \[ \frac{\sec ^{\frac{7}{2}}(c+d x) \left (10 a \left (5 a^2+21 b^2\right ) \cos ^{\frac{7}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-42 b \left (9 a^2+5 b^2\right ) \cos ^{\frac{7}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+63 a^2 b \sin (2 (c+d x))+378 a^2 b \sin (c+d x) \cos ^3(c+d x)+30 a^3 \sin (c+d x)+50 a^3 \sin (c+d x) \cos ^2(c+d x)+210 a b^2 \sin (c+d x) \cos ^2(c+d x)+210 b^3 \sin (c+d x) \cos ^3(c+d x)\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 10.832, size = 847, normalized size = 3.6 \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{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 (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{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 (b \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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