Optimal. Leaf size=189 \[ \frac{6 a \left (a^2+5 b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 b \left (a^2+b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}-\frac{6 a \left (a^2+5 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{8 a^2 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+b)}{5 d} \]
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Rubi [A] time = 0.237839, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 9, 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, 2639, 4046, 2641} \[ \frac{6 a \left (a^2+5 b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 b \left (a^2+b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}-\frac{6 a \left (a^2+5 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{8 a^2 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+b)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3238
Rule 3842
Rule 4047
Rule 3768
Rule 3771
Rule 2639
Rule 4046
Rule 2641
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x) \, dx &=\int \sqrt{\sec (c+d x)} (b+a \sec (c+d x))^3 \, dx\\ &=\frac{2 a^2 \sec ^{\frac{3}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{5 d}+\frac{2}{5} \int \sqrt{\sec (c+d x)} \left (\frac{1}{2} b \left (a^2+5 b^2\right )+\frac{3}{2} a \left (a^2+5 b^2\right ) \sec (c+d x)+6 a^2 b \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 a^2 \sec ^{\frac{3}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{5 d}+\frac{2}{5} \int \sqrt{\sec (c+d x)} \left (\frac{1}{2} b \left (a^2+5 b^2\right )+6 a^2 b \sec ^2(c+d x)\right ) \, dx+\frac{1}{5} \left (3 a \left (a^2+5 b^2\right )\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{6 a \left (a^2+5 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{8 a^2 b \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a^2 \sec ^{\frac{3}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{5 d}+\left (b \left (a^2+b^2\right )\right ) \int \sqrt{\sec (c+d x)} \, dx-\frac{1}{5} \left (3 a \left (a^2+5 b^2\right )\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{6 a \left (a^2+5 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{8 a^2 b \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a^2 \sec ^{\frac{3}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{5 d}+\left (b \left (a^2+b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\frac{1}{5} \left (3 a \left (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{6 a \left (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 b \left (a^2+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)}}{d}+\frac{6 a \left (a^2+5 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{8 a^2 b \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a^2 \sec ^{\frac{3}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 1.51934, size = 134, normalized size = 0.71 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (5 b \left (a^2+b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-3 a \left (a^2+5 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{a \sin (c+d x) \left (3 \left (a^2+5 b^2\right ) \cos (2 (c+d x))+5 \left (a^2+3 b^2\right )+10 a b \cos (c+d x)\right )}{2 \cos ^{\frac{5}{2}}(c+d x)}\right )}{5 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 9.477, size = 738, 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{7}{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{7}{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{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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