Optimal. Leaf size=180 \[ \frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 (3 a A+5 b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 (a B+A b) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 (3 a A+5 b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d} \]
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Rubi [A] time = 0.223326, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {2960, 3997, 3787, 3768, 3771, 2639, 2641} \[ \frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 (3 a A+5 b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 (a B+A b) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 (3 a A+5 b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 2960
Rule 3997
Rule 3787
Rule 3768
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx &=\int \sec ^{\frac{3}{2}}(c+d x) (b+a \sec (c+d x)) (B+A \sec (c+d x)) \, dx\\ &=\frac{2 a A \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2}{5} \int \sec ^{\frac{3}{2}}(c+d x) \left (\frac{1}{2} (3 a A+5 b B)+\frac{5}{2} (A b+a B) \sec (c+d x)\right ) \, dx\\ &=\frac{2 a A \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+(A b+a B) \int \sec ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{5} (3 a A+5 b B) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 (3 a A+5 b B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (A b+a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac{2 a A \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{1}{3} (A b+a B) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{5} (-3 a A-5 b B) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 (3 a A+5 b B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (A b+a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac{2 a A \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{1}{3} \left ((A b+a B) \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 A-5 b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 (3 a A+5 b B) \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 b+a B) \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{2 (3 a A+5 b B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (A b+a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac{2 a A \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 1.82451, size = 132, normalized size = 0.73 \[ \frac{\sec ^{\frac{5}{2}}(c+d x) \left (20 (a B+A b) \cos ^{\frac{5}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-12 (3 a A+5 b B) \cos ^{\frac{5}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) (10 (a B+A b) \cos (c+d x)+3 (3 a A+5 b B) \cos (2 (c+d x))+15 (a A+b B))\right )}{30 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 10.268, size = 663, normalized size = 3.7 \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 )}{\left (b \cos \left (d x + c\right ) + a\right )} \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 b \cos \left (d x + c\right )^{2} + A a +{\left (B a + A b\right )} \cos \left (d x + c\right )\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 )}{\left (b \cos \left (d x + c\right ) + a\right )} \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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