Optimal. Leaf size=172 \[ \frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a (3 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 a (3 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d} \]
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Rubi [A] time = 0.222072, antiderivative size = 172, 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} \[ \frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a (3 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 a (3 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d} \]
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+a \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) (a+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} a (3 A+5 B)+\frac{5}{2} a (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 (A+B)) \int \sec ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{5} (a (3 A+5 B)) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 a (3 A+5 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a (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 (A+B)) \int \sqrt{\sec (c+d x)} \, dx-\frac{1}{5} (a (3 A+5 B)) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a (3 A+5 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a (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 (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 (a (3 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 a (3 A+5 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 (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 a (3 A+5 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a (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 [C] time = 1.89684, size = 292, normalized size = 1.7 \[ \frac{a e^{-i c} \left (-1+e^{2 i c}\right ) \csc (c) (\cos (c+d x)+1) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{\sec (c+d x)} \left ((3 A+5 B) e^{i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{5/2} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )-5 i (A+B) \left (1+e^{2 i (c+d x)}\right )^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-3 A e^{i (c+d x)}-24 A e^{3 i (c+d x)}-5 A e^{4 i (c+d x)}-9 A e^{5 i (c+d x)}+5 A-15 B e^{i (c+d x)}-30 B e^{3 i (c+d x)}-5 B e^{4 i (c+d x)}-15 B e^{5 i (c+d x)}+5 B\right )}{30 d \left (1+e^{2 i (c+d x)}\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 10.024, size = 661, normalized size = 3.8 \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 (a \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 a \cos \left (d x + c\right )^{2} +{\left (A + B\right )} a \cos \left (d x + c\right ) + A a\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 (a \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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