Optimal. Leaf size=140 \[ \frac{2 \left (a^2 B+2 a A b+3 b^2 B\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}+\frac{2 \left (3 a^2 A+5 b (2 a B+A b)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a (5 a B+7 A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 a A \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)}{5 d} \]
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Rubi [A] time = 0.332282, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2954, 2990, 3023, 2748, 2641, 2639} \[ \frac{2 \left (a^2 B+2 a A b+3 b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 \left (3 a^2 A+5 b (2 a B+A b)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a (5 a B+7 A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 a A \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)}{5 d} \]
Antiderivative was successfully verified.
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Rule 2954
Rule 2990
Rule 3023
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx &=\int \frac{(b+a \cos (c+d x))^2 (B+A \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 a A \sqrt{\cos (c+d x)} (b+a \cos (c+d x)) \sin (c+d x)}{5 d}+\frac{2}{5} \int \frac{\frac{1}{2} b (a A+5 b B)+\frac{1}{2} \left (3 a^2 A+5 b (A b+2 a B)\right ) \cos (c+d x)+\frac{1}{2} a (7 A b+5 a B) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 a (7 A b+5 a B) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a A \sqrt{\cos (c+d x)} (b+a \cos (c+d x)) \sin (c+d x)}{5 d}+\frac{4}{15} \int \frac{\frac{5}{4} \left (2 a A b+a^2 B+3 b^2 B\right )+\frac{3}{4} \left (3 a^2 A+5 b (A b+2 a B)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 a (7 A b+5 a B) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a A \sqrt{\cos (c+d x)} (b+a \cos (c+d x)) \sin (c+d x)}{5 d}+\frac{1}{3} \left (2 a A b+a^2 B+3 b^2 B\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{5} \left (3 a^2 A+5 b (A b+2 a B)\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (3 a^2 A+5 b (A b+2 a B)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (2 a A b+a^2 B+3 b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 a (7 A b+5 a B) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a A \sqrt{\cos (c+d x)} (b+a \cos (c+d x)) \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.621987, size = 106, normalized size = 0.76 \[ \frac{2 \left (5 \left (a^2 B+2 a A b+3 b^2 B\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+3 \left (3 a^2 A+10 a b B+5 A b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+a \sin (c+d x) \sqrt{\cos (c+d x)} (3 a A \cos (c+d x)+5 a B+10 A b)\right )}{15 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.096, size = 487, normalized size = 3.5 \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(-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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b^{2} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{3} + A a^{2} \cos \left (d x + c\right )^{2} +{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )\right )} \sqrt{\cos \left (d x + c\right )}, 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 \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{2} \cos \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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