Optimal. Leaf size=103 \[ \frac{2 (3 a A+b B) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}-\frac{2 (a B+A b) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 (a B+A b) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.212193, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {2954, 2968, 3021, 2748, 2636, 2639, 2641} \[ \frac{2 (3 a A+b B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 (a B+A b) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 (a B+A b) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2954
Rule 2968
Rule 3021
Rule 2748
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \sec (c+d x)) (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx &=\int \frac{(b+a \cos (c+d x)) (B+A \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\int \frac{b B+(A b+a B) \cos (c+d x)+a A \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 b B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2}{3} \int \frac{\frac{3}{2} (A b+a B)+\frac{1}{2} (3 a A+b B) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 b B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+(A b+a B) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{3} (3 a A+b B) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 (3 a A+b B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 b B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (A b+a B) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+(-A b-a B) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 (A b+a B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 (3 a A+b B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 b B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (A b+a B) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.49008, size = 107, normalized size = 1.04 \[ \frac{2 \left ((3 a A+b B) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-3 (a B+A b) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+3 a B \sin (c+d x)+3 A b \sin (c+d x)+b B \tan (c+d x)\right )}{3 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.615, size = 428, normalized size = 4.2 \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 \frac{{\left (B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}}{\sqrt{\cos \left (d x + c\right )}}\,{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 (\frac{B b \sec \left (d x + c\right )^{2} + A a +{\left (B a + A b\right )} \sec \left (d x + c\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B \sec{\left (c + d x \right )}\right ) \left (a + b \sec{\left (c + d x \right )}\right )}{\sqrt{\cos{\left (c + d x \right )}}}\, dx \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 \frac{{\left (B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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