Optimal. Leaf size=161 \[ \frac{2 \left (3 a^2 B+6 a A b+b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 \left (a^2 A-b (2 a B+A b)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 b^2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.320127, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {2960, 4024, 4047, 3771, 2641, 4046, 2639} \[ \frac{2 \left (3 a^2 B+6 a A b+b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 \left (a^2 A-b (2 a B+A b)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 b^2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2960
Rule 4024
Rule 4047
Rule 3771
Rule 2641
Rule 4046
Rule 2639
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx &=\int \frac{(b+a \sec (c+d x))^2 (B+A \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 b^2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}-\frac{2}{3} \int \frac{-\frac{3}{2} b (A b+2 a B)+\left (-3 a A b+\left (-\frac{3 a^2}{2}-\frac{b^2}{2}\right ) B\right ) \sec (c+d x)-\frac{3}{2} a^2 A \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 b^2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}-\frac{2}{3} \int \frac{-\frac{3}{2} b (A b+2 a B)-\frac{3}{2} a^2 A \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx-\frac{1}{3} \left (-6 a A b-3 a^2 B-b^2 B\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 b^2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 A \sqrt{\sec (c+d x)} \sin (c+d x)}{d}-\left (a^2 A-b (A b+2 a B)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx-\frac{1}{3} \left (\left (-6 a A b-3 a^2 B-b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (6 a A b+3 a^2 B+b^2 B\right ) \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 b^2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 A \sqrt{\sec (c+d x)} \sin (c+d x)}{d}-\left (\left (a^2 A-b (A b+2 a B)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 \left (a^2 A-b (A b+2 a B)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{d}+\frac{2 \left (6 a A b+3 a^2 B+b^2 B\right ) \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 b^2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 A \sqrt{\sec (c+d x)} \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.728235, size = 124, normalized size = 0.77 \[ \frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (2 \left (3 a^2 B+6 a A b+b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\left (-6 a^2 A+12 a b B+6 A b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{2 \sin (c+d x) \left (3 a^2 A+b^2 B \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}}\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.925, size = 404, normalized size = 2.5 \begin{align*} -{\frac{2}{3\,d} \left ( 4\,B{b}^{2}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+6\,Aab\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +3\,A\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ){a}^{2}-3\,A\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ){b}^{2}-6\,A{a}^{2}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+3\,B{a}^{2}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +{b}^{2}B\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,\sqrt{2} \right ) -6\,B\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) ab-2\,B{b}^{2}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \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 )}^{2} \sec \left (d x + c\right )^{\frac{3}{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^{2} \cos \left (d x + c\right )^{3} + A a^{2} +{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sec \left (d x + c\right )^{\frac{3}{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 )}^{2} \sec \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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