Optimal. Leaf size=121 \[ \frac{2 \left (a^2 A+6 a b B+3 A b^2\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}+\frac{2 \left (a^2 B+2 a A b-b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^2 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b^2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.317962, antiderivative size = 121, 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, 2988, 3023, 2748, 2641, 2639} \[ \frac{2 \left (a^2 A+6 a b B+3 A b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 \left (a^2 B+2 a A b-b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^2 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b^2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2954
Rule 2988
Rule 3023
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \cos ^{\frac{3}{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))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 b^2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-2 \int \frac{-\frac{1}{2} b (A b+2 a B)-\frac{1}{2} \left (2 a A b+a^2 B-b^2 B\right ) \cos (c+d x)-\frac{1}{2} a^2 A \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b^2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a^2 A \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}-\frac{4}{3} \int \frac{\frac{1}{4} \left (-a^2 A-3 b (A b+2 a B)\right )-\frac{3}{4} \left (2 a A b+a^2 B-b^2 B\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b^2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a^2 A \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}-\frac{1}{3} \left (-a^2 A-3 A b^2-6 a b B\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\left (-2 a A b-a^2 B+b^2 B\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (2 a A b+a^2 B-b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 \left (a^2 A+3 A b^2+6 a b B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 b^2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a^2 A \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.662784, size = 102, normalized size = 0.84 \[ \frac{2 \left (\left (a^2 A+6 a b B+3 A b^2\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+3 \left (a^2 B+2 a A b-b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{\sin (c+d x) \left (a^2 A \cos (c+d x)+3 b^2 B\right )}{\sqrt{\cos (c+d x)}}\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.04, size = 404, normalized size = 3.3 \begin{align*} -{\frac{2}{3\,d} \left ( 4\,A{a}^{2}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+{a}^{2}A\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 ) +3\,A{b}^{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 ) -6\,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 ) ab-2\,A{a}^{2}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+6\,Bab\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\,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 ){a}^{2}+3\,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 ){b}^{2}-6\,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 \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{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 ) \sec \left (d x + c\right )^{3} + A a^{2} \cos \left (d x + c\right ) +{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right ) \sec \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right ) \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{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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