Optimal. Leaf size=163 \[ -\frac{(3 A-5 B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{(A-B) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)}-\frac{(3 A-5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{3 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d} \]
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Rubi [A] time = 0.28072, antiderivative size = 163, 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, 4020, 3787, 3769, 3771, 2641, 2639} \[ -\frac{(3 A-5 B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{(A-B) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)}-\frac{(3 A-5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{3 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d} \]
Antiderivative was successfully verified.
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Rule 2960
Rule 4020
Rule 3787
Rule 3769
Rule 3771
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx &=\int \frac{B+A \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx\\ &=\frac{(A-B) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))}+\frac{\int \frac{-\frac{1}{2} a (3 A-5 B)+\frac{3}{2} a (A-B) \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{a^2}\\ &=\frac{(A-B) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))}-\frac{(3 A-5 B) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{2 a}+\frac{(3 (A-B)) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{2 a}\\ &=-\frac{(3 A-5 B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{(A-B) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))}-\frac{(3 A-5 B) \int \sqrt{\sec (c+d x)} \, dx}{6 a}+\frac{\left (3 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{2 a}\\ &=\frac{3 (A-B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a d}-\frac{(3 A-5 B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{(A-B) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))}-\frac{\left ((3 A-5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 a}\\ &=\frac{3 (A-B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a d}-\frac{(3 A-5 B) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 a d}-\frac{(3 A-5 B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{(A-B) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))}\\ \end{align*}
Mathematica [C] time = 4.93313, size = 444, normalized size = 2.72 \[ \frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \left (-\frac{\csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \left ((12 A-13 B) \cos \left (\frac{1}{2} (c-d x)\right )+(6 A-5 B) \cos \left (\frac{1}{2} (3 c+d x)\right )-2 B \sin (c) \sin \left (\frac{3}{2} (c+d x)\right )\right )}{\sqrt{\sec (c+d x)}}-6 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right )-12 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+6 \sqrt{2} B \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right )+20 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{6 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 3.086, size = 262, normalized size = 1.6 \begin{align*}{\frac{1}{3\,da}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 3\,A{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +9\,A{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -5\,B{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -9\,B{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ) +8\,B \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+ \left ( 6\,A-18\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -3\,A+7\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \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 \frac{B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )} \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 (\frac{B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\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 \frac{B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )} \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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