Optimal. Leaf size=232 \[ -\frac{(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(7 A+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{5 (7 A+9 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{5 (7 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a d}-\frac{3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d} \]
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Rubi [A] time = 0.314185, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4221, 3042, 2748, 2635, 2639, 2641} \[ -\frac{(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(7 A+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{5 (7 A+9 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{5 (7 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a d}-\frac{3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3042
Rule 2748
Rule 2635
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx\\ &=-\frac{(A+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{5}{2}}(c+d x) \left (-\frac{1}{2} a (5 A+7 C)+\frac{1}{2} a (7 A+9 C) \cos (c+d x)\right ) \, dx}{a^2}\\ &=-\frac{(A+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}-\frac{\left ((5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx}{2 a}+\frac{\left ((7 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{7}{2}}(c+d x) \, dx}{2 a}\\ &=-\frac{(A+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{(7 A+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{\left (3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{10 a}+\frac{\left (5 (7 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx}{14 a}\\ &=-\frac{3 (5 A+7 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 a d}-\frac{(A+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{(7 A+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{5 (7 A+9 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}+\frac{\left (5 (7 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{42 a}\\ &=-\frac{3 (5 A+7 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 a d}+\frac{5 (7 A+9 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 a d}-\frac{(A+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{(7 A+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{5 (7 A+9 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 3.8355, size = 542, normalized size = 2.34 \[ \frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \left (\sqrt{\sec (c+d x)} \left (20 (14 A+27 C) \sin (2 c) \cos (2 d x)-84 (20 A+33 C) \cos (c) \sin (d x)+20 (14 A+27 C) \cos (2 c) \sin (2 d x)-840 (A+C) \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right )+21 \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \cos (d x) ((20 A+33 C) \cos (2 c)+40 A+51 C)-840 (A+C) \tan \left (\frac{c}{2}\right )-84 C \sin (3 c) \cos (3 d x)+30 C \sin (4 c) \cos (4 d x)-84 C \cos (3 c) \sin (3 d x)+30 C \cos (4 c) \sin (4 d x)\right )+420 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right )+1400 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+588 \sqrt{2} C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right )+1800 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{420 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.041, size = 295, normalized size = 1.3 \begin{align*} -{\frac{1}{105\,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 ( 175\,A{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +315\,A{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +225\,C{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +441\,C{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ) -480\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+864\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+ \left ( -280\,A-888\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}+ \left ( 630\,A+930\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -245\,A-321\,C \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{C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )} \sec \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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac{5}{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{C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )} \sec \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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