Optimal. Leaf size=204 \[ \frac{2 a^2 (35 A+33 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (7 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^2 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{8 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{35 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.509097, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4221, 3046, 2976, 2968, 3023, 2748, 2641, 2639} \[ \frac{2 a^2 (35 A+33 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (7 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^2 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{8 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{35 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3046
Rule 2976
Rule 2968
Rule 3023
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^2 \left (\frac{1}{2} a (7 A+C)+2 a C \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{7 a}\\ &=\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{8 C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt{\sec (c+d x)}}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x)) \left (\frac{1}{4} a^2 (35 A+9 C)+\frac{1}{4} a^2 (35 A+33 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{35 a}\\ &=\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{8 C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt{\sec (c+d x)}}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{4} a^3 (35 A+9 C)+\left (\frac{1}{4} a^3 (35 A+9 C)+\frac{1}{4} a^3 (35 A+33 C)\right ) \cos (c+d x)+\frac{1}{4} a^3 (35 A+33 C) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{35 a}\\ &=\frac{2 a^2 (35 A+33 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{8 C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt{\sec (c+d x)}}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{5}{2} a^3 (7 A+3 C)+\frac{21}{4} a^3 (5 A+3 C) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{105 a}\\ &=\frac{2 a^2 (35 A+33 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{8 C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt{\sec (c+d x)}}+\frac{1}{5} \left (2 a^2 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (4 a^2 (7 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a^2 (5 A+3 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a^2 (7 A+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 d}+\frac{2 a^2 (35 A+33 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{8 C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 1.83422, size = 189, normalized size = 0.93 \[ \frac{a^2 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (-56 i (5 A+3 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )+\cos (c+d x) (5 (28 A+51 C) \sin (c+d x)+840 i A+84 C \sin (2 (c+d x))+15 C \sin (3 (c+d x))+504 i C)+80 (7 A+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{210 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.911, size = 380, normalized size = 1.9 \begin{align*} -{\frac{4\,{a}^{2}}{105\,d}\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 ( 120\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}\cos \left ( 1/2\,dx+c/2 \right ) -348\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) + \left ( 70\,A+378\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -35\,A-117\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +70\,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 EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -105\,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 ) +30\,C\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 ) -63\,C\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 ) \right ){\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{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{2} \sqrt{\sec \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 ({\left (C a^{2} \cos \left (d x + c\right )^{4} + 2 \, C a^{2} \cos \left (d x + c\right )^{3} +{\left (A + C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \, A a^{2} \cos \left (d x + c\right ) + A a^{2}\right )} \sqrt{\sec \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 (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{2} \sqrt{\sec \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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