Optimal. Leaf size=313 \[ \frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)} \left (3 a^2 (3 A+5 C)+35 a b B+24 A b^2\right )}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 b (A+3 C)+a^3 B+9 a b^2 B+b^3 (3 A+C)\right )}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^3 (3 A+5 C)+15 a^2 b B+15 a b^2 (A-C)-5 b^3 B\right )}{5 d}-\frac{2 b^2 \sin (c+d x) (5 a B+9 A b-5 b C)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (5 a B+6 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{5 d} \]
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Rubi [A] time = 0.962522, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4221, 3047, 3031, 3023, 2748, 2641, 2639} \[ \frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)} \left (3 a^2 (3 A+5 C)+35 a b B+24 A b^2\right )}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 b (A+3 C)+a^3 B+9 a b^2 B+b^3 (3 A+C)\right )}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^3 (3 A+5 C)+15 a^2 b B+15 a b^2 (A-C)-5 b^3 B\right )}{5 d}-\frac{2 b^2 \sin (c+d x) (5 a B+9 A b-5 b C)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (5 a B+6 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{5 d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3047
Rule 3031
Rule 3023
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{1}{5} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^2 \left (\frac{1}{2} (6 A b+5 a B)+\frac{1}{2} (3 a A+5 b B+5 a C) \cos (c+d x)-\frac{1}{2} b (3 A-5 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 (6 A b+5 a B) (a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 A (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{1}{15} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{4} \left (24 A b^2+35 a b B+3 a^2 (3 A+5 C)\right )+\frac{1}{4} \left (5 a^2 B+15 b^2 B+6 a b (A+5 C)\right ) \cos (c+d x)-\frac{3}{4} b (9 A b+5 a B-5 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a \left (24 A b^2+35 a b B+3 a^2 (3 A+5 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 (6 A b+5 a B) (a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 A (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}-\frac{1}{15} \left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{8} \left (-24 A b^3-5 a^3 B-50 a b^2 B-15 a^2 b (A+3 C)\right )+\frac{3}{8} \left (15 a^2 b B-5 b^3 B+15 a b^2 (A-C)+a^3 (3 A+5 C)\right ) \cos (c+d x)+\frac{3}{8} b^2 (9 A b+5 a B-5 b C) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b^2 (9 A b+5 a B-5 b C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a \left (24 A b^2+35 a b B+3 a^2 (3 A+5 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 (6 A b+5 a B) (a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 A (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}-\frac{1}{45} \left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{15}{16} \left (a^3 B+9 a b^2 B+b^3 (3 A+C)+3 a^2 b (A+3 C)\right )+\frac{9}{16} \left (15 a^2 b B-5 b^3 B+15 a b^2 (A-C)+a^3 (3 A+5 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b^2 (9 A b+5 a B-5 b C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a \left (24 A b^2+35 a b B+3 a^2 (3 A+5 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 (6 A b+5 a B) (a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 A (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{1}{3} \left (\left (a^3 B+9 a b^2 B+b^3 (3 A+C)+3 a^2 b (A+3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\frac{1}{5} \left (\left (15 a^2 b B-5 b^3 B+15 a b^2 (A-C)+a^3 (3 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 \left (15 a^2 b B-5 b^3 B+15 a b^2 (A-C)+a^3 (3 A+5 C)\right ) \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{2 \left (a^3 B+9 a b^2 B+b^3 (3 A+C)+3 a^2 b (A+3 C)\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 (9 A b+5 a B-5 b C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a \left (24 A b^2+35 a b B+3 a^2 (3 A+5 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 (6 A b+5 a B) (a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 A (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 2.22688, size = 276, normalized size = 0.88 \[ \frac{\sqrt{\sec (c+d x)} \left (\frac{2}{5} a \sin (c+d x) \left (3 a^2 A+5 a^2 C+15 a b B+15 A b^2\right )+\frac{2}{3} \sec (c+d x) \left (3 a^2 A b \sin (c+d x)+a^3 B \sin (c+d x)\right )+\frac{2}{5} a^3 A \tan (c+d x) \sec (c+d x)+\frac{1}{3} b^3 C \sin (2 (c+d x))\right )}{d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (15 a^2 A b+45 a^2 b C+5 a^3 B+45 a b^2 B+15 A b^3+5 b^3 C\right )+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-9 a^3 A-45 a^2 b B-15 a^3 C-45 a A b^2+45 a b^2 C+15 b^3 B\right )}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}}{15 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.409, size = 1419, normalized size = 4.5 \begin{align*} \text{result too large to display} \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} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac{7}{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 (C b^{3} \cos \left (d x + c\right )^{5} +{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} + A a^{3} +{\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sec \left (d x + c\right )^{\frac{7}{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 (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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