Optimal. Leaf size=426 \[ -\frac{2 b^2 \sin (c+d x) \left (3 a^2 (3 A+5 C)+50 a b B+b^2 (59 A-3 C)\right )}{15 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sin (c+d x) \left (6 a^3 (3 A+5 C)+105 a^2 b B+4 a b^2 (33 A-5 C)-5 b^3 B\right )}{15 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (a^2 (3 A+5 C)+15 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (4 a^3 b (A+3 C)+18 a^2 b^2 B+a^4 B+4 a b^3 (3 A+C)+b^4 B\right )}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (30 a^2 b^2 (A-C)+a^4 (3 A+5 C)+20 a^3 b B-20 a b^3 B-b^4 (5 A+3 C)\right )}{5 d}+\frac{2 (5 a B+8 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^4}{5 d} \]
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Rubi [A] time = 1.47176, antiderivative size = 426, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4221, 3047, 3033, 3023, 2748, 2641, 2639} \[ -\frac{2 b^2 \sin (c+d x) \left (3 a^2 (3 A+5 C)+50 a b B+b^2 (59 A-3 C)\right )}{15 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sin (c+d x) \left (6 a^3 (3 A+5 C)+105 a^2 b B+4 a b^2 (33 A-5 C)-5 b^3 B\right )}{15 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (a^2 (3 A+5 C)+15 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (4 a^3 b (A+3 C)+18 a^2 b^2 B+a^4 B+4 a b^3 (3 A+C)+b^4 B\right )}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (30 a^2 b^2 (A-C)+a^4 (3 A+5 C)+20 a^3 b B-20 a b^3 B-b^4 (5 A+3 C)\right )}{5 d}+\frac{2 (5 a B+8 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^4}{5 d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3047
Rule 3033
Rule 3023
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^4 \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))^4 \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))^4 \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))^3 \left (\frac{1}{2} (8 A b+5 a B)+\frac{1}{2} (3 a A+5 b B+5 a C) \cos (c+d x)-\frac{5}{2} b (A-C) \cos ^2(c+d x)\right )}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 (8 A b+5 a B) (a+b \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 A (a+b \cos (c+d x))^4 \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))^2 \left (\frac{3}{4} \left (16 A b^2+15 a b B+a^2 (3 A+5 C)\right )+\frac{1}{4} \left (5 a^2 B+15 b^2 B+2 a b (A+15 C)\right ) \cos (c+d x)-\frac{5}{4} b (11 A b+5 a B-3 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (16 A b^2+15 a b B+a^2 (3 A+5 C)\right ) (a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (8 A b+5 a B) (a+b \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 A (a+b \cos (c+d x))^4 \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{(a+b \cos (c+d x)) \left (\frac{1}{8} \left (192 A b^3+5 a^3 B+195 a b^2 B+a^2 (38 A b+90 b C)\right )-\frac{1}{8} \left (65 a^2 b B-15 b^3 B+a b^2 (101 A-45 C)+3 a^3 (3 A+5 C)\right ) \cos (c+d x)-\frac{5}{8} b \left (50 a b B+b^2 (59 A-3 C)+3 a^2 (3 A+5 C)\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b^2 \left (50 a b B+b^2 (59 A-3 C)+3 a^2 (3 A+5 C)\right ) \sin (c+d x)}{15 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (16 A b^2+15 a b B+a^2 (3 A+5 C)\right ) (a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (8 A b+5 a B) (a+b \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{1}{75} \left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{5}{16} a \left (192 A b^3+5 a^3 B+195 a b^2 B+a^2 (38 A b+90 b C)\right )-\frac{15}{16} \left (20 a^3 b B-20 a b^3 B+30 a^2 b^2 (A-C)-b^4 (5 A+3 C)+a^4 (3 A+5 C)\right ) \cos (c+d x)-\frac{15}{16} b \left (105 a^2 b B-5 b^3 B+4 a b^2 (33 A-5 C)+6 a^3 (3 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b^2 \left (50 a b B+b^2 (59 A-3 C)+3 a^2 (3 A+5 C)\right ) \sin (c+d x)}{15 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b \left (105 a^2 b B-5 b^3 B+4 a b^2 (33 A-5 C)+6 a^3 (3 A+5 C)\right ) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 \left (16 A b^2+15 a b B+a^2 (3 A+5 C)\right ) (a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (8 A b+5 a B) (a+b \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{1}{225} \left (32 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{75}{32} \left (a^4 B+18 a^2 b^2 B+b^4 B+4 a b^3 (3 A+C)+4 a^3 b (A+3 C)\right )-\frac{45}{32} \left (20 a^3 b B-20 a b^3 B+30 a^2 b^2 (A-C)-b^4 (5 A+3 C)+a^4 (3 A+5 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b^2 \left (50 a b B+b^2 (59 A-3 C)+3 a^2 (3 A+5 C)\right ) \sin (c+d x)}{15 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b \left (105 a^2 b B-5 b^3 B+4 a b^2 (33 A-5 C)+6 a^3 (3 A+5 C)\right ) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 \left (16 A b^2+15 a b B+a^2 (3 A+5 C)\right ) (a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (8 A b+5 a B) (a+b \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{1}{3} \left (\left (a^4 B+18 a^2 b^2 B+b^4 B+4 a b^3 (3 A+C)+4 a^3 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 (20 a^3 b B-20 a b^3 B+30 a^2 b^2 (A-C)-b^4 (5 A+3 C)+a^4 (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 (20 a^3 b B-20 a b^3 B+30 a^2 b^2 (A-C)-b^4 (5 A+3 C)+a^4 (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^4 B+18 a^2 b^2 B+b^4 B+4 a b^3 (3 A+C)+4 a^3 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 \left (50 a b B+b^2 (59 A-3 C)+3 a^2 (3 A+5 C)\right ) \sin (c+d x)}{15 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b \left (105 a^2 b B-5 b^3 B+4 a b^2 (33 A-5 C)+6 a^3 (3 A+5 C)\right ) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 \left (16 A b^2+15 a b B+a^2 (3 A+5 C)\right ) (a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (8 A b+5 a B) (a+b \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 4.99125, size = 307, normalized size = 0.72 \[ \frac{\sqrt{\sec (c+d x)} \left (20 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (4 a^3 b (A+3 C)+18 a^2 b^2 B+a^4 B+4 a b^3 (3 A+C)+b^4 B\right )-12 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (30 a^2 b^2 (A-C)+a^4 (3 A+5 C)+20 a^3 b B-20 a b^3 B-b^4 (5 A+3 C)\right )+360 a^2 A b^2 \sin (c+d x)+80 a^3 A b \tan (c+d x)+36 a^4 A \sin (c+d x)+12 a^4 A \tan (c+d x) \sec (c+d x)+240 a^3 b B \sin (c+d x)+20 a^4 B \tan (c+d x)+60 a^4 C \sin (c+d x)+40 a b^3 C \sin (2 (c+d x))+10 b^4 B \sin (2 (c+d x))+3 b^4 C \sin (c+d x)+3 b^4 C \sin (3 (c+d x))\right )}{30 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 6.056, size = 1884, normalized size = 4.4 \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 )}^{4} \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^{4} \cos \left (d x + c\right )^{6} +{\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + A a^{4} +{\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{4} + 4 \, A a^{3} 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 )}^{4} \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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