Optimal. Leaf size=426 \[ -\frac {2 b^2 \sin (c+d x) \left (3 a^2 (3 A+5 C)+50 a b B+b^2 (59 A-3 C)\right )}{15 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 (3 A+5 C)+15 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{5 d}-\frac {2 b \sin (c+d x) \left (6 a^3 (3 A+5 C)+105 a^2 b B+4 a b^2 (33 A-5 C)-5 b^3 B\right )}{15 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 B+4 a^3 b (A+3 C)+18 a^2 b^2 B+4 a b^3 (3 A+C)+b^4 B\right )}{3 d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 (3 A+5 C)+20 a^3 b B+30 a^2 b^2 (A-C)-20 a b^3 B-b^4 (5 A+3 C)\right )}{5 d}+\frac {2 (5 a B+8 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{15 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^4}{5 d} \]
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Rubi [A] time = 1.47, antiderivative size = 426, normalized size of antiderivative = 1.00, 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 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3033
Rule 3047
Rule 4221
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*}
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Mathematica [A] time = 4.84, size = 307, normalized size = 0.72 \[ \frac {\sqrt {\sec (c+d x)} \left (36 a^4 A \sin (c+d x)+12 a^4 A \tan (c+d x) \sec (c+d x)+20 a^4 B \tan (c+d x)+60 a^4 C \sin (c+d x)+80 a^3 A b \tan (c+d x)+240 a^3 b B \sin (c+d x)+360 a^2 A b^2 \sin (c+d x)+20 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 B+4 a^3 b (A+3 C)+18 a^2 b^2 B+4 a b^3 (3 A+C)+b^4 B\right )-12 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 (3 A+5 C)+20 a^3 b B+30 a^2 b^2 (A-C)-20 a b^3 B-b^4 (5 A+3 C)\right )+40 a b^3 C \sin (2 (c+d x))+10 b^4 B \sin (2 (c+d x))+3 b^4 C \sin (c+d x)+3 b^4 C \sin (3 (c+d x))\right )}{30 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{4} \cos \left (d x + c\right )^{6} + {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + A a^{4} + {\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sec \left (d x + c\right )^{\frac {7}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 13.04, size = 1884, normalized size = 4.42 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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