Optimal. Leaf size=360 \[ -\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}+\frac {8 a b \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 \left (15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}-\frac {2 b (9 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt {\sec (c+d x)}}-\frac {2 a b (21 A-5 C) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt {\sec (c+d x)}}+\frac {2 A \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \cos (c+d x))^4}{d} \]
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Rubi [A] time = 1.34, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4221, 3048, 3049, 3033, 3023, 2748, 2641, 2639} \[ -\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}+\frac {8 a b \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 \left (-18 a^2 b^2 (5 A+3 C)+15 a^4 (A-C)-b^4 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}-\frac {2 b (9 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt {\sec (c+d x)}}-\frac {2 a b (21 A-5 C) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt {\sec (c+d x)}}+\frac {2 A \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \cos (c+d x))^4}{d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3033
Rule 3048
Rule 3049
Rule 4221
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{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+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 A (a+b \cos (c+d x))^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^3 \left (4 A b-\frac {1}{2} a (A-C) \cos (c+d x)-\frac {1}{2} b (9 A-C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b (9 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {1}{9} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^2 \left (\frac {1}{4} a b (63 A+C)-\frac {1}{4} \left (9 a^2 (A-C)-b^2 (9 A+7 C)\right ) \cos (c+d x)-\frac {3}{4} a b (21 A-5 C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 a b (21 A-5 C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b (9 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {1}{63} \left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{4} a^2 b (189 A+11 C)-\frac {1}{8} a \left (63 a^2 (A-C)-b^2 (189 A+131 C)\right ) \cos (c+d x)-\frac {1}{8} b \left (a^2 (315 A-123 C)-7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 a b (21 A-5 C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b (9 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {1}{315} \left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {5}{8} a^3 b (189 A+11 C)-\frac {21}{16} \left (15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) \cos (c+d x)-\frac {15}{8} a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}-\frac {2 a b (21 A-5 C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b (9 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {1}{945} \left (32 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {45}{8} a b \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right )-\frac {63}{32} \left (15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}-\frac {2 a b (21 A-5 C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b (9 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {1}{21} \left (4 a b \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\frac {1}{15} \left (\left (15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 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^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {8 a b \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \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 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}-\frac {2 a b (21 A-5 C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b (9 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 1.77, size = 252, normalized size = 0.70 \[ \frac {\sqrt {\sec (c+d x)} \left (960 a b \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-336 \left (15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) \left (2520 a^4 A+120 a b \left (28 a^2 C+28 A b^2+29 b^2 C\right ) \cos (c+d x)+84 \left (18 a^2 b^2 C+3 A b^4+4 b^4 C\right ) \cos (2 (c+d x))+1512 a^2 b^2 C+360 a b^3 C \cos (3 (c+d x))+252 A b^4+35 b^4 C \cos (4 (c+d x))+301 b^4 C\right )\right )}{2520 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{4} \cos \left (d x + c\right )^{6} + 4 \, C a b^{3} \cos \left (d x + c\right )^{5} + 4 \, A a^{3} b \cos \left (d x + c\right ) + A a^{4} + {\left (6 \, C a^{2} b^{2} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (C a^{3} b + A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sec \left (d x + c\right )^{\frac {3}{2}}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.89, size = 1209, normalized size = 3.36 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4 \,d x \]
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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