Integrand size = 35, antiderivative size = 360 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=-\frac {2 \left (15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {8 a b \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}-\frac {2 a b (21 A-5 C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b (9 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{d} \]
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Time = 1.35 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4306, 3127, 3128, 3112, 3102, 2827, 2720, 2719} \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}+\frac {8 a b \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}-\frac {2 \left (15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}-\frac {2 b (9 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt {\sec (c+d x)}}-\frac {2 a b (21 A-5 C) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt {\sec (c+d x)}}+\frac {2 A \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \cos (c+d x))^4}{d} \]
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Rule 2719
Rule 2720
Rule 2827
Rule 3102
Rule 3112
Rule 3127
Rule 3128
Rule 4306
Rubi steps \begin{align*} \text {integral}& = \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)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),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*}
Time = 4.08 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.70 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\frac {\sqrt {\sec (c+d x)} \left (-336 \left (15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+960 a b \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \left (2520 a^4 A+252 A b^4+1512 a^2 b^2 C+301 b^4 C+120 a b \left (28 A b^2+28 a^2 C+29 b^2 C\right ) \cos (c+d x)+84 \left (3 A b^4+18 a^2 b^2 C+4 b^4 C\right ) \cos (2 (c+d x))+360 a b^3 C \cos (3 (c+d x))+35 b^4 C \cos (4 (c+d x))\right ) \sin (c+d x)\right )}{2520 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1208\) vs. \(2(382)=764\).
Time = 10.71 (sec) , antiderivative size = 1209, normalized size of antiderivative = 3.36
method | result | size |
default | \(\text {Expression too large to display}\) | \(1209\) |
parts | \(\text {Expression too large to display}\) | \(1298\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.95 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=-\frac {60 \, \sqrt {2} {\left (7 i \, {\left (3 \, A + C\right )} a^{3} b + i \, {\left (7 \, A + 5 \, C\right )} a b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 60 \, \sqrt {2} {\left (-7 i \, {\left (3 \, A + C\right )} a^{3} b - i \, {\left (7 \, A + 5 \, C\right )} a b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (15 i \, {\left (A - C\right )} a^{4} - 18 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} - i \, {\left (9 \, A + 7 \, C\right )} b^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-15 i \, {\left (A - C\right )} a^{4} + 18 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} + i \, {\left (9 \, A + 7 \, C\right )} b^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (35 \, C b^{4} \cos \left (d x + c\right )^{4} + 180 \, C a b^{3} \cos \left (d x + c\right )^{3} + 315 \, A a^{4} + 7 \, {\left (54 \, C a^{2} b^{2} + {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 60 \, {\left (7 \, C a^{3} b + {\left (7 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d} \]
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Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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\[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4 \,d x \]
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