Optimal. Leaf size=284 \[ -\frac {2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}-\frac {2 a b^2 (35 A-11 C) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b (7 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt {\sec (c+d x)}}+\frac {2 A \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \cos (c+d x))^3}{d} \]
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Rubi [A] time = 0.90, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}-\frac {2 a b^2 (35 A-11 C) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b (7 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt {\sec (c+d x)}}+\frac {2 A \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \cos (c+d x))^3}{d} \]
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))^3 \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))^3 \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))^3 \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))^2 \left (3 A b-\frac {1}{2} a (A-C) \cos (c+d x)-\frac {1}{2} b (7 A-C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b (7 A-C) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {1}{7} \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} a b (35 A+C)-\frac {1}{4} \left (7 a^2 (A-C)-b^2 (7 A+5 C)\right ) \cos (c+d x)-\frac {1}{4} a b (35 A-11 C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 a b^2 (35 A-11 C) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b (7 A-C) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {1}{35} \left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {5}{8} a^2 b (35 A+C)-\frac {7}{8} a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) \cos (c+d x)-\frac {5}{8} b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 a b^2 (35 A-11 C) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b (7 A-C) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {1}{105} \left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {5}{16} b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right )-\frac {21}{16} a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 a b^2 (35 A-11 C) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b (7 A-C) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}-\frac {1}{5} \left (a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (b \left (21 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 {2 a \left (5 a^2 (A-C)-3 b^2 (5 A+3 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 b \left (21 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 a b^2 (35 A-11 C) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b (7 A-C) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 2.05, size = 193, normalized size = 0.68 \[ \frac {\sqrt {\sec (c+d x)} \left (40 b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-168 a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) \left (3 \left (140 a^3 A+42 a b^2 C \cos (2 (c+d x))+42 a b^2 C+5 b^3 C \cos (3 (c+d x))\right )+5 b \left (84 a^2 C+28 A b^2+29 b^2 C\right ) \cos (c+d x)\right )\right )}{420 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{3} \cos \left (d x + c\right )^{5} + 3 \, C a b^{2} \cos \left (d x + c\right )^{4} + 3 \, A a^{2} b \cos \left (d x + c\right ) + A a^{3} + {\left (3 \, C a^{2} b + A b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{3} + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sec \left (d x + c\right )^{\frac {3}{2}}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.49, size = 943, normalized size = 3.32 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]
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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