Optimal. Leaf size=237 \[ \frac {2 a^2 (5 a A-b B) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (3 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}-\frac {2 \left (5 a^3 A-15 a^2 b B-15 a A b^2-3 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b^2 (9 a B+5 A b) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{5 d \sec ^{\frac {3}{2}}(c+d x)} \]
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Rubi [A] time = 0.53, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {2960, 4025, 4074, 4047, 3771, 2641, 4046, 2639} \[ \frac {2 \left (9 a^2 A b+3 a^3 B+3 a b^2 B+A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}-\frac {2 \left (5 a^3 A-15 a^2 b B-15 a A b^2-3 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a^2 (5 a A-b B) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 b^2 (9 a B+5 A b) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{5 d \sec ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2960
Rule 3771
Rule 4025
Rule 4046
Rule 4047
Rule 4074
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x) \, dx &=\int \frac {(b+a \sec (c+d x))^3 (B+A \sec (c+d x))}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 b B (b+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2}{5} \int \frac {(b+a \sec (c+d x)) \left (-\frac {1}{2} b (5 A b+9 a B)-\frac {1}{2} \left (10 a A b+5 a^2 B+3 b^2 B\right ) \sec (c+d x)-\frac {1}{2} a (5 a A-b B) \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 b^2 (5 A b+9 a B) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 b B (b+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4}{15} \int \frac {\frac {3}{4} b \left (15 a A b+14 a^2 B+3 b^2 B\right )+\frac {5}{4} \left (9 a^2 A b+A b^3+3 a^3 B+3 a b^2 B\right ) \sec (c+d x)+\frac {3}{4} a^2 (5 a A-b B) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {2 b^2 (5 A b+9 a B) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 b B (b+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4}{15} \int \frac {\frac {3}{4} b \left (15 a A b+14 a^2 B+3 b^2 B\right )+\frac {3}{4} a^2 (5 a A-b B) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{3} \left (9 a^2 A b+A b^3+3 a^3 B+3 a b^2 B\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {2 b^2 (5 A b+9 a B) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 a^2 (5 a A-b B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 b B (b+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} \left (-5 a^3 A+15 a A b^2+15 a^2 b B+3 b^3 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{3} \left (\left (9 a^2 A b+A b^3+3 a^3 B+3 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (9 a^2 A b+A b^3+3 a^3 B+3 a b^2 B\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 (5 A b+9 a B) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 a^2 (5 a A-b B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 b B (b+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} \left (\left (-5 a^3 A+15 a A b^2+15 a^2 b B+3 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {2 \left (5 a^3 A-15 a A b^2-15 a^2 b B-3 b^3 B\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 (9 a^2 A b+A b^3+3 a^3 B+3 a b^2 B\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 (5 A b+9 a B) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 a^2 (5 a A-b B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 b B (b+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 1.44, size = 172, normalized size = 0.73 \[ \frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \sin (c+d x) \left (3 \left (10 a^3 A+b^3 B \cos (2 (c+d x))+b^3 B\right )+10 b^2 (3 a B+A b) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}}+20 \left (3 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+12 \left (-5 a^3 A+15 a^2 b B+15 a A b^2+3 b^3 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{30 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b^{3} \cos \left (d x + c\right )^{4} + A a^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sec \left (d x + c\right )^{\frac {3}{2}}, x\right ) \]
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 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.66, size = 867, normalized size = 3.66 \[ \text {result too large to display} \]
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 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int \left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]
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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