Optimal. Leaf size=244 \[ \frac {4 a^3 (41 A+42 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {2 (11 A+7 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{35 d}+\frac {4 a^3 (7 A+9 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {4 a^3 (7 A+9 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d} \]
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Rubi [A] time = 0.51, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {2960, 4018, 3997, 3787, 3771, 2641, 3768, 2639} \[ \frac {4 a^3 (41 A+42 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {2 (11 A+7 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{35 d}+\frac {4 a^3 (7 A+9 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {4 a^3 (7 A+9 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2960
Rule 3768
Rule 3771
Rule 3787
Rule 3997
Rule 4018
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx &=\int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 (B+A \sec (c+d x)) \, dx\\ &=\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2}{7} \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2 \left (\frac {1}{2} a (A+7 B)+\frac {1}{2} a (11 A+7 B) \sec (c+d x)\right ) \, dx\\ &=\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {4}{35} \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x)) \left (\frac {1}{2} a^2 (8 A+21 B)+\frac {1}{2} a^2 (41 A+42 B) \sec (c+d x)\right ) \, dx\\ &=\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {8}{105} \int \sqrt {\sec (c+d x)} \left (\frac {5}{4} a^3 (13 A+21 B)+\frac {21}{4} a^3 (7 A+9 B) \sec (c+d x)\right ) \, dx\\ &=\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {1}{5} \left (2 a^3 (7 A+9 B)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{21} \left (2 a^3 (13 A+21 B)\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {4 a^3 (7 A+9 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}-\frac {1}{5} \left (2 a^3 (7 A+9 B)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (2 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {4 a^3 (13 A+21 B) \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 {4 a^3 (7 A+9 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}-\frac {1}{5} \left (2 a^3 (7 A+9 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {4 a^3 (7 A+9 B) \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 {4 a^3 (13 A+21 B) \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 {4 a^3 (7 A+9 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}\\ \end {align*}
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Mathematica [C] time = 4.16, size = 435, normalized size = 1.78 \[ \frac {a^3 \csc (c) e^{-i d x} (\cos (c+d x)+1)^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (7 \sqrt {2} \left (-1+e^{2 i c}\right ) (7 A+9 B) e^{2 i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-\frac {\left (-1+e^{2 i c}\right ) e^{-i (c-d x)} \sqrt {\sec (c+d x)} \left (10 i (13 A+21 B) \left (1+e^{2 i (c+d x)}\right )^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 A \left (84 e^{i (c+d x)}-95 e^{2 i (c+d x)}+441 e^{3 i (c+d x)}+95 e^{4 i (c+d x)}+504 e^{5 i (c+d x)}+65 e^{6 i (c+d x)}+147 e^{7 i (c+d x)}-65\right )+21 B \left (16 e^{i (c+d x)}-5 e^{2 i (c+d x)}+54 e^{3 i (c+d x)}+5 e^{4 i (c+d x)}+56 e^{5 i (c+d x)}+5 e^{6 i (c+d x)}+18 e^{7 i (c+d x)}-5\right )\right )}{2 \left (1+e^{2 i (c+d x)}\right )^3}\right )}{420 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B a^{3} \cos \left (d x + c\right )^{4} + {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 3 \, {\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{2} + {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + A a^{3}\right )} \sec \left (d x + c\right )^{\frac {9}{2}}, x\right ) \]
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 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 5.51, size = 929, normalized size = 3.81 \[ \text {result too large to display} \]
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 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
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 \[ \int \left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]
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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