Optimal. Leaf size=217 \[ \frac {2 a (5 A+7 (B+C)) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 a (3 A+3 B+5 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 a (5 A+7 (B+C)) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 a (3 A+3 B+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a (A+B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 a A \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d} \]
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Rubi [A] time = 0.35, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4221, 3031, 3021, 2748, 2636, 2641, 2639} \[ \frac {2 a (5 A+7 (B+C)) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 a (3 A+3 B+5 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 a (5 A+7 (B+C)) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 a (3 A+3 B+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a (A+B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 a A \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2641
Rule 2748
Rule 3021
Rule 3031
Rule 4221
Rubi steps
\begin {align*} \int (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 a A \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac {1}{7} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {7}{2} a (A+B)-\frac {1}{2} a (5 A+7 (B+C)) \cos (c+d x)-\frac {7}{2} a C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 a (A+B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a A \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac {1}{35} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {5}{4} a (5 A+7 (B+C))-\frac {7}{4} a (3 A+3 B+5 C) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a (A+B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a A \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{5} \left (a (3 A+3 B+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx+\frac {1}{7} \left (a (5 A+7 (B+C)) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a (3 A+3 B+5 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a (5 A+7 (B+C)) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a (A+B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a A \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac {1}{5} \left (a (3 A+3 B+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (a (5 A+7 (B+C)) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 a (3 A+3 B+5 C) \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 a (5 A+7 (B+C)) \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 (3 A+3 B+5 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a (5 A+7 (B+C)) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a (A+B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a A \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 2.24, size = 172, normalized size = 0.79 \[ \frac {a \sec ^{\frac {7}{2}}(c+d x) \left (40 (5 A+7 (B+C)) \cos ^{\frac {7}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-168 (3 A+3 B+5 C) \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) (21 (13 A+13 B+15 C) \cos (c+d x)+10 (5 A+7 (B+C)) \cos (2 (c+d x))+63 A \cos (3 (c+d x))+110 A+63 B \cos (3 (c+d x))+70 B+105 C \cos (3 (c+d x))+70 C)\right )}{420 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C a \cos \left (d x + c\right )^{3} + {\left (B + C\right )} a \cos \left (d x + c\right )^{2} + {\left (A + B\right )} a \cos \left (d x + c\right ) + A a\right )} \sec \left (d x + c\right )^{\frac {9}{2}}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 10.55, size = 849, normalized size = 3.91 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
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 \[ \int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,\left (a+a\,\cos \left (c+d\,x\right )\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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