Optimal. Leaf size=318 \[ \frac {2 b^2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a^4 d (a+b)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{21 a^3 d}+\frac {2 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{5 a^4 d}-\frac {2 \sin (c+d x) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{5 a^4 d \sqrt {\cos (c+d x)}}+\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)} \]
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Rubi [A] time = 1.74, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3055, 3059, 2639, 3002, 2641, 2805} \[ \frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{21 a^3 d}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^2 b (3 A+5 C)-3 a^3 B-5 a b^2 B+5 A b^3\right )}{5 a^4 d}+\frac {2 b^2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a^4 d (a+b)}+\frac {2 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \sin (c+d x) \left (a^2 b (3 A+5 C)-3 a^3 B-5 a b^2 B+5 A b^3\right )}{5 a^4 d \sqrt {\cos (c+d x)}}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2805
Rule 3002
Rule 3055
Rule 3059
Rubi steps
\begin {align*} \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx &=\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \int \frac {-\frac {7}{2} (A b-a B)+\frac {1}{2} a (5 A+7 C) \cos (c+d x)+\frac {5}{2} A b \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{7 a}\\ &=\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 \int \frac {\frac {5}{4} \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right )+\frac {1}{4} a (4 A b+21 a B) \cos (c+d x)-\frac {21}{4} b (A b-a B) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{35 a^2}\\ &=\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 \int \frac {-\frac {21}{8} \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right )-\frac {1}{8} a \left (28 A b^2-28 a b B-5 a^2 (5 A+7 C)\right ) \cos (c+d x)+\frac {5}{8} b \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{105 a^3}\\ &=\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt {\cos (c+d x)}}+\frac {16 \int \frac {\frac {5}{16} \left (21 A b^4-7 a^3 b B-21 a b^3 B+7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right )+\frac {1}{16} a \left (140 A b^3-63 a^3 B-140 a b^2 B+4 a^2 b (22 A+35 C)\right ) \cos (c+d x)+\frac {21}{16} b \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{105 a^4}\\ &=\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt {\cos (c+d x)}}-\frac {16 \int \frac {-\frac {5}{16} b \left (21 A b^4-7 a^3 b B-21 a b^3 B+7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right )-\frac {5}{16} a b^2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{105 a^4 b}+\frac {\left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 a^4}\\ &=\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^4 d}+\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt {\cos (c+d x)}}+\frac {\left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 a^3}+\frac {\left (b^2 \left (A b^2-a (b B-a C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{a^4}\\ &=\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^4 d}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a^3 d}+\frac {2 b^2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a^4 (a+b) d}+\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 4.84, size = 416, normalized size = 1.31 \[ \frac {\frac {4 a \left (-63 a^3 B+4 a^2 b (22 A+35 C)-140 a b^2 B+140 A b^3\right ) \left ((a+b) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-a \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{b (a+b)}+\frac {2 \left (5 \left (6 a^3 A \tan (c+d x)+a \sin (2 (c+d x)) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )\right )+42 \sin (c+d x) \left (a^2 (a B-A b)+\cos ^2(c+d x) \left (3 a^3 B-a^2 b (3 A+5 C)+5 a b^2 B-5 A b^3\right )\right )\right )}{\cos ^{\frac {5}{2}}(c+d x)}-\frac {42 \sin (c+d x) \left (3 a^3 B-a^2 b (3 A+5 C)+5 a b^2 B-5 A b^3\right ) \left (\left (b^2-2 a^2\right ) \Pi \left (-\frac {b}{a};\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) F\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )\right )}{a b \sqrt {\sin ^2(c+d x)}}+\frac {2 \left (10 a^4 (5 A+7 C)-133 a^3 b B+7 a^2 b^2 (19 A+45 C)-315 a b^3 B+315 A b^4\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}}{210 a^4 d} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 12.59, size = 1003, normalized size = 3.15 \[ \text {result too large to display} \]
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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^{9/2}\,\left (a+b\,\cos \left (c+d\,x\right )\right )} \,d x \]
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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