Optimal. Leaf size=345 \[ -\frac {2 a^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^5 d (a+b)}+\frac {2 \sin (c+d x) \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{21 b^3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right )}{5 b^4 d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-21 a^4 C+21 a^3 b B-7 a^2 b^2 (3 A+C)+7 a b^3 B-b^4 (7 A+5 C)\right )}{21 b^5 d}+\frac {2 (b B-a C) \sin (c+d x)}{5 b^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C \sin (c+d x)}{7 b d \sec ^{\frac {5}{2}}(c+d x)} \]
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Rubi [A] time = 1.48, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4221, 3049, 3059, 2639, 3002, 2641, 2805} \[ \frac {2 \sin (c+d x) \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{21 b^3 d \sqrt {\sec (c+d x)}}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-7 a^2 b^2 (3 A+C)+21 a^3 b B-21 a^4 C+7 a b^3 B-b^4 (7 A+5 C)\right )}{21 b^5 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^2 b B-5 a^3 C-a b^2 (5 A+3 C)+3 b^3 B\right )}{5 b^4 d}-\frac {2 a^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^5 d (a+b)}+\frac {2 (b B-a C) \sin (c+d x)}{5 b^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C \sin (c+d x)}{7 b d \sec ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2805
Rule 3002
Rule 3049
Rule 3059
Rule 4221
Rubi steps
\begin {align*} \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx\\ &=\frac {2 C \sin (c+d x)}{7 b d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (\frac {5 a C}{2}+\frac {1}{2} b (7 A+5 C) \cos (c+d x)+\frac {7}{2} (b B-a C) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{7 b}\\ &=\frac {2 C \sin (c+d x)}{7 b d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (b B-a C) \sin (c+d x)}{5 b^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\cos (c+d x)} \left (\frac {21}{4} a (b B-a C)+\frac {1}{4} b (21 b B+4 a C) \cos (c+d x)+\frac {5}{4} \left (7 A b^2-7 a b B+7 a^2 C+5 b^2 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{35 b^2}\\ &=\frac {2 C \sin (c+d x)}{7 b d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (b B-a C) \sin (c+d x)}{5 b^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+7 a^2 C+5 b^2 C\right ) \sin (c+d x)}{21 b^3 d \sqrt {\sec (c+d x)}}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {5}{8} a \left (7 A b^2-7 a b B+7 a^2 C+5 b^2 C\right )+\frac {1}{8} b \left (35 A b^2+28 a b B-28 a^2 C+25 b^2 C\right ) \cos (c+d x)+\frac {21}{8} \left (5 a^2 b B+3 b^3 B-5 a^3 C-a b^2 (5 A+3 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{105 b^3}\\ &=\frac {2 C \sin (c+d x)}{7 b d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (b B-a C) \sin (c+d x)}{5 b^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+7 a^2 C+5 b^2 C\right ) \sin (c+d x)}{21 b^3 d \sqrt {\sec (c+d x)}}-\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {5}{8} a b \left (7 A b^2-7 a b B+7 a^2 C+5 b^2 C\right )+\frac {5}{8} \left (21 a^3 b B+7 a b^3 B-21 a^4 C-7 a^2 b^2 (3 A+C)-b^4 (7 A+5 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{105 b^4}+\frac {\left (\left (5 a^2 b B+3 b^3 B-5 a^3 C-a 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}{5 b^4}\\ &=\frac {2 \left (5 a^2 b B+3 b^3 B-5 a^3 C-a 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 b^4 d}+\frac {2 C \sin (c+d x)}{7 b d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (b B-a C) \sin (c+d x)}{5 b^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+7 a^2 C+5 b^2 C\right ) \sin (c+d x)}{21 b^3 d \sqrt {\sec (c+d x)}}-\frac {\left (\left (21 a^3 b B+7 a b^3 B-21 a^4 C-7 a^2 b^2 (3 A+C)-b^4 (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}{21 b^5}-\frac {\left (a^3 \left (A b^2-a (b B-a C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{b^5}\\ &=\frac {2 \left (5 a^2 b B+3 b^3 B-5 a^3 C-a 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 b^4 d}-\frac {2 \left (21 a^3 b B+7 a b^3 B-21 a^4 C-7 a^2 b^2 (3 A+C)-b^4 (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 b^5 d}-\frac {2 a^3 \left (A b^2-a (b B-a C)\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{b^5 (a+b) d}+\frac {2 C \sin (c+d x)}{7 b d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (b B-a C) \sin (c+d x)}{5 b^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+7 a^2 C+5 b^2 C\right ) \sin (c+d x)}{21 b^3 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 6.67, size = 532, normalized size = 1.54 \[ \frac {4 b^2 \sin (c+d x) \left (70 a^2 C+42 b (b B-a C) \cos (c+d x)-70 a b B+70 A b^2+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right )-\frac {2 \cos (c+d x) \cot (c+d x) (a \sec (c+d x)+b) \left (-4 a b^2 \sqrt {-\tan ^2(c+d x)} \sqrt {\sec (c+d x)} \left (-28 a^2 C+28 a b B+35 A b^2+25 b^2 C\right ) \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )-2 b^2 \sqrt {-\tan ^2(c+d x)} \sqrt {\sec (c+d x)} \left (-35 a^3 C+35 a^2 b B-a b^2 (35 A+13 C)+63 b^3 B\right ) \left (F\left (\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )-\Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )\right )-21 \left (5 a^3 C-5 a^2 b B+a b^2 (5 A+3 C)-3 b^3 B\right ) \left (4 a^2 \sqrt {-\tan ^2(c+d x)} \sqrt {\sec (c+d x)} \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )-2 b^2 \sqrt {-\tan ^2(c+d x)} \sqrt {\sec (c+d x)} \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )-4 a b \sec ^2(c+d x)-2 b (2 a-b) \sqrt {-\tan ^2(c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )+4 a b \sqrt {-\tan ^2(c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )+4 a b\right )\right )}{a (a+b \cos (c+d x))}}{420 b^5 d \sqrt {\sec (c+d x)}} \]
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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 9.50, size = 1097, normalized size = 3.18 \[ \text {result too large to display} \]
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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (a+b\,\cos \left (c+d\,x\right )\right )} \,d x \]
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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