Optimal. Leaf size=579 \[ -\frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sec ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {a \left (-21 a^4 C-5 a^2 b^2 (A-7 C)+b^4 (11 A-8 C)\right ) \sin (c+d x)}{4 b^4 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)}}+\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x)}{4 b^2 d \left (a^2-b^2\right )^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}+\frac {\left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x)}{20 b^3 d \left (a^2-b^2\right )^2 \sec ^{\frac {3}{2}}(c+d x)}+\frac {a \left (-63 a^6 C-3 a^4 b^2 (5 A-43 C)+a^2 b^4 (33 A-64 C)-8 b^6 (3 A+C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^6 d \left (a^2-b^2\right )^2}+\frac {a^2 \left (63 a^6 C+15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+35 A b^6\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^6 d (a-b)^2 (a+b)^3}-\frac {\left (-315 a^6 C-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-8 b^6 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{20 b^5 d \left (a^2-b^2\right )^2} \]
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Rubi [A] time = 2.43, antiderivative size = 579, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4221, 3048, 3047, 3049, 3059, 2639, 3002, 2641, 2805} \[ \frac {\left (a^2 b^2 (15 A-101 C)+63 a^4 C-b^4 (45 A-8 C)\right ) \sin (c+d x)}{20 b^3 d \left (a^2-b^2\right )^2 \sec ^{\frac {3}{2}}(c+d x)}+\frac {a \left (-5 a^2 b^2 (A-7 C)-21 a^4 C+b^4 (11 A-8 C)\right ) \sin (c+d x)}{4 b^4 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)}}+\frac {\left (-a^2 b^2 (A-15 C)-9 a^4 C+7 A b^4\right ) \sin (c+d x)}{4 b^2 d \left (a^2-b^2\right )^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sec ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {a \left (-3 a^4 b^2 (5 A-43 C)+a^2 b^4 (33 A-64 C)-63 a^6 C-8 b^6 (3 A+C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^6 d \left (a^2-b^2\right )^2}-\frac {\left (-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{20 b^5 d \left (a^2-b^2\right )^2}+\frac {a^2 \left (15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+63 a^6 C+35 A b^6\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^6 d (a-b)^2 (a+b)^3} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2805
Rule 3002
Rule 3047
Rule 3048
Rule 3049
Rule 3059
Rule 4221
Rubi steps
\begin {align*} \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx\\ &=-\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (\frac {7}{2} \left (A b^2+a^2 C\right )-2 a b (A+C) \cos (c+d x)-\frac {1}{2} \left (5 A b^2+9 a^2 C-4 b^2 C\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (\frac {5}{4} \left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right )-a b \left (3 A b^2-\left (a^2-4 b^2\right ) C\right ) \cos (c+d x)+\frac {1}{4} \left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\cos (c+d x)} \left (\frac {3}{8} a \left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right )-\frac {1}{2} b \left (9 a^4 C-2 b^4 (5 A+3 C)-a^2 b^2 (5 A+18 C)\right ) \cos (c+d x)+\frac {15}{8} a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{5 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {15}{16} a^2 \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right )+\frac {3}{4} a b \left (a^2 b^2 (5 A-32 C)+21 a^4 C-4 b^4 (5 A+C)\right ) \cos (c+d x)-\frac {3}{16} \left (a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}-\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {15}{16} a^2 b \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right )-\frac {15}{16} a \left (a^2 b^4 (33 A-64 C)-3 a^4 b^2 (5 A-43 C)-63 a^6 C-8 b^6 (3 A+C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^5 \left (a^2-b^2\right )^2}-\frac {\left (\left (a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{40 b^5 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (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)}}{20 b^5 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {\left (a^2 \left (35 A b^6-a^2 b^4 (38 A-99 C)+15 a^4 b^2 (A-10 C)+63 a^6 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}{8 b^6 \left (a^2-b^2\right )^2}+\frac {\left (a \left (a^2 b^4 (33 A-64 C)-3 a^4 b^2 (5 A-43 C)-63 a^6 C-8 b^6 (3 A+C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{8 b^6 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (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)}}{20 b^5 \left (a^2-b^2\right )^2 d}+\frac {a \left (a^2 b^4 (33 A-64 C)-3 a^4 b^2 (5 A-43 C)-63 a^6 C-8 b^6 (3 A+C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 b^6 \left (a^2-b^2\right )^2 d}+\frac {a^2 \left (35 A b^6-a^2 b^4 (38 A-99 C)+15 a^4 b^2 (A-10 C)+63 a^6 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)}}{4 (a-b)^2 b^6 (a+b)^3 d}-\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 7.51, size = 925, normalized size = 1.60 \[ \frac {\frac {2 \left (105 C a^6+25 A b^2 a^4-211 b^2 C a^4-35 A b^4 a^2+112 b^4 C a^2+40 A b^6+24 b^6 C\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 ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (168 b C a^5+40 A b^3 a^3-256 b^3 C a^3-160 A b^5 a-32 b^5 C a\right ) \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (315 C a^6+75 A b^2 a^4-561 b^2 C a^4-145 A b^4 a^2+192 b^4 C a^2+40 A b^6+24 b^6 C\right ) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left (-4 \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left (\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left (\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 b^2 \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{80 (a-b)^2 b^4 (a+b)^2 d}+\frac {\sqrt {\sec (c+d x)} \left (-\frac {\left (75 C a^6+35 A b^2 a^4-107 b^2 C a^4-65 A b^4 a^2+4 b^4 C a^2-2 b^6 C\right ) \sin (c+d x)}{20 b^5 \left (a^2-b^2\right )^2}-\frac {-C \sin (c+d x) a^6-A b^2 \sin (c+d x) a^4}{2 b^5 \left (b^2-a^2\right ) (a+b \cos (c+d x))^2}+\frac {17 C \sin (c+d x) a^7+9 A b^2 \sin (c+d x) a^5-23 b^2 C \sin (c+d x) a^5-15 A b^4 \sin (c+d x) a^3}{4 b^5 \left (b^2-a^2\right )^2 (a+b \cos (c+d x))}-\frac {a C \sin (2 (c+d x))}{b^4}+\frac {C \sin (3 (c+d x))}{10 b^3}\right )}{d} \]
Warning: Unable to verify antiderivative.
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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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 16.49, size = 2466, normalized size = 4.26 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]
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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