Optimal. Leaf size=363 \[ \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)}}+\frac {\left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^3 d \left (a^2-b^2\right )}+\frac {a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^4 d (a-b) (a+b)^2}-\frac {\left (-15 a^4 B+9 a^3 A b+16 a^2 b^2 B-12 a A b^3+2 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b^4 d \left (a^2-b^2\right )} \]
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Rubi [A] time = 0.96, antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {2960, 4030, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)}}-\frac {\left (9 a^3 A b+16 a^2 b^2 B-15 a^4 B-12 a A b^3+2 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b^4 d \left (a^2-b^2\right )}+\frac {\left (3 a^2 A b-5 a^3 B+4 a b^2 B-2 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^3 d \left (a^2-b^2\right )}+\frac {a^2 \left (3 a^2 A b-5 a^3 B+7 a b^2 B-5 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^4 d (a-b) (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2805
Rule 2960
Rule 3771
Rule 3787
Rule 3849
Rule 4030
Rule 4104
Rule 4106
Rubi steps
\begin {align*} \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x)} \, dx &=\int \frac {B+A \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))^2} \, dx\\ &=\frac {a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}+\frac {\int \frac {\frac {1}{2} \left (-3 a A b+5 a^2 B-2 b^2 B\right )-b (A b-a B) \sec (c+d x)+\frac {3}{2} a (A b-a B) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac {\left (3 a A b-5 a^2 B+2 b^2 B\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}-\frac {2 \int \frac {-\frac {3}{4} \left (3 a^2 A b-2 A b^3-5 a^3 B+4 a b^2 B\right )-\frac {1}{2} b \left (3 a A b-2 a^2 B-b^2 B\right ) \sec (c+d x)+\frac {1}{4} a \left (3 a A b-5 a^2 B+2 b^2 B\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (b+a \sec (c+d x))} \, dx}{3 b^2 \left (a^2-b^2\right )}\\ &=-\frac {\left (3 a A b-5 a^2 B+2 b^2 B\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}-\frac {2 \int \frac {-\frac {3}{4} b \left (3 a^2 A b-2 A b^3-5 a^3 B+4 a b^2 B\right )-\left (\frac {1}{2} b^2 \left (3 a A b-2 a^2 B-b^2 B\right )-\frac {3}{4} a \left (3 a^2 A b-2 A b^3-5 a^3 B+4 a b^2 B\right )\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{3 b^4 \left (a^2-b^2\right )}+\frac {\left (a^2 \left (3 a^2 A b-5 A b^3-5 a^3 B+7 a b^2 B\right )\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{b+a \sec (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )}\\ &=-\frac {\left (3 a A b-5 a^2 B+2 b^2 B\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}+\frac {\left (3 a^2 A b-2 A b^3-5 a^3 B+4 a b^2 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{2 b^3 \left (a^2-b^2\right )}-\frac {\left (9 a^3 A b-12 a A b^3-15 a^4 B+16 a^2 b^2 B+2 b^4 B\right ) \int \sqrt {\sec (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )}+\frac {\left (a^2 \left (3 a^2 A b-5 A b^3-5 a^3 B+7 a b^2 B\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}{2 b^4 \left (a^2-b^2\right )}\\ &=\frac {a^2 \left (3 a^2 A b-5 A b^3-5 a^3 B+7 a b^2 B\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)}}{(a-b) b^4 (a+b)^2 d}-\frac {\left (3 a A b-5 a^2 B+2 b^2 B\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}+\frac {\left (\left (3 a^2 A b-2 A b^3-5 a^3 B+4 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )}-\frac {\left (\left (9 a^3 A b-12 a A b^3-15 a^4 B+16 a^2 b^2 B+2 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{6 b^4 \left (a^2-b^2\right )}\\ &=\frac {\left (3 a^2 A b-2 A b^3-5 a^3 B+4 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{b^3 \left (a^2-b^2\right ) d}-\frac {\left (9 a^3 A b-12 a A b^3-15 a^4 B+16 a^2 b^2 B+2 b^4 B\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{3 b^4 \left (a^2-b^2\right ) d}+\frac {a^2 \left (3 a^2 A b-5 A b^3-5 a^3 B+7 a b^2 B\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)}}{(a-b) b^4 (a+b)^2 d}-\frac {\left (3 a A b-5 a^2 B+2 b^2 B\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}\\ \end {align*}
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Mathematica [A] time = 6.99, size = 701, normalized size = 1.93 \[ \frac {\sqrt {\sec (c+d x)} \left (\frac {a^2 (a B-A b) \sin (c+d x)}{b^3 \left (a^2-b^2\right )}-\frac {a^3 A b \sin (c+d x)-a^4 B \sin (c+d x)}{b^3 \left (b^2-a^2\right ) (a+b \cos (c+d x))}+\frac {B \sin (2 (c+d x))}{3 b^2}\right )}{d}-\frac {\frac {2 \left (8 a^2 b B-12 a A b^2+4 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )}{b \left (1-\cos ^2(c+d x)\right ) (a+b \cos (c+d x))}+\frac {2 \left (5 a^3 B-3 a^2 A b-8 a b^2 B+6 A b^3\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left (F\left (\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )-\Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )\right )}{a \left (1-\cos ^2(c+d x)\right ) (a+b \cos (c+d x))}+\frac {\left (15 a^3 B-9 a^2 A b-12 a b^2 B+6 A b^3\right ) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left (-4 a^2 \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)} \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )+2 b^2 \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)} \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)} F\left (\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )-4 a b \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )-4 a b\right )}{a b^2 \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right ) (a+b \cos (c+d x))}}{12 b^2 d (a-b) (a+b)} \]
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 \[ \int \frac {B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 5.32, size = 1066, normalized size = 2.94 \[ \text {result too large to display} \]
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 \[ \int \frac {B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {A+B\,\cos \left (c+d\,x\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2} \,d x \]
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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