Optimal. Leaf size=523 \[ \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}+\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{4 b^2 d \left (a^2-b^2\right )^2 \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac {\left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{12 b^3 d \left (a^2-b^2\right )^2}-\frac {\left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{12 b^3 d \left (a^2-b^2\right )^2 \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^4 d \left (a^2-b^2\right )^2}-\frac {a \left (-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^4 d (a-b)^2 (a+b)^3}+\frac {\left (-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right ) \sin (c+d x)}{4 b^4 d \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)}} \]
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Rubi [A] time = 1.98, antiderivative size = 523, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {2954, 3000, 3055, 3059, 2639, 3002, 2641, 2805} \[ -\frac {\left (15 a^3 A b+61 a^2 b^2 B-35 a^4 B-33 a A b^3-8 b^4 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{12 b^3 d \left (a^2-b^2\right )^2}-\frac {\left (-29 a^2 A b^3+15 a^4 A b+65 a^3 b^2 B-35 a^5 B-24 a b^4 B+8 A b^5\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^4 d \left (a^2-b^2\right )^2}-\frac {a \left (-38 a^2 A b^3+15 a^4 A b+86 a^3 b^2 B-35 a^5 B-63 a b^4 B+35 A b^5\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^4 d (a-b)^2 (a+b)^3}+\frac {a \left (3 a^2 A b-7 a^3 B+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{4 b^2 d \left (a^2-b^2\right )^2 \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac {\left (15 a^3 A b+61 a^2 b^2 B-35 a^4 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{12 b^3 d \left (a^2-b^2\right )^2 \cos ^{\frac {3}{2}}(c+d x)}+\frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}+\frac {\left (-29 a^2 A b^3+15 a^4 A b+65 a^3 b^2 B-35 a^5 B-24 a b^4 B+8 A b^5\right ) \sin (c+d x)}{4 b^4 d \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2805
Rule 2954
Rule 3000
Rule 3002
Rule 3055
Rule 3059
Rubi steps
\begin {align*} \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx &=\int \frac {B+A \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (b+a \cos (c+d x))^3} \, dx\\ &=\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2}-\frac {\int \frac {\frac {1}{2} \left (3 a A b-7 a^2 B+4 b^2 B\right )+2 b (A b-a B) \cos (c+d x)-\frac {5}{2} a (A b-a B) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (b+a \cos (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2}+\frac {a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (-15 a^3 A b+33 a A b^3+35 a^4 B-61 a^2 b^2 B+8 b^4 B\right )+b \left (a^2 A b+2 A b^3+a^3 B-4 a b^2 B\right ) \cos (c+d x)+\frac {3}{4} a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (b+a \cos (c+d x))} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2}+\frac {a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))}+\frac {\int \frac {\frac {3}{8} \left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right )+\frac {1}{2} b \left (3 a^3 A b-12 a A b^3-7 a^4 B+14 a^2 b^2 B+2 b^4 B\right ) \cos (c+d x)-\frac {1}{8} a \left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))} \, dx}{3 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2}+\frac {a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))}+\frac {2 \int \frac {\frac {1}{16} \left (-45 a^5 A b+99 a^3 A b^3-72 a A b^5+105 a^6 B-223 a^4 b^2 B+128 a^2 b^4 B+8 b^6 B\right )-\frac {1}{4} b \left (15 a^4 A b-30 a^2 A b^3+6 A b^5-35 a^5 B+64 a^3 b^2 B-20 a b^4 B\right ) \cos (c+d x)-\frac {3}{16} a \left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{3 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2}+\frac {a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))}-\frac {2 \int \frac {\frac {1}{16} a \left (45 a^5 A b-99 a^3 A b^3+72 a A b^5-105 a^6 B+223 a^4 b^2 B-128 a^2 b^4 B-8 b^6 B\right )+\frac {1}{16} a^2 b \left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{3 a b^4 \left (a^2-b^2\right )^2}-\frac {\left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right ) \int \sqrt {\cos (c+d x)} \, dx}{8 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2}+\frac {a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))}-\frac {\left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{24 b^3 \left (a^2-b^2\right )^2}-\frac {\left (a \left (15 a^4 A b-38 a^2 A b^3+35 A b^5-35 a^5 B+86 a^3 b^2 B-63 a b^4 B\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{8 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac {a \left (15 a^4 A b-38 a^2 A b^3+35 A b^5-35 a^5 B+86 a^3 b^2 B-63 a b^4 B\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 (a-b)^2 b^4 (a+b)^3 d}-\frac {\left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2}+\frac {a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))}\\ \end {align*}
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Mathematica [A] time = 7.36, size = 570, normalized size = 1.09 \[ \frac {\sqrt {\cos (c+d x)} \left (\frac {a^4 B \sin (c+d x)-a^3 A b \sin (c+d x)}{2 b^3 \left (b^2-a^2\right ) (a \cos (c+d x)+b)^2}+\frac {-11 a^6 B \sin (c+d x)+7 a^5 A b \sin (c+d x)+17 a^4 b^2 B \sin (c+d x)-13 a^3 A b^3 \sin (c+d x)}{4 b^4 \left (b^2-a^2\right )^2 (a \cos (c+d x)+b)}+\frac {2 \sec (c+d x) (A b \sin (c+d x)-3 a B \sin (c+d x))}{b^4}+\frac {2 B \tan (c+d x) \sec (c+d x)}{3 b^3}\right )}{d}+\frac {\frac {\left (280 a^5 b B-120 a^4 A b^2-512 a^3 b^3 B+240 a^2 A b^4+160 a b^5 B-48 A b^6\right ) \left (2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-\frac {2 b \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}\right )}{a}+\frac {2 \left (105 a^6 B-45 a^5 A b-195 a^4 b^2 B+87 a^3 A b^3+72 a^2 b^4 B-24 a A b^5\right ) \sin (c+d x) \cos (2 (c+d x)) \left (\left (a^2-2 b^2\right ) \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) F\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )\right )}{a^2 b \sqrt {1-\cos ^2(c+d x)} \left (2 \cos ^2(c+d x)-1\right )}+\frac {2 \left (315 a^6 B-135 a^5 A b-641 a^4 b^2 B+285 a^3 A b^3+328 a^2 b^4 B-168 a A b^5+16 b^6 B\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}}{48 b^4 d (a-b)^2 (a+b)^2} \]
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 \[ \int \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 35.26, size = 2178, normalized size = 4.16 \[ \text {Expression too large to display} \]
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 \[ \int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^{9/2}\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]
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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