Optimal. Leaf size=480 \[ \frac{b (A b-a B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}+\frac{b \left (11 a^2 A b-7 a^3 B+a b^2 B-5 A b^3\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left (a^2-b^2\right )^2 (a \sec (c+d x)+b)}+\frac{\left (-29 a^2 A b^2+8 a^4 A+9 a^3 b B-3 a b^3 B+15 A b^4\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^3 d \left (a^2-b^2\right )^2}+\frac{\left (11 a^2 A b-7 a^3 B+a b^2 B-5 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^2 d \left (a^2-b^2\right )^2}-\frac{\left (-29 a^2 A b^2+8 a^4 A+9 a^3 b B-3 a b^3 B+15 A b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^3 d \left (a^2-b^2\right )^2}-\frac{\left (-38 a^2 A b^3+35 a^4 A b+6 a^3 b^2 B-15 a^5 B-3 a b^4 B+15 A b^5\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^3 d (a-b)^2 (a+b)^3} \]
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Rubi [A] time = 1.45029, antiderivative size = 480, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2960, 4029, 4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{b (A b-a B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}+\frac{b \left (11 a^2 A b-7 a^3 B+a b^2 B-5 A b^3\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left (a^2-b^2\right )^2 (a \sec (c+d x)+b)}+\frac{\left (-29 a^2 A b^2+8 a^4 A+9 a^3 b B-3 a b^3 B+15 A b^4\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^3 d \left (a^2-b^2\right )^2}+\frac{\left (11 a^2 A b-7 a^3 B+a b^2 B-5 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^2 d \left (a^2-b^2\right )^2}-\frac{\left (-29 a^2 A b^2+8 a^4 A+9 a^3 b B-3 a b^3 B+15 A b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^3 d \left (a^2-b^2\right )^2}-\frac{\left (-38 a^2 A b^3+35 a^4 A b+6 a^3 b^2 B-15 a^5 B-3 a b^4 B+15 A b^5\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^3 d (a-b)^2 (a+b)^3} \]
Antiderivative was successfully verified.
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Rule 2960
Rule 4029
Rule 4098
Rule 4102
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx &=\int \frac{\sec ^{\frac{7}{2}}(c+d x) (B+A \sec (c+d x))}{(b+a \sec (c+d x))^3} \, dx\\ &=\frac{b (A b-a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}-\frac{\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left (-\frac{3}{2} b (A b-a B)+2 a (A b-a B) \sec (c+d x)-\frac{1}{2} \left (4 a^2 A-5 A b^2+a b B\right ) \sec ^2(c+d x)\right )}{(b+a \sec (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac{b (A b-a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{b \left (11 a^2 A b-5 A b^3-7 a^3 B+a b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}-\frac{\int \frac{\sqrt{\sec (c+d x)} \left (-\frac{1}{4} b \left (11 a^2 A b-5 A b^3-7 a^3 B+a b^2 B\right )+a \left (4 a^2 A b-A b^3-2 a^3 B-a b^2 B\right ) \sec (c+d x)-\frac{1}{4} \left (8 a^4 A-29 a^2 A b^2+15 A b^4+9 a^3 b B-3 a b^3 B\right ) \sec ^2(c+d x)\right )}{b+a \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (8 a^4 A-29 a^2 A b^2+15 A b^4+9 a^3 b B-3 a b^3 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac{b (A b-a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{b \left (11 a^2 A b-5 A b^3-7 a^3 B+a b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}-\frac{\int \frac{\frac{1}{8} b \left (8 a^4 A-29 a^2 A b^2+15 A b^4+9 a^3 b B-3 a b^3 B\right )+\frac{1}{2} a \left (2 a^4 A-10 a^2 A b^2+5 A b^4+4 a^3 b B-a b^3 B\right ) \sec (c+d x)+\frac{1}{8} \left (24 a^4 A b-33 a^2 A b^3+15 A b^5-8 a^5 B+5 a^3 b^2 B-3 a b^4 B\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (b+a \sec (c+d x))} \, dx}{a^3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (8 a^4 A-29 a^2 A b^2+15 A b^4+9 a^3 b B-3 a b^3 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac{b (A b-a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{b \left (11 a^2 A b-5 A b^3-7 a^3 B+a b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}-\frac{\int \frac{\frac{1}{8} b^2 \left (8 a^4 A-29 a^2 A b^2+15 A b^4+9 a^3 b B-3 a b^3 B\right )-\left (\frac{1}{8} a b \left (8 a^4 A-29 a^2 A b^2+15 A b^4+9 a^3 b B-3 a b^3 B\right )-\frac{1}{2} a b \left (2 a^4 A-10 a^2 A b^2+5 A b^4+4 a^3 b B-a b^3 B\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{a^3 b^2 \left (a^2-b^2\right )^2}-\frac{\left (35 a^4 A b-38 a^2 A b^3+15 A b^5-15 a^5 B+6 a^3 b^2 B-3 a b^4 B\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{b+a \sec (c+d x)} \, dx}{8 a^3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (8 a^4 A-29 a^2 A b^2+15 A b^4+9 a^3 b B-3 a b^3 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac{b (A b-a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{b \left (11 a^2 A b-5 A b^3-7 a^3 B+a b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac{\left (11 a^2 A b-5 A b^3-7 a^3 B+a b^2 B\right ) \int \sqrt{\sec (c+d x)} \, dx}{8 a^2 \left (a^2-b^2\right )^2}-\frac{\left (8 a^4 A-29 a^2 A b^2+15 A b^4+9 a^3 b B-3 a b^3 B\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{8 a^3 \left (a^2-b^2\right )^2}-\frac{\left (\left (35 a^4 A b-38 a^2 A b^3+15 A b^5-15 a^5 B+6 a^3 b^2 B-3 a b^4 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}{8 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (35 a^4 A b-38 a^2 A b^3+15 A b^5-15 a^5 B+6 a^3 b^2 B-3 a b^4 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)}}{4 a^3 (a-b)^2 (a+b)^3 d}+\frac{\left (8 a^4 A-29 a^2 A b^2+15 A b^4+9 a^3 b B-3 a b^3 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac{b (A b-a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{b \left (11 a^2 A b-5 A b^3-7 a^3 B+a b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac{\left (\left (11 a^2 A b-5 A b^3-7 a^3 B+a b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{8 a^2 \left (a^2-b^2\right )^2}-\frac{\left (\left (8 a^4 A-29 a^2 A b^2+15 A b^4+9 a^3 b B-3 a b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{8 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (8 a^4 A-29 a^2 A b^2+15 A b^4+9 a^3 b B-3 a b^3 B\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (11 a^2 A b-5 A b^3-7 a^3 B+a b^2 B\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 a^2 \left (a^2-b^2\right )^2 d}-\frac{\left (35 a^4 A b-38 a^2 A b^3+15 A b^5-15 a^5 B+6 a^3 b^2 B-3 a b^4 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)}}{4 a^3 (a-b)^2 (a+b)^3 d}+\frac{\left (8 a^4 A-29 a^2 A b^2+15 A b^4+9 a^3 b B-3 a b^3 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac{b (A b-a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{b \left (11 a^2 A b-5 A b^3-7 a^3 B+a b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 7.16484, size = 850, normalized size = 1.77 \[ \frac{\sqrt{\sec (c+d x)} \left (\frac{\left (8 A a^4+9 b B a^3-29 A b^2 a^2-3 b^3 B a+15 A b^4\right ) \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2}+\frac{A b^2 \sin (c+d x)-a b B \sin (c+d x)}{2 a \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac{-5 A \sin (c+d x) b^4+a B \sin (c+d x) b^3+11 a^2 A \sin (c+d x) b^2-7 a^3 B \sin (c+d x) b}{4 a^2 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}\right )}{d}-\frac{-\frac{2 \left (16 A a^5+32 b B a^4-80 A b^2 a^3-8 b^3 B a^2+40 A b^4 a\right ) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac{2 \left (-16 B a^5+56 A b a^4+19 b^2 B a^3-95 A b^3 a^2-9 b^4 B a+45 A b^5\right ) \left (F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+\Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right ) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac{\left (15 A b^5-3 a B b^4-29 a^2 A b^3+9 a^3 B b^2+8 a^4 A b\right ) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left (4 \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}-2 b^2 \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{16 a^3 (a-b)^2 (a+b)^2 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 19.237, size = 2002, normalized size = 4.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac{3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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