Optimal. Leaf size=402 \[ \frac{\left (3 a^2 A b+a^3 B-7 a b^2 B+3 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{4 a b d \left (a^2-b^2\right )^2}+\frac{a (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac{a \left (a^2 A b+3 a^3 B-9 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac{\left (a^2 A b+3 a^3 B-9 a b^2 B+5 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^2 d \left (a^2-b^2\right )^2}+\frac{\left (-10 a^2 A b^3+a^4 A b-6 a^3 b^2 B+3 a^5 B+15 a b^4 B-3 A b^5\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a b^2 d (a-b)^2 (a+b)^3} \]
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Rubi [A] time = 0.914029, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4029, 4098, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{a (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac{a \left (a^2 A b+3 a^3 B-9 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac{\left (3 a^2 A b+a^3 B-7 a b^2 B+3 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a b d \left (a^2-b^2\right )^2}+\frac{\left (a^2 A b+3 a^3 B-9 a b^2 B+5 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^2 d \left (a^2-b^2\right )^2}+\frac{\left (-10 a^2 A b^3+a^4 A b-6 a^3 b^2 B+3 a^5 B+15 a b^4 B-3 A b^5\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a b^2 d (a-b)^2 (a+b)^3} \]
Antiderivative was successfully verified.
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Rule 4029
Rule 4098
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx &=\frac{a (A b-a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\int \frac{\sqrt{\sec (c+d x)} \left (\frac{1}{2} a (A b-a B)-2 b (A b-a B) \sec (c+d x)+\frac{1}{2} \left (a A b+3 a^2 B-4 b^2 B\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac{a (A b-a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac{a \left (a^2 A b+5 A b^3+3 a^3 B-9 a b^2 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\int \frac{-\frac{1}{4} a \left (a^2 A b+5 A b^3+3 a^3 B-9 a b^2 B\right )-b \left (a^2 A b+2 A b^3+a^3 B-4 a b^2 B\right ) \sec (c+d x)-\frac{1}{4} \left (a^3 A b-7 a A b^3+3 a^4 B-5 a^2 b^2 B+8 b^4 B\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{a (A b-a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac{a \left (a^2 A b+5 A b^3+3 a^3 B-9 a b^2 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\int \frac{-\frac{1}{4} a^2 \left (a^2 A b+5 A b^3+3 a^3 B-9 a b^2 B\right )-\left (-\frac{1}{4} a b \left (a^2 A b+5 A b^3+3 a^3 B-9 a b^2 B\right )+a b \left (a^2 A b+2 A b^3+a^3 B-4 a b^2 B\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{2 a^2 b^2 \left (a^2-b^2\right )^2}+\frac{\left (a^4 A b-10 a^2 A b^3-3 A b^5+3 a^5 B-6 a^3 b^2 B+15 a b^4 B\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{8 a b^2 \left (a^2-b^2\right )^2}\\ &=\frac{a (A b-a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac{a \left (a^2 A b+5 A b^3+3 a^3 B-9 a b^2 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\left (a^2 A b+5 A b^3+3 a^3 B-9 a b^2 B\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{8 b^2 \left (a^2-b^2\right )^2}+\frac{\left (3 a^2 A b+3 A b^3+a^3 B-7 a b^2 B\right ) \int \sqrt{\sec (c+d x)} \, dx}{8 a b \left (a^2-b^2\right )^2}+\frac{\left (\left (a^4 A b-10 a^2 A b^3-3 A b^5+3 a^5 B-6 a^3 b^2 B+15 a b^4 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{8 a b^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (a^4 A b-10 a^2 A b^3-3 A b^5+3 a^5 B-6 a^3 b^2 B+15 a b^4 B\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 a (a-b)^2 b^2 (a+b)^3 d}+\frac{a (A b-a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac{a \left (a^2 A b+5 A b^3+3 a^3 B-9 a b^2 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\left (\left (a^2 A b+5 A b^3+3 a^3 B-9 a b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{8 b^2 \left (a^2-b^2\right )^2}+\frac{\left (\left (3 a^2 A b+3 A b^3+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)}} \, dx}{8 a b \left (a^2-b^2\right )^2}\\ &=\frac{\left (a^2 A b+5 A b^3+3 a^3 B-9 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)}}{4 b^2 \left (a^2-b^2\right )^2 d}+\frac{\left (3 a^2 A b+3 A b^3+a^3 B-7 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 b \left (a^2-b^2\right )^2 d}+\frac{\left (a^4 A b-10 a^2 A b^3-3 A b^5+3 a^5 B-6 a^3 b^2 B+15 a b^4 B\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 a (a-b)^2 b^2 (a+b)^3 d}+\frac{a (A b-a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac{a \left (a^2 A b+5 A b^3+3 a^3 B-9 a b^2 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 7.01683, size = 800, normalized size = 1.99 \[ \frac{-\frac{2 \left (16 A b^4-32 a B b^3+8 a^2 A b^2+8 a^3 B b\right ) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac{2 \left (9 B a^4+3 A b a^3-19 b^2 B a^2-9 A b^3 a+16 b^4 B\right ) \left (\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )+\Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right ) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}-\frac{2 \left (3 B a^4+A b a^3-9 b^2 B a^2+5 A b^3 a\right ) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (\Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2-2 b \sec ^2(c+d x) a+2 b a+2 b E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+(a-2 b) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 b^2 \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{16 (a-b)^2 b^2 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left (-\frac{\left (3 B a^3+A b a^2-9 b^2 B a+5 A b^3\right ) \sin (c+d x)}{4 b^2 \left (b^2-a^2\right )^2}+\frac{A b \sin (c+d x)-a B \sin (c+d x)}{2 \left (b^2-a^2\right ) (b+a \cos (c+d x))^2}+\frac{B \sin (c+d x) a^3+3 A b \sin (c+d x) a^2-7 b^2 B \sin (c+d x) a+3 A b^3 \sin (c+d x)}{4 b \left (b^2-a^2\right )^2 (b+a \cos (c+d x))}\right )}{d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 8.45, size = 1768, normalized size = 4.4 \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 \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}{{\left (b \sec \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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