Optimal. Leaf size=365 \[ \frac{\left (16 a^2 A b^2+2 a^4 A-12 a^3 b B+9 a b^3 B-15 A b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 a^4 d \left (a^2-b^2\right )}+\frac{b (A b-a B) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{\left (2 a^2 A+3 a b B-5 A b^2\right ) \sin (c+d x)}{3 a^2 d \left (a^2-b^2\right ) \sqrt{\sec (c+d x)}}-\frac{\left (4 a^2 A b-2 a^3 B+3 a b^2 B-5 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 d \left (a^2-b^2\right )}-\frac{b^2 \left (7 a^2 A b-5 a^3 B+3 a b^2 B-5 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^4 d (a-b) (a+b)^2} \]
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Rubi [A] time = 0.859338, antiderivative size = 365, 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 = {4030, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{b (A b-a B) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{\left (2 a^2 A+3 a b B-5 A b^2\right ) \sin (c+d x)}{3 a^2 d \left (a^2-b^2\right ) \sqrt{\sec (c+d x)}}+\frac{\left (16 a^2 A b^2+2 a^4 A-12 a^3 b B+9 a b^3 B-15 A b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^4 d \left (a^2-b^2\right )}-\frac{\left (4 a^2 A b-2 a^3 B+3 a b^2 B-5 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 d \left (a^2-b^2\right )}-\frac{b^2 \left (7 a^2 A b-5 a^3 B+3 a b^2 B-5 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^4 d (a-b) (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 4030
Rule 4104
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx &=\frac{b (A b-a B) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}-\frac{\int \frac{\frac{1}{2} \left (-2 a^2 A+5 A b^2-3 a b B\right )+a (A b-a B) \sec (c+d x)-\frac{3}{2} b (A b-a B) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx}{a \left (a^2-b^2\right )}\\ &=\frac{\left (2 a^2 A-5 A b^2+3 a b B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)}}+\frac{b (A b-a B) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{2 \int \frac{-\frac{3}{4} \left (4 a^2 A b-5 A b^3-2 a^3 B+3 a b^2 B\right )+\frac{1}{2} a \left (a^2 A+2 A b^2-3 a b B\right ) \sec (c+d x)+\frac{1}{4} b \left (2 a^2 A-5 A b^2+3 a b B\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{3 a^2 \left (a^2-b^2\right )}\\ &=\frac{\left (2 a^2 A-5 A b^2+3 a b B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)}}+\frac{b (A b-a B) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{2 \int \frac{-\frac{3}{4} a \left (4 a^2 A b-5 A b^3-2 a^3 B+3 a b^2 B\right )-\left (-\frac{1}{2} a^2 \left (a^2 A+2 A b^2-3 a b B\right )-\frac{3}{4} b \left (4 a^2 A b-5 A b^3-2 a^3 B+3 a b^2 B\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{3 a^4 \left (a^2-b^2\right )}-\frac{\left (b^2 \left (7 a^2 A b-5 A b^3-5 a^3 B+3 a b^2 B\right )\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^4 \left (a^2-b^2\right )}\\ &=\frac{\left (2 a^2 A-5 A b^2+3 a b B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)}}+\frac{b (A b-a B) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}-\frac{\left (4 a^2 A b-5 A b^3-2 a^3 B+3 a b^2 B\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{2 a^3 \left (a^2-b^2\right )}+\frac{\left (2 a^4 A+16 a^2 A b^2-15 A b^4-12 a^3 b B+9 a b^3 B\right ) \int \sqrt{\sec (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )}-\frac{\left (b^2 \left (7 a^2 A b-5 A b^3-5 a^3 B+3 a b^2 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}{2 a^4 \left (a^2-b^2\right )}\\ &=-\frac{b^2 \left (7 a^2 A b-5 A b^3-5 a^3 B+3 a b^2 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)}}{a^4 (a-b) (a+b)^2 d}+\frac{\left (2 a^2 A-5 A b^2+3 a b B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)}}+\frac{b (A b-a B) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}-\frac{\left (\left (4 a^2 A b-5 A b^3-2 a^3 B+3 a b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )}+\frac{\left (\left (2 a^4 A+16 a^2 A b^2-15 A b^4-12 a^3 b B+9 a b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 a^4 \left (a^2-b^2\right )}\\ &=-\frac{\left (4 a^2 A b-5 A b^3-2 a^3 B+3 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)}}{a^3 \left (a^2-b^2\right ) d}+\frac{\left (2 a^4 A+16 a^2 A b^2-15 A b^4-12 a^3 b B+9 a b^3 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 a^4 \left (a^2-b^2\right ) d}-\frac{b^2 \left (7 a^2 A b-5 A b^3-5 a^3 B+3 a b^2 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)}}{a^4 (a-b) (a+b)^2 d}+\frac{\left (2 a^2 A-5 A b^2+3 a b B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)}}+\frac{b (A b-a B) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 7.14556, size = 704, normalized size = 1.93 \[ \frac{\frac{2 \left (-8 a^2 A b+6 a^3 B-3 a b^2 B+5 A b^3\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left (\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )+\Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )}{b \left (1-\cos ^2(c+d x)\right ) (a \cos (c+d x)+b)}-\frac{2 \left (-12 a^2 A b+6 a^3 B-9 a b^2 B+15 A b^3\right ) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )+a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-2 a b \sec ^2(c+d x)+2 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+2 a b\right )}{a^2 b \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right ) (a \cos (c+d x)+b)}-\frac{2 \left (4 a^3 A-12 a^2 b B+8 a A b^2\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )}{a \left (1-\cos ^2(c+d x)\right ) (a \cos (c+d x)+b)}}{12 a^2 d (a-b) (a+b)}+\frac{\sqrt{\sec (c+d x)} \left (\frac{b^2 (A b-a B) \sin (c+d x)}{a^3 \left (b^2-a^2\right )}-\frac{a b^3 B \sin (c+d x)-A b^4 \sin (c+d x)}{a^3 \left (a^2-b^2\right ) (a \cos (c+d x)+b)}+\frac{A \sin (2 (c+d x))}{3 a^2}\right )}{d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 7.161, size = 1059, normalized size = 2.9 \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{B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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