Optimal. Leaf size=196 \[ \frac{2 \left (a^2 A-3 a b B+3 A b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 a^3 d}-\frac{2 b^2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 d (a+b)}-\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.467163, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {4034, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{2 \left (a^2 A-3 a b B+3 A b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^3 d}-\frac{2 b^2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 d (a+b)}-\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4034
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))} \, dx &=\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{2 \int \frac{\frac{3}{2} (A b-a B)-\frac{1}{2} a A \sec (c+d x)-\frac{1}{2} A b \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{3 a}\\ &=\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{2 \int \frac{\frac{3}{2} a (A b-a B)-\left (\frac{a^2 A}{2}+\frac{3}{2} b (A b-a B)\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{3 a^3}-\frac{\left (b^2 (A b-a B)\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{a^3}\\ &=\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A b-a B) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{a^2}+\frac{\left (a^2 A+3 A b^2-3 a b B\right ) \int \sqrt{\sec (c+d x)} \, dx}{3 a^3}-\frac{\left (b^2 (A b-a B) \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}{a^3}\\ &=-\frac{2 b^2 (A b-a B) \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^3 (a+b) d}+\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{\left ((A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{a^2}+\frac{\left (\left (a^2 A+3 A b^2-3 a b B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 a^3}\\ &=-\frac{2 (A b-a B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{2 \left (a^2 A+3 A b^2-3 a b 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^3 d}-\frac{2 b^2 (A b-a B) \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^3 (a+b) d}+\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 6.52404, size = 282, normalized size = 1.44 \[ \frac{2 \csc (c+d x) \left (a (a A+3 a B-3 A b) \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )+a^2 A \sin (c+d x) \tan (c+d x)+3 a^2 B \sec ^2(c+d x)-3 a^2 B-3 A b^2 \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-3 a (a B-A b) \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-3 a A b \sec ^2(c+d x)+3 a A b+3 a b B \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )}{3 a^3 d \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.248, size = 786, normalized size = 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] 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 )} \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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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 )} \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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