Optimal. Leaf size=317 \[ \frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (-a^2 b (4 A+C)+2 a^3 B-a b^2 B+3 A b^3\right )}{a^3 d \left (a^2-b^2\right )}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 (-(2 A-C))-a b B+3 A b^2\right )}{a^2 d \left (a^2-b^2\right )}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (-a^2 b^2 (5 A+C)+3 a^3 b B+a^4 (-C)-a b^3 B+3 A b^4\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 d (a-b) (a+b)^2} \]
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Rubi [A] time = 0.678562, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-a^2 b (4 A+C)+2 a^3 B-a b^2 B+3 A b^3\right )}{a^3 d \left (a^2-b^2\right )}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 (-(2 A-C))-a b B+3 A b^2\right )}{a^2 d \left (a^2-b^2\right )}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (-a^2 b^2 (5 A+C)+3 a^3 b B+a^4 (-C)-a b^3 B+3 A b^4\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 d (a-b) (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 4100
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)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2} \, dx &=\frac{\left (A b^2-a (b B-a C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{\int \frac{\frac{1}{2} \left (3 A b^2-a b B-a^2 (2 A-C)\right )+a (A b-a B+b C) \sec (c+d x)-\frac{1}{2} \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{a \left (a^2-b^2\right )}\\ &=\frac{\left (A b^2-a (b B-a C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{\int \frac{\frac{1}{2} a \left (3 A b^2-a b B-a^2 (2 A-C)\right )-\left (\frac{1}{2} b \left (3 A b^2-a b B-a^2 (2 A-C)\right )-a^2 (A b-a B+b C)\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{a^3 \left (a^2-b^2\right )}-\frac{\left (3 A b^4+3 a^3 b B-a b^3 B-a^4 C-a^2 b^2 (5 A+C)\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )}\\ &=\frac{\left (A b^2-a (b B-a C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{\left (3 A b^2-a b B-a^2 (2 A-C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{2 a^2 \left (a^2-b^2\right )}+\frac{\left (3 A b^3+2 a^3 B-a b^2 B-a^2 b (4 A+C)\right ) \int \sqrt{\sec (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )}-\frac{\left (\left (3 A b^4+3 a^3 b B-a b^3 B-a^4 C-a^2 b^2 (5 A+C)\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^3 \left (a^2-b^2\right )}\\ &=-\frac{\left (3 A b^4+3 a^3 b B-a b^3 B-a^4 C-a^2 b^2 (5 A+C)\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^3 (a-b) (a+b)^2 d}+\frac{\left (A b^2-a (b B-a C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{\left (\left (3 A b^2-a b B-a^2 (2 A-C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )}+\frac{\left (\left (3 A b^3+2 a^3 B-a b^2 B-a^2 b (4 A+C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{2 a^3 \left (a^2-b^2\right )}\\ &=-\frac{\left (3 A b^2-a b B-a^2 (2 A-C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^2 \left (a^2-b^2\right ) d}+\frac{\left (3 A b^3+2 a^3 B-a b^2 B-a^2 b (4 A+C)\right ) \sqrt{\cos (c+d x)} F\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 (3 A b^4+3 a^3 b B-a b^3 B-a^4 C-a^2 b^2 (5 A+C)\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^3 (a-b) (a+b)^2 d}+\frac{\left (A b^2-a (b B-a C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [B] time = 7.13084, size = 835, normalized size = 2.63 \[ \frac{\left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (-\frac{2 \left (-4 B a^2+4 A b a+4 b C a\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 (-2 A a^2-C a^2+b B a+A b^2\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 (-2 A a^2+C a^2-b B a+3 A b^2\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 )}\right ) (b+a \cos (c+d x))^2}{2 a (b-a) (a+b) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2}+\frac{\sqrt{\sec (c+d x)} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac{2 \left (C a^2-b B a+A b^2\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right )}-\frac{2 \left (A \sin (c+d x) b^3-a B \sin (c+d x) b^2+a^2 C \sin (c+d x) b\right )}{a^2 \left (a^2-b^2\right ) (b+a \cos (c+d x))}\right ) (b+a \cos (c+d x))^2}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 7.458, size = 856, normalized size = 2.7 \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{C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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