Optimal. Leaf size=277 \[ \frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 b^2 d}-\frac{2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 b^3 d}+\frac{2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^3 d}+\frac{2 a^2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 d (a+b)}+\frac{2 (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b^2 d}+\frac{2 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 b d} \]
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Rubi [A] time = 1.01417, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4033, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ -\frac{2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 b^3 d}+\frac{2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^3 d}+\frac{2 a^2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 d (a+b)}+\frac{2 (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b^2 d}+\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^2 d}+\frac{2 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 b d} \]
Antiderivative was successfully verified.
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Rule 4033
Rule 4102
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx &=\frac{2 B \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 b d}+\frac{2 \int \frac{\sec ^{\frac{3}{2}}(c+d x) \left (\frac{3 a B}{2}+\frac{3}{2} b B \sec (c+d x)+\frac{5}{2} (A b-a B) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{5 b}\\ &=\frac{2 (A b-a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 d}+\frac{2 B \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 b d}+\frac{4 \int \frac{\sqrt{\sec (c+d x)} \left (\frac{5}{4} a (A b-a B)+\frac{1}{4} b (5 A b+4 a B) \sec (c+d x)-\frac{3}{4} \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{15 b^2}\\ &=-\frac{2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 b^3 d}+\frac{2 (A b-a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 d}+\frac{2 B \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 b d}+\frac{8 \int \frac{\frac{3}{8} a \left (5 a A b-5 a^2 B-3 b^2 B\right )+\frac{1}{8} b \left (20 a A b-20 a^2 B-9 b^2 B\right ) \sec (c+d x)+\frac{5}{8} \left (3 a^2+b^2\right ) (A b-a B) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{15 b^3}\\ &=-\frac{2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 b^3 d}+\frac{2 (A b-a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 d}+\frac{2 B \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 b d}+\frac{8 \int \frac{\frac{3}{8} a^2 \left (5 a A b-5 a^2 B-3 b^2 B\right )-\left (-\frac{1}{8} a b \left (20 a A b-20 a^2 B-9 b^2 B\right )+\frac{3}{8} a b \left (5 a A b-5 a^2 B-3 b^2 B\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{15 a^2 b^3}+\frac{\left (a^2 (A b-a B)\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{b^3}\\ &=-\frac{2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 b^3 d}+\frac{2 (A b-a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 d}+\frac{2 B \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 b d}+\frac{(A b-a B) \int \sqrt{\sec (c+d x)} \, dx}{3 b^2}+\frac{\left (5 a A b-5 a^2 B-3 b^2 B\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{5 b^3}+\frac{\left (a^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}{b^3}\\ &=\frac{2 a^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)}}{b^3 (a+b) d}-\frac{2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 b^3 d}+\frac{2 (A b-a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 d}+\frac{2 B \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 b d}+\frac{\left ((A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 b^2}+\frac{\left (\left (5 a A b-5 a^2 B-3 b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 b^3}\\ &=\frac{2 \left (5 a A b-5 a^2 B-3 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)}}{5 b^3 d}+\frac{2 (A b-a B) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 b^2 d}+\frac{2 a^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)}}{b^3 (a+b) d}-\frac{2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 b^3 d}+\frac{2 (A b-a B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 d}+\frac{2 B \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 b d}\\ \end{align*}
Mathematica [B] time = 6.96241, size = 669, normalized size = 2.42 \[ \frac{\sqrt{\sec (c+d x)} \left (\frac{2 \left (5 a^2 B-5 a A b+3 b^2 B\right ) \sin (c+d x)}{5 b^3}+\frac{2 \sec (c+d x) (A b \sin (c+d x)-a B \sin (c+d x))}{3 b^2}+\frac{2 B \tan (c+d x) \sec (c+d x)}{5 b}\right )}{d}-\frac{\frac{2 \left (-45 a^2 A b+45 a^3 B+19 a b^2 B-10 A b^3\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left (\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )+\Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )}{b \left (1-\cos ^2(c+d x)\right ) (a \cos (c+d x)+b)}-\frac{2 \left (-15 a^2 A b+15 a^3 B+9 a b^2 B\right ) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )+a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-2 a b \sec ^2(c+d x)+2 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+2 a b\right )}{a^2 b \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right ) (a \cos (c+d x)+b)}-\frac{2 \left (40 a^2 b B-40 a A b^2+18 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )}{a \left (1-\cos ^2(c+d x)\right ) (a \cos (c+d x)+b)}}{30 b^3 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 6.75, size = 785, normalized size = 2.8 \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{7}{2}}}{b \sec \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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