Optimal. Leaf size=342 \[ \frac{\left (5 a^2-2 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 b^2 d \left (a^2-b^2\right )}-\frac{a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac{\left (5 a^2-2 b^2\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b^2 d \left (a^2-b^2\right )}-\frac{a \left (5 a^2-4 b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{b^3 d \left (a^2-b^2\right )}+\frac{a \left (5 a^2-4 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 d \left (a^2-b^2\right )}+\frac{a^2 \left (5 a^2-7 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 d (a-b) (a+b)^2} \]
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Rubi [A] time = 0.944133, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {3845, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ -\frac{a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac{\left (5 a^2-2 b^2\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b^2 d \left (a^2-b^2\right )}-\frac{a \left (5 a^2-4 b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{b^3 d \left (a^2-b^2\right )}+\frac{\left (5 a^2-2 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^2 d \left (a^2-b^2\right )}+\frac{a \left (5 a^2-4 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 d \left (a^2-b^2\right )}+\frac{a^2 \left (5 a^2-7 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 d (a-b) (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 3845
Rule 4102
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{9}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=-\frac{a^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left (\frac{3 a^2}{2}-a b \sec (c+d x)-\frac{1}{2} \left (5 a^2-2 b^2\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{b \left (a^2-b^2\right )}\\ &=\frac{\left (5 a^2-2 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac{a^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{2 \int \frac{\sqrt{\sec (c+d x)} \left (-\frac{1}{4} a \left (5 a^2-2 b^2\right )+\frac{1}{2} b \left (2 a^2+b^2\right ) \sec (c+d x)+\frac{3}{4} a \left (5 a^2-4 b^2\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{3 b^2 \left (a^2-b^2\right )}\\ &=-\frac{a \left (5 a^2-4 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (5 a^2-2 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac{a^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{4 \int \frac{-\frac{3}{8} a^2 \left (5 a^2-4 b^2\right )-\frac{1}{4} a b \left (10 a^2-7 b^2\right ) \sec (c+d x)-\frac{1}{8} \left (15 a^4-16 a^2 b^2-2 b^4\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{3 b^3 \left (a^2-b^2\right )}\\ &=-\frac{a \left (5 a^2-4 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (5 a^2-2 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac{a^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{4 \int \frac{-\frac{3}{8} a^3 \left (5 a^2-4 b^2\right )-\left (\frac{1}{4} a^2 b \left (10 a^2-7 b^2\right )-\frac{3}{8} a^2 b \left (5 a^2-4 b^2\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{3 a^2 b^3 \left (a^2-b^2\right )}+\frac{\left (a^2 \left (5 a^2-7 b^2\right )\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )}\\ &=-\frac{a \left (5 a^2-4 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (5 a^2-2 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac{a^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac{\left (a \left (5 a^2-4 b^2\right )\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{2 b^3 \left (a^2-b^2\right )}+\frac{\left (5 a^2-2 b^2\right ) \int \sqrt{\sec (c+d x)} \, dx}{6 b^2 \left (a^2-b^2\right )}+\frac{\left (a^2 \left (5 a^2-7 b^2\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 b^3 \left (a^2-b^2\right )}\\ &=\frac{a^2 \left (5 a^2-7 b^2\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-b) b^3 (a+b)^2 d}-\frac{a \left (5 a^2-4 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (5 a^2-2 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac{a^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac{\left (a \left (5 a^2-4 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )}+\frac{\left (\left (5 a^2-2 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 b^2 \left (a^2-b^2\right )}\\ &=\frac{a \left (5 a^2-4 b^2\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (5 a^2-2 b^2\right ) \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 \left (a^2-b^2\right ) d}+\frac{a^2 \left (5 a^2-7 b^2\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-b) b^3 (a+b)^2 d}-\frac{a \left (5 a^2-4 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (5 a^2-2 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac{a^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 6.3972, size = 319, normalized size = 0.93 \[ \frac{\frac{\cot (c+d x) \left (2 \left (-16 a^2 b^2+15 a^3 b+15 a^4-12 a b^3-2 b^4\right ) \sqrt{-\tan ^2(c+d x)} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )-6 a b \left (5 a^2-4 b^2\right ) \sqrt{-\tan ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+6 a \left (b \left (5 a^2-4 b^2\right ) \sin ^2(c+d x) \sec ^{\frac{3}{2}}(c+d x)+a \left (5 a^2-7 b^2\right ) \sqrt{-\tan ^2(c+d x)} \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )\right )}{(a-b) (a+b)}+\frac{b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (20 a b \left (a^2-b^2\right ) \cos (c+d x)+3 \left (5 a^4-4 a^2 b^2\right ) \cos (2 (c+d x))-16 a^2 b^2+15 a^4+4 b^4\right )}{\left (b^2-a^2\right ) (a \cos (c+d x)+b)}}{6 b^4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 6.195, size = 1002, 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{\sec \left (d x + c\right )^{\frac{9}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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