Optimal. Leaf size=277 \[ \frac{b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}+\frac{\left (2 a^2-3 b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d \left (a^2-b^2\right )}+\frac{b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d \left (a^2-b^2\right )}-\frac{\left (2 a^2-3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d \left (a^2-b^2\right )}-\frac{b \left (5 a^2-3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d (a-b) (a+b)^2} \]
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Rubi [A] time = 0.7112, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {3238, 3845, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}+\frac{\left (2 a^2-3 b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d \left (a^2-b^2\right )}+\frac{b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d \left (a^2-b^2\right )}-\frac{\left (2 a^2-3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d \left (a^2-b^2\right )}-\frac{b \left (5 a^2-3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d (a-b) (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 3238
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{3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx &=\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(b+a \sec (c+d x))^2} \, dx\\ &=\frac{b^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (b+a \sec (c+d x))}+\frac{\int \frac{\sqrt{\sec (c+d x)} \left (\frac{b^2}{2}-a b \sec (c+d x)+\frac{1}{2} \left (2 a^2-3 b^2\right ) \sec ^2(c+d x)\right )}{b+a \sec (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=\frac{\left (2 a^2-3 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (b+a \sec (c+d x))}+\frac{2 \int \frac{-\frac{1}{4} b \left (2 a^2-3 b^2\right )-\frac{1}{2} a \left (a^2-2 b^2\right ) \sec (c+d x)-\frac{1}{4} b \left (4 a^2-3 b^2\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (b+a \sec (c+d x))} \, dx}{a^2 \left (a^2-b^2\right )}\\ &=\frac{\left (2 a^2-3 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (b+a \sec (c+d x))}+\frac{2 \int \frac{-\frac{1}{4} b^2 \left (2 a^2-3 b^2\right )-\left (-\frac{1}{4} a b \left (2 a^2-3 b^2\right )+\frac{1}{2} a b \left (a^2-2 b^2\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{a^2 b^2 \left (a^2-b^2\right )}-\frac{\left (b \left (5 a^2-3 b^2\right )\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{b+a \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )}\\ &=\frac{\left (2 a^2-3 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (b+a \sec (c+d x))}+\frac{b \int \sqrt{\sec (c+d x)} \, dx}{2 a \left (a^2-b^2\right )}-\frac{\left (2 a^2-3 b^2\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{2 a^2 \left (a^2-b^2\right )}-\frac{\left (b \left (5 a^2-3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{2 a^2 \left (a^2-b^2\right )}\\ &=-\frac{b \left (5 a^2-3 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^2 (a-b) (a+b)^2 d}+\frac{\left (2 a^2-3 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (b+a \sec (c+d x))}+\frac{\left (b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{2 a \left (a^2-b^2\right )}-\frac{\left (\left (2 a^2-3 b^2\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 (2 a^2-3 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)}}{a^2 \left (a^2-b^2\right ) d}+\frac{b \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a \left (a^2-b^2\right ) d}-\frac{b \left (5 a^2-3 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^2 (a-b) (a+b)^2 d}+\frac{\left (2 a^2-3 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (b+a \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 4.26419, size = 355, normalized size = 1.28 \[ \frac{\frac{2 a \sin (c+d x) \left (2 a \left (a^2-b^2\right ) \sec (c+d x)+2 a^2 b-3 b^3\right )}{\left (a^2-b^2\right ) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\cot (c+d x) \left (-2 \left (4 a^2 b+2 a^3-3 a b^2-3 b^3\right ) \sqrt{-\tan ^2(c+d x)} F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+2 a \left (2 a^2-3 b^2\right ) \sqrt{-\tan ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-10 a^2 b \sqrt{-\tan ^2(c+d x)} \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-2 a^3 \sec ^{\frac{3}{2}}(c+d x)+2 a^3 \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)+3 a b^2 \sec ^{\frac{3}{2}}(c+d x)-3 a b^2 \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)+6 b^3 \sqrt{-\tan ^2(c+d x)} \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )}{(a-b) (a+b)}}{2 a^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 9.095, size = 874, normalized size = 3.2 \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{3}{2}}}{{\left (b \cos \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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