Optimal. Leaf size=172 \[ \frac{2 \left (a^2+3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 a^3 d}-\frac{2 b^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 d (a+b)}-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.366478, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {3853, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{2 \left (a^2+3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^3 d}-\frac{2 b^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 d (a+b)}-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3853
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx &=\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{2 \int \frac{-\frac{3 b}{2}+\frac{1}{2} a \sec (c+d x)+\frac{1}{2} b \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{3 a}\\ &=\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{2 \int \frac{-\frac{3 a b}{2}-\left (-\frac{a^2}{2}-\frac{3 b^2}{2}\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{3 a^3}-\frac{b^3 \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{a^3}\\ &=\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{b \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{a^2}+\frac{\left (a^2+3 b^2\right ) \int \sqrt{\sec (c+d x)} \, dx}{3 a^3}-\frac{\left (b^3 \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^3 \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 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{\left (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+3 b^2\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 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+3 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 a^3 d}-\frac{2 b^3 \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 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 6.08485, size = 196, normalized size = 1.14 \[ -\frac{\cot (c+d x) \left (-4 a (a-3 b) \sqrt{-\tan ^2(c+d x)} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )-a^2 \sqrt{\sec (c+d x)}+a^2 \cos (3 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)+12 b^2 \sqrt{-\tan ^2(c+d x)} \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+6 a b \sec ^{\frac{3}{2}}(c+d x)-6 a b \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)-12 a b \sqrt{-\tan ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )}{6 a^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.543, size = 516, normalized size = 3. \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{1}{{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \sec{\left (c + d x \right )}\right ) \sec ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx \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{1}{{\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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