Optimal. Leaf size=112 \[ \frac{2 \left (a^2+3 b^2\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 a^3 d}-\frac{2 b^3 \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 d (a+b)}-\frac{2 b E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d} \]
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Rubi [A] time = 0.390238, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {4264, 3853, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{2 \left (a^2+3 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^3 d}-\frac{2 b^3 \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 d (a+b)}-\frac{2 b E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d} \]
Antiderivative was successfully verified.
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Rule 4264
Rule 3853
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx\\ &=\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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{\left (b^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{a^3}\\ &=\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d}-\frac{b^3 \int \frac{1}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{a^3}-\frac{\left (b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\sec (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 \sqrt{\sec (c+d x)} \, dx}{3 a^3}\\ &=-\frac{2 b^3 \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 (a+b) d}+\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d}-\frac{b \int \sqrt{\cos (c+d x)} \, dx}{a^2}+\frac{\left (a^2+3 b^2\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 a^3}\\ &=-\frac{2 b E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{2 \left (a^2+3 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^3 d}-\frac{2 b^3 \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 (a+b) d}+\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 1.6599, size = 160, normalized size = 1.43 \[ \frac{\frac{6 \sin (c+d x) \left (-2 b (a+b) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right ),-1\right )+\left (a^2-2 b^2\right ) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )+2 a b E\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )\right )}{a^2 \sqrt{\sin ^2(c+d x)}}+4 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-\frac{6 b \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}+4 \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a d} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.714, size = 516, normalized size = 4.6 \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{\cos \left (d x + c\right )^{\frac{3}{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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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{\cos \left (d x + c\right )^{\frac{3}{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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