Optimal. Leaf size=95 \[ \frac{2 \left (3 a^2+b^2\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}-\frac{4 a b E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{4 a b \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.144892, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4264, 3788, 3768, 3771, 2639, 4046, 2641} \[ \frac{2 \left (3 a^2+b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{4 a b E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{4 a b \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4264
Rule 3788
Rule 3768
Rule 3771
Rule 2639
Rule 4046
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \sec (c+d x))^2}{\sqrt{\cos (c+d x)}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2 \, dx\\ &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \left (a^2+b^2 \sec ^2(c+d x)\right ) \, dx+\left (2 a b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 b^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a b \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\left (2 a b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (\left (3 a^2+b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 b^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a b \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-(2 a b) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{3} \left (3 a^2+b^2\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{4 a b E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 \left (3 a^2+b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 b^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a b \sin (c+d x)}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.5771, size = 73, normalized size = 0.77 \[ \frac{2 \left (\left (3 a^2+b^2\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-6 a b E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{b \sin (c+d x) (6 a \cos (c+d x)+b)}{\cos ^{\frac{3}{2}}(c+d x)}\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.614, size = 514, normalized size = 5.4 \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{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \sec \left (d x + c\right )^{2} + 2 \, a b \sec \left (d x + c\right ) + a^{2}}{\sqrt{\cos \left (d x + c\right )}}, x\right ) \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{\left (a + b \sec{\left (c + d x \right )}\right )^{2}}{\sqrt{\cos{\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{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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