Optimal. Leaf size=135 \[ \frac{4 a b \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}-\frac{2 \left (5 a^2+3 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (5 a^2+3 b^2\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 a b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)} \]
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Rubi [A] time = 0.164317, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4264, 3788, 3768, 3771, 2641, 4046, 2639} \[ -\frac{2 \left (5 a^2+3 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (5 a^2+3 b^2\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 a b F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{4 a b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4264
Rule 3788
Rule 3768
Rule 3771
Rule 2641
Rule 4046
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \sec (c+d x))^2}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx\\ &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(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{5}{2}}(c+d x) \, dx\\ &=\frac{2 b^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{1}{3} \left (2 a b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{5} \left (\left (5 a^2+3 b^2\right ) \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)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (5 a^2+3 b^2\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{1}{3} (2 a b) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\frac{1}{5} \left (\left (5 a^2+3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{4 a b F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 b^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (5 a^2+3 b^2\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}-\frac{1}{5} \left (5 a^2+3 b^2\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 \left (5 a^2+3 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{4 a b F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 b^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (5 a^2+3 b^2\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.355176, size = 124, normalized size = 0.92 \[ \frac{20 a b \cos ^{\frac{3}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-6 \left (5 a^2+3 b^2\right ) \cos ^{\frac{3}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+15 a^2 \sin (2 (c+d x))+20 a b \sin (c+d x)+9 b^2 \sin (2 (c+d x))+6 b^2 \tan (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.947, size = 660, normalized size = 4.9 \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] 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}}{\cos \left (d x + c\right )^{\frac{3}{2}}}, x\right ) \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{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}}{\cos \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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