Optimal. Leaf size=149 \[ \frac{2 a \left (a^2+b^2\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{d}-\frac{6 b \left (5 a^2+b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{6 b \left (5 a^2+b^2\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{8 a b^2 \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 \sin (c+d x) (a+b \sec (c+d x))}{5 d \cos ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.245708, antiderivative size = 149, 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 = {4264, 3842, 4047, 3768, 3771, 2639, 4046, 2641} \[ \frac{2 a \left (a^2+b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}-\frac{6 b \left (5 a^2+b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{6 b \left (5 a^2+b^2\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{8 a b^2 \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 \sin (c+d x) (a+b \sec (c+d x))}{5 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4264
Rule 3842
Rule 4047
Rule 3768
Rule 3771
Rule 2639
Rule 4046
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \sec (c+d x))^3}{\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))^3 \, dx\\ &=\frac{2 b^2 (a+b \sec (c+d x)) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{1}{5} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \left (\frac{1}{2} a \left (5 a^2+b^2\right )+\frac{3}{2} b \left (5 a^2+b^2\right ) \sec (c+d x)+6 a b^2 \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 b^2 (a+b \sec (c+d x)) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{1}{5} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \left (\frac{1}{2} a \left (5 a^2+b^2\right )+6 a b^2 \sec ^2(c+d x)\right ) \, dx+\frac{1}{5} \left (3 b \left (5 a^2+b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{8 a b^2 \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 b \left (5 a^2+b^2\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 (a+b \sec (c+d x)) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\left (a \left (a^2+b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \, dx-\frac{1}{5} \left (3 b \left (5 a^2+b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{8 a b^2 \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 b \left (5 a^2+b^2\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 (a+b \sec (c+d x)) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\left (a \left (a^2+b^2\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\frac{1}{5} \left (3 b \left (5 a^2+b^2\right )\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{6 b \left (5 a^2+b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a \left (a^2+b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{8 a b^2 \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 b \left (5 a^2+b^2\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 (a+b \sec (c+d x)) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.8762, size = 125, normalized size = 0.84 \[ \frac{10 a \left (a^2+b^2\right ) \cos ^{\frac{3}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+3 \left (5 a^2 b+b^3\right ) \sin (2 (c+d x))-6 b \left (5 a^2+b^2\right ) \cos ^{\frac{3}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+10 a b^2 \sin (c+d x)+2 b^3 \tan (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.534, size = 738, normalized size = 5. \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^{3} \sec \left (d x + c\right )^{3} + 3 \, a b^{2} \sec \left (d x + c\right )^{2} + 3 \, a^{2} b \sec \left (d x + c\right ) + a^{3}}{\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 )^{3}}{\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 )}^{3}}{\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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