Optimal. Leaf size=234 \[ \frac{2 b \left (27 a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (7 a^2+15 b^2\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a \left (7 a^2+15 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 b \left (27 a^2+7 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b^2 \sin (c+d x) (a \sec (c+d x)+b)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)} \]
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Rubi [A] time = 0.276859, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {3238, 3841, 4047, 3769, 3771, 2639, 4045, 2641} \[ \frac{2 b \left (27 a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (7 a^2+15 b^2\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a \left (7 a^2+15 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 b \left (27 a^2+7 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b^2 \sin (c+d x) (a \sec (c+d x)+b)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3238
Rule 3841
Rule 4047
Rule 3769
Rule 3771
Rule 2639
Rule 4045
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^3}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\int \frac{(b+a \sec (c+d x))^3}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 b^2 (b+a \sec (c+d x)) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2}{9} \int \frac{10 a b^2+\frac{1}{2} b \left (27 a^2+7 b^2\right ) \sec (c+d x)+\frac{1}{2} a \left (9 a^2+5 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 b^2 (b+a \sec (c+d x)) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2}{9} \int \frac{10 a b^2+\frac{1}{2} a \left (9 a^2+5 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx+\frac{1}{9} \left (b \left (27 a^2+7 b^2\right )\right ) \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left (27 a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 (b+a \sec (c+d x)) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{15} \left (b \left (27 a^2+7 b^2\right )\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{7} \left (a \left (7 a^2+15 b^2\right )\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left (27 a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (7 a^2+15 b^2\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b^2 (b+a \sec (c+d x)) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{21} \left (a \left (7 a^2+15 b^2\right )\right ) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{15} \left (b \left (27 a^2+7 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 b \left (27 a^2+7 b^2\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left (27 a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (7 a^2+15 b^2\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b^2 (b+a \sec (c+d x)) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{21} \left (a \left (7 a^2+15 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b \left (27 a^2+7 b^2\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 a \left (7 a^2+15 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)}}{21 d}+\frac{40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left (27 a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (7 a^2+15 b^2\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b^2 (b+a \sec (c+d x)) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 1.12106, size = 159, normalized size = 0.68 \[ \frac{\sqrt{\sec (c+d x)} \left (\sin (2 (c+d x)) \left (7 b \left (108 a^2+43 b^2\right ) \cos (c+d x)+5 \left (84 a^3+54 a b^2 \cos (2 (c+d x))+234 a b^2+7 b^3 \cos (3 (c+d x))\right )\right )+120 a \left (7 a^2+15 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+168 b \left (27 a^2+7 b^2\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{1260 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.982, size = 470, normalized size = 2. \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 \cos \left (d x + c\right ) + a\right )}^{3}}{\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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}}{\sec \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \cos{\left (c + d x \right )}\right )^{3}}{\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{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\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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