Optimal. Leaf size=187 \[ \frac{68 a^3 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{44 a^3 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{44 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{68 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d} \]
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Rubi [A] time = 0.234638, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3238, 3791, 3769, 3771, 2639, 2641} \[ \frac{68 a^3 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{44 a^3 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{44 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{68 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d} \]
Antiderivative was successfully verified.
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Rule 3238
Rule 3791
Rule 3769
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^3}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\int \frac{(a+a \sec (c+d x))^3}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\int \left (\frac{a^3}{\sec ^{\frac{9}{2}}(c+d x)}+\frac{3 a^3}{\sec ^{\frac{7}{2}}(c+d x)}+\frac{3 a^3}{\sec ^{\frac{5}{2}}(c+d x)}+\frac{a^3}{\sec ^{\frac{3}{2}}(c+d x)}\right ) \, dx\\ &=a^3 \int \frac{1}{\sec ^{\frac{9}{2}}(c+d x)} \, dx+a^3 \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx+\left (3 a^3\right ) \int \frac{1}{\sec ^{\frac{7}{2}}(c+d x)} \, dx+\left (3 a^3\right ) \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a^3 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{1}{3} a^3 \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{9} \left (7 a^3\right ) \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x)} \, dx+\frac{1}{5} \left (9 a^3\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{7} \left (15 a^3\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^3 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{68 a^3 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{44 a^3 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{1}{15} \left (7 a^3\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{7} \left (5 a^3\right ) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{3} \left (a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{5} \left (9 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{18 a^3 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 a^3 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{68 a^3 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{44 a^3 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{1}{15} \left (7 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{7} \left (5 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{68 a^3 \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{44 a^3 \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{2 a^3 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{68 a^3 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{44 a^3 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 2.18728, size = 156, normalized size = 0.83 \[ \frac{a^3 \left (\frac{22848 i \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right )}{\sqrt{1+e^{2 i (c+d x)}}}-5280 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right ) \sec (c+d x)+5820 \sin (c+d x)+2044 \sin (2 (c+d x))+540 \sin (3 (c+d x))+70 \sin (4 (c+d x))-11424 i\right )}{2520 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.276, size = 260, normalized size = 1.4 \begin{align*} -{\frac{4\,{a}^{3}}{315\,d}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 560\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{11}-600\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}+212\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}+66\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}-430\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+165\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -357\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +192\,\cos \left ( 1/2\,dx+c/2 \right ) \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \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 (a \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{a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \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*} a^{3} \left (\int \frac{3 \cos{\left (c + d x \right )}}{\sec ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx + \int \frac{3 \cos ^{2}{\left (c + d x \right )}}{\sec ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx + \int \frac{\cos ^{3}{\left (c + d x \right )}}{\sec ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx + \int \frac{1}{\sec ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx\right ) \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 (a \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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