Optimal. Leaf size=131 \[ \frac{20 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}+\frac{2 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a^3 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d} \]
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Rubi [A] time = 0.137819, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3791, 3769, 3771, 2641, 2639, 3768} \[ \frac{2 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a^3 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{20 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{4 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3791
Rule 3769
Rule 3771
Rule 2641
Rule 2639
Rule 3768
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^3}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\int \left (\frac{a^3}{\sec ^{\frac{3}{2}}(c+d x)}+\frac{3 a^3}{\sqrt{\sec (c+d x)}}+3 a^3 \sqrt{\sec (c+d x)}+a^3 \sec ^{\frac{3}{2}}(c+d x)\right ) \, dx\\ &=a^3 \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx+a^3 \int \sec ^{\frac{3}{2}}(c+d x) \, dx+\left (3 a^3\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\left (3 a^3\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 a^3 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{1}{3} a^3 \int \sqrt{\sec (c+d x)} \, dx-a^3 \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\left (3 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\left (3 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{6 a^3 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{d}+\frac{6 a^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{d}+\frac{2 a^3 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\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-\left (a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{4 a^3 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{d}+\frac{20 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)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}\\ \end{align*}
Mathematica [C] time = 1.48802, size = 169, normalized size = 1.29 \[ \frac{a^3 \left (\cos \left (\frac{c}{2}\right )-i \sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}\right )+i \sin \left (\frac{c}{2}\right )\right ) \left (\frac{24 i \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )}{\sqrt{1+e^{2 i (c+d x)}}}+2 \left (-10 i \sqrt{1+e^{2 i (c+d x)}} \sec (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )+\sin (c+d x)+3 \tan (c+d x)-6 i\right )\right )}{3 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.462, size = 172, normalized size = 1.3 \begin{align*} -{\frac{4\,{a}^{3}}{3\,d} \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +5\,{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}-3\,{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}-4\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) \right ) \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 \sec \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} \sec \left (d x + c\right )^{3} + 3 \, a^{3} \sec \left (d x + c\right )^{2} + 3 \, a^{3} \sec \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{1}{\sec ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx + \int \frac{3}{\sqrt{\sec{\left (c + d x \right )}}}\, dx + \int 3 \sqrt{\sec{\left (c + d x \right )}}\, dx + \int \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 \sec \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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