Optimal. Leaf size=118 \[ \frac{2 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a^4 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{40 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d} \]
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Rubi [A] time = 0.210607, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3238, 3791, 3769, 3771, 2641, 2639, 3768} \[ \frac{2 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a^4 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{40 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d} \]
Antiderivative was successfully verified.
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Rule 3238
Rule 3791
Rule 3769
Rule 3771
Rule 2641
Rule 2639
Rule 3768
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^4 \sec ^{\frac{5}{2}}(c+d x) \, dx &=\int \frac{(a+a \sec (c+d x))^4}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\int \left (\frac{a^4}{\sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^4}{\sqrt{\sec (c+d x)}}+6 a^4 \sqrt{\sec (c+d x)}+4 a^4 \sec ^{\frac{3}{2}}(c+d x)+a^4 \sec ^{\frac{5}{2}}(c+d x)\right ) \, dx\\ &=a^4 \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx+a^4 \int \sec ^{\frac{5}{2}}(c+d x) \, dx+\left (4 a^4\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\left (4 a^4\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx+\left (6 a^4\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 a^4 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{8 a^4 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 a^4 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+2 \left (\frac{1}{3} a^4 \int \sqrt{\sec (c+d x)} \, dx\right )-\left (4 a^4\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\left (4 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\left (6 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{8 a^4 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{d}+\frac{12 a^4 \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^4 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{8 a^4 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 a^4 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+2 \left (\frac{1}{3} \left (a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\right )-\left (4 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{40 a^4 \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^4 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{8 a^4 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 a^4 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.334052, size = 70, normalized size = 0.59 \[ \frac{a^4 \sec ^{\frac{3}{2}}(c+d x) \left (5 \sin (c+d x)+24 \sin (2 (c+d x))+\sin (3 (c+d x))+80 \cos ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{6 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.509, size = 292, normalized size = 2.5 \begin{align*}{\frac{8\,{a}^{4}}{3\,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 ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +10\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-14\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -5\,\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}}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +7\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) \right ) \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 ( 4\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-4\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{-1} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}{\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{\left (a \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac{5}{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 ({\left (a^{4} \cos \left (d x + c\right )^{4} + 4 \, a^{4} \cos \left (d x + c\right )^{3} + 6 \, a^{4} \cos \left (d x + c\right )^{2} + 4 \, a^{4} \cos \left (d x + c\right ) + a^{4}\right )} \sec \left (d x + c\right )^{\frac{5}{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{\left (a \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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