Optimal. Leaf size=161 \[ \frac{136 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{8 a^4 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{94 a^4 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{64 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d} \]
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Rubi [A] time = 0.187004, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3791, 3769, 3771, 2641, 2639} \[ \frac{8 a^4 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{94 a^4 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{136 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{64 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d} \]
Antiderivative was successfully verified.
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Rule 3791
Rule 3769
Rule 3771
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^4}{\sec ^{\frac{7}{2}}(c+d x)} \, dx &=\int \left (\frac{a^4}{\sec ^{\frac{7}{2}}(c+d x)}+\frac{4 a^4}{\sec ^{\frac{5}{2}}(c+d x)}+\frac{6 a^4}{\sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^4}{\sqrt{\sec (c+d x)}}+a^4 \sqrt{\sec (c+d x)}\right ) \, dx\\ &=a^4 \int \frac{1}{\sec ^{\frac{7}{2}}(c+d x)} \, dx+a^4 \int \sqrt{\sec (c+d x)} \, dx+\left (4 a^4\right ) \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x)} \, dx+\left (4 a^4\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\left (6 a^4\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^4 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^4 \sin (c+d x)}{d \sqrt{\sec (c+d x)}}+\frac{1}{7} \left (5 a^4\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx+\left (2 a^4\right ) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{5} \left (12 a^4\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\left (a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (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\\ &=\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{2 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)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{94 a^4 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{1}{21} \left (5 a^4\right ) \int \sqrt{\sec (c+d x)} \, dx+\left (2 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{5} \left (12 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{64 a^4 \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{6 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)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{94 a^4 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{1}{21} \left (5 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{64 a^4 \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{136 a^4 \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^4 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{94 a^4 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 1.69522, size = 180, normalized size = 1.12 \[ \frac{a^4 \left (\cos \left (\frac{c}{2}\right )-i \sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}\right )+i \sin \left (\frac{c}{2}\right )\right ) \left (\frac{10752 i \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )}{\sqrt{1+e^{2 i (c+d x)}}}-2720 i \sqrt{1+e^{2 i (c+d x)}} \sec (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )+1910 \sin (c+d x)+336 \sin (2 (c+d x))+30 \sin (3 (c+d x))-5376 i\right )}{420 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.352, size = 272, normalized size = 1.7 \begin{align*} -{\frac{8\,{a}^{4}}{105\,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 ( 60\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}-258\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +448\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +85\,{\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}}-168\,{\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}}-167\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\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(-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{a^{4} \sec \left (d x + c\right )^{4} + 4 \, a^{4} \sec \left (d x + c\right )^{3} + 6 \, a^{4} \sec \left (d x + c\right )^{2} + 4 \, a^{4} \sec \left (d x + c\right ) + a^{4}}{\sec \left (d x + c\right )^{\frac{7}{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 \frac{{\left (a \sec \left (d x + c\right ) + a\right )}^{4}}{\sec \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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