Optimal. Leaf size=136 \[ -\frac{5 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 a^2 d}-\frac{4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{4 \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{5 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^2} \]
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Rubi [A] time = 0.262784, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {4264, 3816, 4019, 3787, 3771, 2641, 3768, 2639} \[ -\frac{5 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d}-\frac{4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{4 \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{5 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 4264
Rule 3816
Rule 4019
Rule 3787
Rule 3771
Rule 2641
Rule 3768
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx\\ &=-\frac{\sin (c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x) \left (\frac{3 a}{2}-\frac{7}{2} a \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{3 a^2}\\ &=-\frac{5 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (1+\sec (c+d x))}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \left (\frac{5 a^2}{2}-6 a^2 \sec (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{5 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (1+\sec (c+d x))}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac{\left (5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \, dx}{6 a^2}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx}{a^2}\\ &=\frac{4 \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{5 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (1+\sec (c+d x))}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac{5 \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 a^2}-\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{a^2}\\ &=-\frac{5 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d}+\frac{4 \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{5 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (1+\sec (c+d x))}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac{2 \int \sqrt{\cos (c+d x)} \, dx}{a^2}\\ &=-\frac{4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}-\frac{5 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d}+\frac{4 \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{5 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (1+\sec (c+d x))}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2}\\ \end{align*}
Mathematica [C] time = 6.78334, size = 393, normalized size = 2.89 \[ \frac{\cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (\frac{2 \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec ^3\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d}+\frac{2 \tan \left (\frac{c}{2}\right ) \sec ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d}+\frac{8 \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}+\frac{8 \sec (c) \sin (d x) \sec (c+d x)}{d}+\frac{8 \cot \left (\frac{c}{2}\right ) \sec (c)}{d}\right )}{\cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2(c+d x) \left (12 \left (-1+e^{2 i c}\right ) \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )-5 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )+12 \left (1+e^{2 i (c+d x)}\right )\right )}{3 \left (-1+e^{2 i c}\right ) d \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.683, size = 405, normalized size = 3. \begin{align*} -{\frac{1}{6\,{a}^{2}d} \left ( 2\,\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}}\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( 5\,{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -12\,{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ) \cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-2\,\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}}\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( 5\,{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -12\,{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ) \cos \left ( 1/2\,dx+c/2 \right ) -48\,\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+86\,\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-37\,\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}{\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{\sqrt{\cos \left (d x + c\right )}}{a^{2} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right ) + a^{2} \cos \left (d x + c\right )^{4}}, 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{1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2} \cos \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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