Optimal. Leaf size=181 \[ \frac{11 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{2 a^3 d}-\frac{119 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{119 \sin (c+d x) \sqrt{\cos (c+d x)}}{30 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.396354, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {4264, 3817, 4020, 3787, 3769, 3771, 2641, 2639} \[ \frac{11 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac{119 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{119 \sin (c+d x) \sqrt{\cos (c+d x)}}{30 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 4264
Rule 3817
Rule 4020
Rule 3787
Rule 3769
Rule 3771
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx\\ &=-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{13 a}{2}+\frac{7}{2} a \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{69 a^2}{2}+25 a^2 \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx}{15 a^4}\\ &=-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac{119 \sqrt{\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{495 a^3}{4}+\frac{357}{4} a^3 \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{15 a^6}\\ &=-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac{119 \sqrt{\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\left (119 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{20 a^3}+\frac{\left (33 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{4 a^3}\\ &=\frac{11 \sqrt{\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac{119 \sqrt{\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{119 \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}+\frac{\left (11 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \, dx}{4 a^3}\\ &=-\frac{119 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{11 \sqrt{\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac{119 \sqrt{\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac{11 \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{4 a^3}\\ &=-\frac{119 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{11 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac{11 \sqrt{\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac{119 \sqrt{\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 2.48146, size = 375, normalized size = 2.07 \[ \frac{\cos ^6\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\frac{1}{4} \sec \left (\frac{c}{2}\right ) \left (-709 \sin \left (c+\frac{d x}{2}\right )+715 \sin \left (c+\frac{3 d x}{2}\right )-170 \sin \left (2 c+\frac{3 d x}{2}\right )+202 \sin \left (2 c+\frac{5 d x}{2}\right )+25 \sin \left (3 c+\frac{5 d x}{2}\right )+5 \sin \left (3 c+\frac{7 d x}{2}\right )+5 \sin \left (4 c+\frac{7 d x}{2}\right )+1061 \sin \left (\frac{d x}{2}\right )\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right )+720 \cot (c)+708 \csc (c)}{3 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \sec ^3(c+d x) \left (119 \left (-1+e^{2 i c}\right ) \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )+55 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )+119 \left (1+e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) d \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{5 a^3 (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.446, size = 283, normalized size = 1.6 \begin{align*} -{\frac{1}{60\,{a}^{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 ( 160\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+468\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+330\,\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 ) \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+714\,\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} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -1058\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+474\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-47\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+3 \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 ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5} \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{\cos \left (d x + c\right )^{\frac{3}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3}}\,{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{\cos \left (d x + c\right )^{\frac{3}{2}}}{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}}, 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{\cos \left (d x + c\right )^{\frac{3}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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