Optimal. Leaf size=221 \[ \frac{11 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{2 a^3 d}+\frac{11 \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{119 \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}-\frac{119 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.35179, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3817, 4020, 3787, 3769, 3771, 2641, 2639} \[ \frac{11 \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{119 \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}+\frac{11 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac{119 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3817
Rule 4020
Rule 3787
Rule 3769
Rule 3771
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx &=-\frac{\sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac{\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{\sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac{\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{\sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac{119 \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\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{\sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac{119 \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}-\frac{119 \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{20 a^3}+\frac{33 \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{4 a^3}\\ &=\frac{11 \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac{119 \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac{11 \int \sqrt{\sec (c+d x)} \, dx}{4 a^3}-\frac{\left (119 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}\\ &=-\frac{119 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{10 a^3 d}+\frac{11 \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac{119 \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac{\left (11 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{4 a^3}\\ &=-\frac{119 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{10 a^3 d}+\frac{11 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{2 a^3 d}+\frac{11 \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac{119 \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 2.41702, size = 285, normalized size = 1.29 \[ \frac{e^{-i d x} \cos \left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left (119 i e^{-\frac{3}{2} i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left (1+e^{i (c+d x)}\right )^5 \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+5280 \cos ^5\left (\frac{1}{2} (c+d x)\right ) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+193 \sin \left (\frac{1}{2} (c+d x)\right )+579 \sin \left (\frac{3}{2} (c+d x)\right )+555 \sin \left (\frac{5}{2} (c+d x)\right )+227 \sin \left (\frac{7}{2} (c+d x)\right )+10 \sin \left (\frac{9}{2} (c+d x)\right )-5355 i \cos \left (\frac{1}{2} (c+d x)\right )-3927 i \cos \left (\frac{3}{2} (c+d x)\right )-1785 i \cos \left (\frac{5}{2} (c+d x)\right )-357 i \cos \left (\frac{7}{2} (c+d x)\right )\right )}{120 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.557, size = 283, normalized size = 1.3 \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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{\sec \left (d x + c\right )}}{a^{3} \sec \left (d x + c\right )^{5} + 3 \, a^{3} \sec \left (d x + c\right )^{4} + 3 \, a^{3} \sec \left (d x + c\right )^{3} + a^{3} \sec \left (d x + c\right )^{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{1}{{\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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