Optimal. Leaf size=155 \[ \frac{\text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{6 a^3 d}-\frac{E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{\sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}+\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^2} \]
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Rubi [A] time = 0.378032, antiderivative size = 155, 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, 3815, 4019, 4020, 3787, 3771, 2639, 2641} \[ \frac{F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac{E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{\sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}+\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 4264
Rule 3815
Rule 4019
Rule 4020
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\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{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx\\ &=\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\sec (c+d x)} \left (\frac{a}{2}+\frac{3}{2} a \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac{\sin (c+d x)}{15 a d \sqrt{\cos (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{\frac{a^2}{2}+3 a^2 \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))} \, dx}{15 a^4}\\ &=\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac{\sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2}+\frac{\sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \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{3 a^3}{4}+\frac{5}{4} a^3 \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{15 a^6}\\ &=\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac{\sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2}+\frac{\sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \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{1}{\sqrt{\sec (c+d x)}} \, dx}{20 a^3}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \, dx}{12 a^3}\\ &=\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac{\sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2}+\frac{\sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\int \sqrt{\cos (c+d x)} \, dx}{20 a^3}+\frac{\int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{12 a^3}\\ &=-\frac{E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac{\sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2}+\frac{\sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 2.01075, size = 342, normalized size = 2.21 \[ \frac{\cos ^6\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \left (14 \cos \left (\frac{1}{2} (c-d x)\right )+16 \cos \left (\frac{1}{2} (3 c+d x)\right )+20 \cos \left (\frac{1}{2} (c+3 d x)\right )-5 \cos \left (\frac{1}{2} (5 c+3 d x)\right )+3 \cos \left (\frac{1}{2} (3 c+5 d x)\right )\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right )}{8 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \sec ^3(c+d x) \left (3 \left (-1+e^{2 i c}\right ) \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )+5 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )+3 \left (1+e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) d \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{15 a^3 (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.563, size = 270, normalized size = 1.7 \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 ( 12\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+10\,\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}+6\,\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 ) -2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}-24\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+17\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-3 \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}{\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^{3} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right ) + a^{3} \cos \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} \cos \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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