Optimal. Leaf size=96 \[ -\frac{\text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{a d}-\frac{3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d}+\frac{3 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)} \]
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Rubi [A] time = 0.15654, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4264, 3818, 3787, 3771, 2641, 3768, 2639} \[ -\frac{F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d}-\frac{3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d}+\frac{3 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 4264
Rule 3818
Rule 3787
Rule 3771
Rule 2641
Rule 3768
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{5}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx\\ &=-\frac{\sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \left (\frac{a}{2}-\frac{3}{2} a \sec (c+d x)\right ) \, dx}{a^2}\\ &=-\frac{\sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \, dx}{2 a}+\frac{\left (3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx}{2 a}\\ &=\frac{3 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}-\frac{\int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{2 a}-\frac{\left (3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{2 a}\\ &=-\frac{F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d}+\frac{3 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}-\frac{3 \int \sqrt{\cos (c+d x)} \, dx}{2 a}\\ &=-\frac{3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d}-\frac{F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d}+\frac{3 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}\\ \end{align*}
Mathematica [C] time = 2.01631, size = 303, normalized size = 3.16 \[ \frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \left (2 \cos \left (\frac{1}{2} (c-d x)\right )+\cos \left (\frac{1}{2} (3 c+d x)\right )+3 \cos \left (\frac{1}{2} (c+3 d x)\right )\right ) \sec \left (\frac{1}{2} (c+d x)\right )}{2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sec (c+d x) \left (3 \left (-1+e^{2 i c}\right ) \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )-\left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )+3 \left (1+e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) d \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{a (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.639, size = 253, normalized size = 2.6 \begin{align*} -{\frac{1}{da} \left ( -\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-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 ({\it EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,\sqrt{2} \right ) -3\,{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ) +6\,\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}-5\,\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 ) ^{-1}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac{5}{2}}}\,{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{\sqrt{\cos \left (d x + c\right )}}{a \cos \left (d x + c\right )^{3} \sec \left (d x + c\right ) + a \cos \left (d x + c\right )^{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{1}{{\left (a \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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