Optimal. Leaf size=135 \[ \frac{4 \sqrt{\sin (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right )}{21 a d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac{2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac{4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}} \]
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Rubi [A] time = 0.250353, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3872, 2839, 2564, 30, 2567, 2636, 2642, 2641} \[ \frac{4 \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{21 a d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac{2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac{4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2839
Rule 2564
Rule 30
Rule 2567
Rule 2636
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sec (c+d x)) (e \sin (c+d x))^{5/2}} \, dx &=-\int \frac{\cos (c+d x)}{(-a-a \cos (c+d x)) (e \sin (c+d x))^{5/2}} \, dx\\ &=\frac{e^2 \int \frac{\cos (c+d x)}{(e \sin (c+d x))^{9/2}} \, dx}{a}-\frac{e^2 \int \frac{\cos ^2(c+d x)}{(e \sin (c+d x))^{9/2}} \, dx}{a}\\ &=\frac{2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}+\frac{2 \int \frac{1}{(e \sin (c+d x))^{5/2}} \, dx}{7 a}+\frac{e \operatorname{Subst}\left (\int \frac{1}{x^{9/2}} \, dx,x,e \sin (c+d x)\right )}{a d}\\ &=-\frac{2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac{2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac{4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}}+\frac{2 \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{21 a e^2}\\ &=-\frac{2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac{2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac{4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}}+\frac{\left (2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{21 a e^2 \sqrt{e \sin (c+d x)}}\\ &=-\frac{2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac{2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac{4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}}+\frac{4 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{21 a d e^2 \sqrt{e \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.25271, size = 91, normalized size = 0.67 \[ -\frac{2 \left (\sin ^{\frac{7}{2}}(c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right ) \text{EllipticF}\left (\frac{1}{4} (-2 c-2 d x+\pi ),2\right )+2 \cos (c+d x)+\cos (2 (c+d x))+4\right )}{21 a d e (\cos (c+d x)+1) (e \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.51, size = 136, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( -{\frac{2\,e}{7\,a} \left ( e\sin \left ( dx+c \right ) \right ) ^{-{\frac{7}{2}}}}-{\frac{2}{21\,a{e}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}\cos \left ( dx+c \right ) } \left ( \sqrt{-\sin \left ( dx+c \right ) +1}\sqrt{2+2\,\sin \left ( dx+c \right ) } \left ( \sin \left ( dx+c \right ) \right ) ^{{\frac{9}{2}}}{\it EllipticF} \left ( \sqrt{-\sin \left ( dx+c \right ) +1},{\frac{\sqrt{2}}{2}} \right ) -2\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}+5\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}-3\,\sin \left ( dx+c \right ) \right ){\frac{1}{\sqrt{e\sin \left ( dx+c \right ) }}}} \right ) } \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{e \sin \left (d x + c\right )}}{{\left (a e^{3} \cos \left (d x + c\right )^{2} - a e^{3} +{\left (a e^{3} \cos \left (d x + c\right )^{2} - a e^{3}\right )} \sec \left (d x + c\right )\right )} \sin \left (d x + c\right )}, 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 )} \left (e \sin \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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