Optimal. Leaf size=190 \[ \frac{20 \sqrt{\sin (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right )}{21 a^2 d \sqrt{e \sin (c+d x)}}+\frac{4 e^3}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos ^3(c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos (c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{4 e}{3 a^2 d (e \sin (c+d x))^{3/2}}+\frac{16 e \cos (c+d x)}{21 a^2 d (e \sin (c+d x))^{3/2}} \]
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Rubi [A] time = 0.590883, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {3872, 2875, 2873, 2567, 2636, 2642, 2641, 2564, 14} \[ \frac{4 e^3}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos ^3(c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos (c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{4 e}{3 a^2 d (e \sin (c+d x))^{3/2}}+\frac{16 e \cos (c+d x)}{21 a^2 d (e \sin (c+d x))^{3/2}}+\frac{20 \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{21 a^2 d \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2875
Rule 2873
Rule 2567
Rule 2636
Rule 2642
Rule 2641
Rule 2564
Rule 14
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sec (c+d x))^2 \sqrt{e \sin (c+d x)}} \, dx &=\int \frac{\cos ^2(c+d x)}{(-a-a \cos (c+d x))^2 \sqrt{e \sin (c+d x)}} \, dx\\ &=\frac{e^4 \int \frac{\cos ^2(c+d x) (-a+a \cos (c+d x))^2}{(e \sin (c+d x))^{9/2}} \, dx}{a^4}\\ &=\frac{e^4 \int \left (\frac{a^2 \cos ^2(c+d x)}{(e \sin (c+d x))^{9/2}}-\frac{2 a^2 \cos ^3(c+d x)}{(e \sin (c+d x))^{9/2}}+\frac{a^2 \cos ^4(c+d x)}{(e \sin (c+d x))^{9/2}}\right ) \, dx}{a^4}\\ &=\frac{e^4 \int \frac{\cos ^2(c+d x)}{(e \sin (c+d x))^{9/2}} \, dx}{a^2}+\frac{e^4 \int \frac{\cos ^4(c+d x)}{(e \sin (c+d x))^{9/2}} \, dx}{a^2}-\frac{\left (2 e^4\right ) \int \frac{\cos ^3(c+d x)}{(e \sin (c+d x))^{9/2}} \, dx}{a^2}\\ &=-\frac{2 e^3 \cos (c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos ^3(c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{\left (2 e^2\right ) \int \frac{1}{(e \sin (c+d x))^{5/2}} \, dx}{7 a^2}-\frac{\left (6 e^2\right ) \int \frac{\cos ^2(c+d x)}{(e \sin (c+d x))^{5/2}} \, dx}{7 a^2}-\frac{\left (2 e^3\right ) \operatorname{Subst}\left (\int \frac{1-\frac{x^2}{e^2}}{x^{9/2}} \, dx,x,e \sin (c+d x)\right )}{a^2 d}\\ &=-\frac{2 e^3 \cos (c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos ^3(c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}+\frac{16 e \cos (c+d x)}{21 a^2 d (e \sin (c+d x))^{3/2}}-\frac{2 \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{21 a^2}+\frac{4 \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{7 a^2}-\frac{\left (2 e^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^{9/2}}-\frac{1}{e^2 x^{5/2}}\right ) \, dx,x,e \sin (c+d x)\right )}{a^2 d}\\ &=\frac{4 e^3}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos (c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos ^3(c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{4 e}{3 a^2 d (e \sin (c+d x))^{3/2}}+\frac{16 e \cos (c+d x)}{21 a^2 d (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^2 \sqrt{e \sin (c+d x)}}+\frac{\left (4 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{7 a^2 \sqrt{e \sin (c+d x)}}\\ &=\frac{4 e^3}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos (c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos ^3(c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{4 e}{3 a^2 d (e \sin (c+d x))^{3/2}}+\frac{16 e \cos (c+d x)}{21 a^2 d (e \sin (c+d x))^{3/2}}+\frac{20 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{21 a^2 d \sqrt{e \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.37354, size = 82, normalized size = 0.43 \[ -\frac{\csc ^3(c+d x) \left (40 \sin ^{\frac{7}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{4} (-2 c-2 d x+\pi ),2\right )+16 \sin ^4\left (\frac{1}{2} (c+d x)\right ) (11 \cos (c+d x)+8)\right )}{42 a^2 d \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.707, size = 148, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{4\,{e}^{3} \left ( 7\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-4 \right ) }{21\,{a}^{2}} \left ( e\sin \left ( dx+c \right ) \right ) ^{-{\frac{7}{2}}}}-{\frac{2}{21\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}\cos \left ( dx+c \right ) } \left ( 5\,\sqrt{-\sin \left ( dx+c \right ) +1}\sqrt{2+2\,\sin \left ( dx+c \right ) } \left ( \sin \left ( dx+c \right ) \right ) ^{9/2}{\it EllipticF} \left ( \sqrt{-\sin \left ( dx+c \right ) +1},1/2\,\sqrt{2} \right ) +11\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}-17\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}+6\,\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^{2} e \sec \left (d x + c\right )^{2} + 2 \, a^{2} e \sec \left (d x + c\right ) + a^{2} e\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 )}^{2} \sqrt{e \sin \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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