Optimal. Leaf size=224 \[ \frac{4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{4 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{15 a^2 d e^2 \sqrt{\sin (c+d x)}}-\frac{4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac{16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac{4 \cos (c+d x)}{15 a^2 d e \sqrt{e \sin (c+d x)}} \]
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Rubi [A] time = 0.6638, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 17, 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, 2640, 2639, 2564, 14} \[ \frac{4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{4 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{15 a^2 d e^2 \sqrt{\sin (c+d x)}}-\frac{4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac{16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac{4 \cos (c+d x)}{15 a^2 d e \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2875
Rule 2873
Rule 2567
Rule 2636
Rule 2640
Rule 2639
Rule 2564
Rule 14
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx &=\int \frac{\cos ^2(c+d x)}{(-a-a \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx\\ &=\frac{e^4 \int \frac{\cos ^2(c+d x) (-a+a \cos (c+d x))^2}{(e \sin (c+d x))^{11/2}} \, dx}{a^4}\\ &=\frac{e^4 \int \left (\frac{a^2 \cos ^2(c+d x)}{(e \sin (c+d x))^{11/2}}-\frac{2 a^2 \cos ^3(c+d x)}{(e \sin (c+d x))^{11/2}}+\frac{a^2 \cos ^4(c+d x)}{(e \sin (c+d x))^{11/2}}\right ) \, dx}{a^4}\\ &=\frac{e^4 \int \frac{\cos ^2(c+d x)}{(e \sin (c+d x))^{11/2}} \, dx}{a^2}+\frac{e^4 \int \frac{\cos ^4(c+d x)}{(e \sin (c+d x))^{11/2}} \, dx}{a^2}-\frac{\left (2 e^4\right ) \int \frac{\cos ^3(c+d x)}{(e \sin (c+d x))^{11/2}} \, dx}{a^2}\\ &=-\frac{2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{\left (2 e^2\right ) \int \frac{1}{(e \sin (c+d x))^{7/2}} \, dx}{9 a^2}-\frac{\left (2 e^2\right ) \int \frac{\cos ^2(c+d x)}{(e \sin (c+d x))^{7/2}} \, dx}{3 a^2}-\frac{\left (2 e^3\right ) \operatorname{Subst}\left (\int \frac{1-\frac{x^2}{e^2}}{x^{11/2}} \, dx,x,e \sin (c+d x)\right )}{a^2 d}\\ &=-\frac{2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}+\frac{16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac{2 \int \frac{1}{(e \sin (c+d x))^{3/2}} \, dx}{15 a^2}+\frac{4 \int \frac{1}{(e \sin (c+d x))^{3/2}} \, dx}{15 a^2}-\frac{\left (2 e^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^{11/2}}-\frac{1}{e^2 x^{7/2}}\right ) \, dx,x,e \sin (c+d x)\right )}{a^2 d}\\ &=\frac{4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac{16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac{4 \cos (c+d x)}{15 a^2 d e \sqrt{e \sin (c+d x)}}+\frac{2 \int \sqrt{e \sin (c+d x)} \, dx}{15 a^2 e^2}-\frac{4 \int \sqrt{e \sin (c+d x)} \, dx}{15 a^2 e^2}\\ &=\frac{4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac{16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac{4 \cos (c+d x)}{15 a^2 d e \sqrt{e \sin (c+d x)}}+\frac{\left (2 \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{15 a^2 e^2 \sqrt{\sin (c+d x)}}-\frac{\left (4 \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{15 a^2 e^2 \sqrt{\sin (c+d x)}}\\ &=\frac{4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac{16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac{4 \cos (c+d x)}{15 a^2 d e \sqrt{e \sin (c+d x)}}-\frac{4 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{15 a^2 d e^2 \sqrt{\sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 1.42102, size = 163, normalized size = 0.73 \[ \frac{\sec ^4\left (\frac{1}{2} (c+d x)\right ) (\cos (c+d x)+i \sin (c+d x)) \left (e^{-2 i (c+d x)} \sqrt{1-e^{2 i (c+d x)}} \left (1+e^{i (c+d x)}\right )^4 \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},e^{2 i (c+d x)}\right )+16 i \sin (c+d x)+13 i \sin (2 (c+d x))-40 \cos (c+d x)-19 \cos (2 (c+d x))-31\right )}{180 a^2 d e \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.753, size = 213, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{4\,{e}^{3} \left ( 9\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-4 \right ) }{45\,{a}^{2}} \left ( e\sin \left ( dx+c \right ) \right ) ^{-{\frac{9}{2}}}}+{\frac{2}{45\,e{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{5}\cos \left ( dx+c \right ) } \left ( 6\,\sqrt{-\sin \left ( dx+c \right ) +1}\sqrt{2+2\,\sin \left ( dx+c \right ) } \left ( \sin \left ( dx+c \right ) \right ) ^{11/2}{\it EllipticE} \left ( \sqrt{-\sin \left ( dx+c \right ) +1},1/2\,\sqrt{2} \right ) -3\,\sqrt{-\sin \left ( dx+c \right ) +1}\sqrt{2+2\,\sin \left ( dx+c \right ) } \left ( \sin \left ( dx+c \right ) \right ) ^{11/2}{\it EllipticF} \left ( \sqrt{-\sin \left ( dx+c \right ) +1},1/2\,\sqrt{2} \right ) +6\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}-19\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}+23\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}-10\,\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 )}}{a^{2} e^{2} \cos \left (d x + c\right )^{2} - a^{2} e^{2} +{\left (a^{2} e^{2} \cos \left (d x + c\right )^{2} - a^{2} e^{2}\right )} \sec \left (d x + c\right )^{2} + 2 \,{\left (a^{2} e^{2} \cos \left (d x + c\right )^{2} - a^{2} e^{2}\right )} \sec \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} \left (e \sin \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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