Optimal. Leaf size=224 \[ -\frac{2 a^2 \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d e^{3/2}}+\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d e^{3/2}}+\frac{3 a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}}{d e^3}-\frac{5 a^2 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{4 a^2}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \sec (c+d x)}{d e \sqrt{e \sin (c+d x)}} \]
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Rubi [A] time = 0.424021, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {3872, 2873, 2636, 2640, 2639, 2564, 325, 329, 298, 203, 206, 2570, 2571} \[ -\frac{2 a^2 \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d e^{3/2}}+\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d e^{3/2}}+\frac{3 a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}}{d e^3}-\frac{5 a^2 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{4 a^2}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \sec (c+d x)}{d e \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2873
Rule 2636
Rule 2640
Rule 2639
Rule 2564
Rule 325
Rule 329
Rule 298
Rule 203
Rule 206
Rule 2570
Rule 2571
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx &=\int \frac{(-a-a \cos (c+d x))^2 \sec ^2(c+d x)}{(e \sin (c+d x))^{3/2}} \, dx\\ &=\int \left (\frac{a^2}{(e \sin (c+d x))^{3/2}}+\frac{2 a^2 \sec (c+d x)}{(e \sin (c+d x))^{3/2}}+\frac{a^2 \sec ^2(c+d x)}{(e \sin (c+d x))^{3/2}}\right ) \, dx\\ &=a^2 \int \frac{1}{(e \sin (c+d x))^{3/2}} \, dx+a^2 \int \frac{\sec ^2(c+d x)}{(e \sin (c+d x))^{3/2}} \, dx+\left (2 a^2\right ) \int \frac{\sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx\\ &=-\frac{2 a^2 \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \sec (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{a^2 \int \sqrt{e \sin (c+d x)} \, dx}{e^2}+\frac{\left (3 a^2\right ) \int \sec ^2(c+d x) \sqrt{e \sin (c+d x)} \, dx}{e^2}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2} \left (1-\frac{x^2}{e^2}\right )} \, dx,x,e \sin (c+d x)\right )}{d e}\\ &=-\frac{4 a^2}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \sec (c+d x)}{d e \sqrt{e \sin (c+d x)}}+\frac{3 a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}}{d e^3}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1-\frac{x^2}{e^2}} \, dx,x,e \sin (c+d x)\right )}{d e^3}-\frac{\left (3 a^2\right ) \int \sqrt{e \sin (c+d x)} \, dx}{2 e^2}-\frac{\left (a^2 \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{e^2 \sqrt{\sin (c+d x)}}\\ &=-\frac{4 a^2}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \sec (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}+\frac{3 a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}}{d e^3}+\frac{\left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-\frac{x^4}{e^2}} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{d e^3}-\frac{\left (3 a^2 \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{2 e^2 \sqrt{\sin (c+d x)}}\\ &=-\frac{4 a^2}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \sec (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{5 a^2 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}+\frac{3 a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}}{d e^3}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{e-x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{d e}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{e+x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{d e}\\ &=-\frac{2 a^2 \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d e^{3/2}}+\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d e^{3/2}}-\frac{4 a^2}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \sec (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{5 a^2 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}+\frac{3 a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}}{d e^3}\\ \end{align*}
Mathematica [C] time = 10.6163, size = 135, normalized size = 0.6 \[ -\frac{2 a^2 \cos ^4\left (\frac{1}{2} (c+d x)\right ) \cot (c+d x) \sqrt{e \sin (c+d x)} \sec ^4\left (\frac{1}{2} \sin ^{-1}(\sin (c+d x))\right ) \left (\sin ^2(c+d x) \text{Hypergeometric2F1}\left (\frac{3}{4},\frac{3}{2},\frac{7}{4},\sin ^2(c+d x)\right )+6 \text{Hypergeometric2F1}\left (-\frac{1}{4},1,\frac{3}{4},\sin ^2(c+d x)\right )+6 \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{3}{2},\frac{3}{4},\sin ^2(c+d x)\right )\right )}{3 d e^2 \sqrt{\cos ^2(c+d x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 2.436, size = 238, normalized size = 1.1 \begin{align*}{\frac{{a}^{2}}{2\,d\cos \left ( dx+c \right ) } \left ( 10\,{e}^{3/2}\sqrt{-\sin \left ( dx+c \right ) +1}\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticE} \left ( \sqrt{-\sin \left ( dx+c \right ) +1},1/2\,\sqrt{2} \right ) -5\,{e}^{3/2}\sqrt{-\sin \left ( dx+c \right ) +1}\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{-\sin \left ( dx+c \right ) +1},1/2\,\sqrt{2} \right ) -10\,{e}^{3/2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}-8\,{e}^{3/2}\cos \left ( dx+c \right ) +4\,{\it Artanh} \left ({\frac{\sqrt{e\sin \left ( dx+c \right ) }}{\sqrt{e}}} \right ) \sqrt{e\sin \left ( dx+c \right ) }e\cos \left ( dx+c \right ) -4\,\sqrt{e\sin \left ( dx+c \right ) }\arctan \left ({\frac{\sqrt{e\sin \left ( dx+c \right ) }}{\sqrt{e}}} \right ) e\cos \left ( dx+c \right ) +2\,{e}^{3/2} \right ){e}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{e\sin \left ( dx+c \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{{\left (a^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \sec \left (d x + c\right ) + a^{2}\right )} \sqrt{e \sin \left (d x + c\right )}}{e^{2} \cos \left (d x + c\right )^{2} - e^{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{{\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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