Optimal. Leaf size=139 \[ \frac{3 a^2 \sqrt{\sin (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right )}{d \sqrt{e \sin (c+d x)}}+\frac{2 a^2 \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}+\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}+\frac{a^2 \sec (c+d x) \sqrt{e \sin (c+d x)}}{d e} \]
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Rubi [A] time = 0.307262, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3872, 2873, 2642, 2641, 2564, 329, 212, 206, 203, 2571} \[ \frac{2 a^2 \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}+\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}+\frac{a^2 \sec (c+d x) \sqrt{e \sin (c+d x)}}{d e}+\frac{3 a^2 \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2873
Rule 2642
Rule 2641
Rule 2564
Rule 329
Rule 212
Rule 206
Rule 203
Rule 2571
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^2}{\sqrt{e \sin (c+d x)}} \, dx &=\int \frac{(-a-a \cos (c+d x))^2 \sec ^2(c+d x)}{\sqrt{e \sin (c+d x)}} \, dx\\ &=\int \left (\frac{a^2}{\sqrt{e \sin (c+d x)}}+\frac{2 a^2 \sec (c+d x)}{\sqrt{e \sin (c+d x)}}+\frac{a^2 \sec ^2(c+d x)}{\sqrt{e \sin (c+d x)}}\right ) \, dx\\ &=a^2 \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx+a^2 \int \frac{\sec ^2(c+d x)}{\sqrt{e \sin (c+d x)}} \, dx+\left (2 a^2\right ) \int \frac{\sec (c+d x)}{\sqrt{e \sin (c+d x)}} \, dx\\ &=\frac{a^2 \sec (c+d x) \sqrt{e \sin (c+d x)}}{d e}+\frac{1}{2} a^2 \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x^2}{e^2}\right )} \, dx,x,e \sin (c+d x)\right )}{d e}+\frac{\left (a^2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{\sqrt{e \sin (c+d x)}}\\ &=\frac{2 a^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{d \sqrt{e \sin (c+d x)}}+\frac{a^2 \sec (c+d x) \sqrt{e \sin (c+d x)}}{d e}+\frac{\left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{x^4}{e^2}} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{d e}+\frac{\left (a^2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{2 \sqrt{e \sin (c+d x)}}\\ &=\frac{3 a^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{d \sqrt{e \sin (c+d x)}}+\frac{a^2 \sec (c+d x) \sqrt{e \sin (c+d x)}}{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}+\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}\\ &=\frac{2 a^2 \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}+\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}+\frac{3 a^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{d \sqrt{e \sin (c+d x)}}+\frac{a^2 \sec (c+d x) \sqrt{e \sin (c+d x)}}{d e}\\ \end{align*}
Mathematica [C] time = 64.3412, size = 164, normalized size = 1.18 \[ \frac{a^2 \sqrt{\sin (c+d x)} \cos ^4\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \sec ^4\left (\frac{1}{2} \sin ^{-1}(\sin (c+d x))\right ) \left (3 \sqrt{\sin (c+d x)} \sqrt{\cos ^2(c+d x)} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},\sin ^2(c+d x)\right )+\sqrt{\sin (c+d x)}+2 \sqrt{\cos ^2(c+d x)} \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )+2 \sqrt{\cos ^2(c+d x)} \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )\right )}{d \sqrt{e \sin (c+d x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 2.059, size = 163, normalized size = 1.2 \begin{align*} -{\frac{{a}^{2}}{2\,d\cos \left ( dx+c \right ) } \left ( 3\,\sqrt{e}\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 ) -4\,\cos \left ( dx+c \right ) \sqrt{e\sin \left ( dx+c \right ) }{\it Artanh} \left ({\frac{\sqrt{e\sin \left ( dx+c \right ) }}{\sqrt{e}}} \right ) -4\,\cos \left ( dx+c \right ) \sqrt{e\sin \left ( dx+c \right ) }\arctan \left ({\frac{\sqrt{e\sin \left ( dx+c \right ) }}{\sqrt{e}}} \right ) -2\,\sqrt{e}\sin \left ( dx+c \right ) \right ){\frac{1}{\sqrt{e}}}{\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 \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] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int \frac{1}{\sqrt{e \sin{\left (c + d x \right )}}}\, dx + \int \frac{2 \sec{\left (c + d x \right )}}{\sqrt{e \sin{\left (c + d x \right )}}}\, dx + \int \frac{\sec ^{2}{\left (c + d x \right )}}{\sqrt{e \sin{\left (c + d x \right )}}}\, dx\right ) \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}}{\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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