Optimal. Leaf size=194 \[ -\frac{2 a^2 e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d}+\frac{2 a^2 e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d}-\frac{9 a^2 e^2 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}-\frac{4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac{2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac{a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d} \]
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Rubi [A] time = 0.382437, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {3872, 2873, 2635, 2640, 2639, 2564, 321, 329, 298, 203, 206, 2566} \[ -\frac{2 a^2 e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d}+\frac{2 a^2 e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d}-\frac{9 a^2 e^2 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}-\frac{4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac{2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac{a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2873
Rule 2635
Rule 2640
Rule 2639
Rule 2564
Rule 321
Rule 329
Rule 298
Rule 203
Rule 206
Rule 2566
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx &=\int (-a-a \cos (c+d x))^2 \sec ^2(c+d x) (e \sin (c+d x))^{5/2} \, dx\\ &=\int \left (a^2 (e \sin (c+d x))^{5/2}+2 a^2 \sec (c+d x) (e \sin (c+d x))^{5/2}+a^2 \sec ^2(c+d x) (e \sin (c+d x))^{5/2}\right ) \, dx\\ &=a^2 \int (e \sin (c+d x))^{5/2} \, dx+a^2 \int \sec ^2(c+d x) (e \sin (c+d x))^{5/2} \, dx+\left (2 a^2\right ) \int \sec (c+d x) (e \sin (c+d x))^{5/2} \, dx\\ &=-\frac{2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac{a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{x^{5/2}}{1-\frac{x^2}{e^2}} \, dx,x,e \sin (c+d x)\right )}{d e}+\frac{1}{5} \left (3 a^2 e^2\right ) \int \sqrt{e \sin (c+d x)} \, dx-\frac{1}{2} \left (3 a^2 e^2\right ) \int \sqrt{e \sin (c+d x)} \, dx\\ &=-\frac{4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac{2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac{a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d}+\frac{\left (2 a^2 e\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1-\frac{x^2}{e^2}} \, dx,x,e \sin (c+d x)\right )}{d}+\frac{\left (3 a^2 e^2 \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{5 \sqrt{\sin (c+d x)}}-\frac{\left (3 a^2 e^2 \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{2 \sqrt{\sin (c+d x)}}\\ &=-\frac{9 a^2 e^2 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}-\frac{4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac{2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac{a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d}+\frac{\left (4 a^2 e\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-\frac{x^4}{e^2}} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{d}\\ &=-\frac{9 a^2 e^2 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}-\frac{4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac{2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac{a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d}+\frac{\left (2 a^2 e^3\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 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{d}\\ &=-\frac{2 a^2 e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d}+\frac{2 a^2 e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d}-\frac{9 a^2 e^2 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}-\frac{4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac{2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac{a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d}\\ \end{align*}
Mathematica [C] time = 16.6173, size = 205, normalized size = 1.06 \[ \frac{2 a^2 \cos ^4\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) (e \sin (c+d x))^{5/2} \sec ^4\left (\frac{1}{2} \sin ^{-1}(\sin (c+d x))\right ) \left (9 \sin ^{\frac{3}{2}}(c+d x) \sqrt{\cos ^2(c+d x)} \text{Hypergeometric2F1}\left (\frac{3}{4},\frac{3}{2},\frac{7}{4},\sin ^2(c+d x)\right )+3 \sin ^{\frac{7}{2}}(c+d x)-9 \sin ^{\frac{3}{2}}(c+d x)-10 \sin ^{\frac{3}{2}}(c+d x) \sqrt{\cos ^2(c+d x)}-15 \sqrt{\cos ^2(c+d x)} \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )+15 \sqrt{\cos ^2(c+d x)} \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )\right )}{15 d \sin ^{\frac{5}{2}}(c+d x)} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 2.388, size = 265, normalized size = 1.4 \begin{align*}{\frac{{a}^{2}}{30\,d\cos \left ( dx+c \right ) } \left ( 60\,{\it Artanh} \left ({\frac{\sqrt{e\sin \left ( dx+c \right ) }}{\sqrt{e}}} \right ) \sqrt{e\sin \left ( dx+c \right ) }{e}^{5/2}\cos \left ( dx+c \right ) -60\,\sqrt{e\sin \left ( dx+c \right ) }\arctan \left ({\frac{\sqrt{e\sin \left ( dx+c \right ) }}{\sqrt{e}}} \right ){e}^{5/2}\cos \left ( dx+c \right ) +54\,\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 ){e}^{3}-27\,\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 ){e}^{3}+12\,{e}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}+40\,{e}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}-42\,{e}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}-40\,{e}^{3}\cos \left ( dx+c \right ) +30\,{e}^{3} \right ){\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 (-{\left (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 )\right )} \sqrt{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(-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{\left (a \sec \left (d x + c\right ) + a\right )}^{2} \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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