Optimal. Leaf size=270 \[ \frac{7 a^2 e^2 \sqrt{\sin (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right ) \sqrt{e \csc (c+d x)}}{3 d}-\frac{4 a^2 e^2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 d}+\frac{5 a^2 e^2 \tan (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \csc (c+d x) \sec (c+d x) \sqrt{e \csc (c+d x)}}{3 d}+\frac{2 a^2 e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d}+\frac{2 a^2 e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d} \]
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Rubi [A] time = 0.333941, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {3878, 3872, 2873, 2636, 2641, 2564, 325, 329, 212, 206, 203, 2570, 2571} \[ -\frac{4 a^2 e^2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 d}+\frac{5 a^2 e^2 \tan (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \csc (c+d x) \sec (c+d x) \sqrt{e \csc (c+d x)}}{3 d}+\frac{2 a^2 e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d}+\frac{2 a^2 e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d}+\frac{7 a^2 e^2 \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \csc (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3878
Rule 3872
Rule 2873
Rule 2636
Rule 2641
Rule 2564
Rule 325
Rule 329
Rule 212
Rule 206
Rule 203
Rule 2570
Rule 2571
Rubi steps
\begin{align*} \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx &=\left (e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^2}{\sin ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\left (e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{(-a-a \cos (c+d x))^2 \sec ^2(c+d x)}{\sin ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\left (e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \left (\frac{a^2}{\sin ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 \sec (c+d x)}{\sin ^{\frac{5}{2}}(c+d x)}+\frac{a^2 \sec ^2(c+d x)}{\sin ^{\frac{5}{2}}(c+d x)}\right ) \, dx\\ &=\left (a^2 e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sin ^{\frac{5}{2}}(c+d x)} \, dx+\left (a^2 e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{\sec ^2(c+d x)}{\sin ^{\frac{5}{2}}(c+d x)} \, dx+\left (2 a^2 e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{\sec (c+d x)}{\sin ^{\frac{5}{2}}(c+d x)} \, dx\\ &=-\frac{2 a^2 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \csc (c+d x) \sqrt{e \csc (c+d x)} \sec (c+d x)}{3 d}+\frac{1}{3} \left (a^2 e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx+\frac{1}{3} \left (5 a^2 e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{\sec ^2(c+d x)}{\sqrt{\sin (c+d x)}} \, dx+\frac{\left (2 a^2 e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{5/2} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{2 a^2 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{4 a^2 e^2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \csc (c+d x) \sqrt{e \csc (c+d x)} \sec (c+d x)}{3 d}+\frac{2 a^2 e^2 \sqrt{e \csc (c+d x)} F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 d}+\frac{5 a^2 e^2 \sqrt{e \csc (c+d x)} \tan (c+d x)}{3 d}+\frac{1}{6} \left (5 a^2 e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx+\frac{\left (2 a^2 e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{2 a^2 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{4 a^2 e^2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \csc (c+d x) \sqrt{e \csc (c+d x)} \sec (c+d x)}{3 d}+\frac{7 a^2 e^2 \sqrt{e \csc (c+d x)} F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 d}+\frac{5 a^2 e^2 \sqrt{e \csc (c+d x)} \tan (c+d x)}{3 d}+\frac{\left (4 a^2 e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d}\\ &=-\frac{2 a^2 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{4 a^2 e^2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \csc (c+d x) \sqrt{e \csc (c+d x)} \sec (c+d x)}{3 d}+\frac{7 a^2 e^2 \sqrt{e \csc (c+d x)} F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 d}+\frac{5 a^2 e^2 \sqrt{e \csc (c+d x)} \tan (c+d x)}{3 d}+\frac{\left (2 a^2 e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d}+\frac{\left (2 a^2 e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d}\\ &=-\frac{2 a^2 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{4 a^2 e^2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \csc (c+d x) \sqrt{e \csc (c+d x)} \sec (c+d x)}{3 d}+\frac{2 a^2 e^2 \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right ) \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}{d}+\frac{2 a^2 e^2 \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right ) \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}{d}+\frac{7 a^2 e^2 \sqrt{e \csc (c+d x)} F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 d}+\frac{5 a^2 e^2 \sqrt{e \csc (c+d x)} \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [C] time = 3.74815, size = 195, normalized size = 0.72 \[ -\frac{a^2 e^2 \cos ^4\left (\frac{1}{2} (c+d x)\right ) \tan (c+d x) \sqrt{e \csc (c+d x)} \sec ^4\left (\frac{1}{2} \csc ^{-1}(\csc (c+d x))\right ) \left (7 \sqrt{-\cot ^2(c+d x)} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},\csc ^2(c+d x)\right )+4 \csc ^2(c+d x)+4 \sqrt{\cos ^2(c+d x)} \csc ^2(c+d x)+6 \sqrt{\cos ^2(c+d x)} \sqrt{\csc (c+d x)} \tan ^{-1}\left (\sqrt{\csc (c+d x)}\right )-6 \sqrt{\cos ^2(c+d x)} \sqrt{\csc (c+d x)} \tanh ^{-1}\left (\sqrt{\csc (c+d x)}\right )-7\right )}{3 d} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.272, size = 730, normalized size = 2.7 \begin{align*} \text{result too large to display} \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} \csc \left (d x + c\right )^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} e^{2} \csc \left (d x + c\right )^{2} \sec \left (d x + c\right ) + a^{2} e^{2} \csc \left (d x + c\right )^{2}\right )} \sqrt{e \csc \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 (e \csc \left (d x + c\right )\right )^{\frac{5}{2}}{\left (a \sec \left (d x + c\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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