Optimal. Leaf size=120 \[ \frac{2 \sin (c+d x)}{3 a d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 \sin (c+d x) \cos (c+d x)}{5 a d e^2 \sqrt{e \csc (c+d x)}}-\frac{4 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{5 a d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}} \]
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Rubi [A] time = 0.222214, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3878, 3872, 2839, 2564, 30, 2569, 2639} \[ \frac{2 \sin (c+d x)}{3 a d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 \sin (c+d x) \cos (c+d x)}{5 a d e^2 \sqrt{e \csc (c+d x)}}-\frac{4 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{5 a d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3878
Rule 3872
Rule 2839
Rule 2564
Rule 30
Rule 2569
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(e \csc (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx &=\frac{\int \frac{\sin ^{\frac{5}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx}{e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{\int \frac{\cos (c+d x) \sin ^{\frac{5}{2}}(c+d x)}{-a-a \cos (c+d x)} \, dx}{e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=\frac{\int \cos (c+d x) \sqrt{\sin (c+d x)} \, dx}{a e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}-\frac{\int \cos ^2(c+d x) \sqrt{\sin (c+d x)} \, dx}{a e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{2 \cos (c+d x) \sin (c+d x)}{5 a d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 \int \sqrt{\sin (c+d x)} \, dx}{5 a e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{\operatorname{Subst}\left (\int \sqrt{x} \, dx,x,\sin (c+d x)\right )}{a d e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{4 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right )}{5 a d e^2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{2 \sin (c+d x)}{3 a d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 \cos (c+d x) \sin (c+d x)}{5 a d e^2 \sqrt{e \csc (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.874674, size = 100, normalized size = 0.83 \[ \frac{8 \sqrt{1-e^{2 i (c+d x)}} (\cot (c+d x)+i) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},e^{2 i (c+d x)}\right )+20 \sin (c+d x)-6 (\sin (2 (c+d x))+4 i)}{30 a d e^2 \sqrt{e \csc (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.227, size = 563, normalized size = 4.7 \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 (\frac{\sqrt{e \csc \left (d x + c\right )}}{a e^{3} \csc \left (d x + c\right )^{3} \sec \left (d x + c\right ) + a e^{3} \csc \left (d x + c\right )^{3}}, 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 (e \csc \left (d x + c\right )\right )^{\frac{5}{2}}{\left (a \sec \left (d x + c\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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