Optimal. Leaf size=149 \[ -\frac{4 \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right )}{21 a d e^3 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 \cos ^3(c+d x)}{7 a d e^3 \sqrt{e \csc (c+d x)}}-\frac{2 \cos (c+d x)}{21 a d e^3 \sqrt{e \csc (c+d x)}}+\frac{2 \sin ^2(c+d x)}{5 a d e^3 \sqrt{e \csc (c+d x)}} \]
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Rubi [A] time = 0.253985, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3878, 3872, 2839, 2564, 30, 2568, 2569, 2641} \[ \frac{2 \cos ^3(c+d x)}{7 a d e^3 \sqrt{e \csc (c+d x)}}-\frac{2 \cos (c+d x)}{21 a d e^3 \sqrt{e \csc (c+d x)}}+\frac{2 \sin ^2(c+d x)}{5 a d e^3 \sqrt{e \csc (c+d x)}}-\frac{4 F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{21 a d e^3 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3878
Rule 3872
Rule 2839
Rule 2564
Rule 30
Rule 2568
Rule 2569
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx &=\frac{\int \frac{\sin ^{\frac{7}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx}{e^3 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{\int \frac{\cos (c+d x) \sin ^{\frac{7}{2}}(c+d x)}{-a-a \cos (c+d x)} \, dx}{e^3 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=\frac{\int \cos (c+d x) \sin ^{\frac{3}{2}}(c+d x) \, dx}{a e^3 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}-\frac{\int \cos ^2(c+d x) \sin ^{\frac{3}{2}}(c+d x) \, dx}{a e^3 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=\frac{2 \cos ^3(c+d x)}{7 a d e^3 \sqrt{e \csc (c+d x)}}-\frac{\int \frac{\cos ^2(c+d x)}{\sqrt{\sin (c+d x)}} \, dx}{7 a e^3 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{\operatorname{Subst}\left (\int x^{3/2} \, dx,x,\sin (c+d x)\right )}{a d e^3 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{2 \cos (c+d x)}{21 a d e^3 \sqrt{e \csc (c+d x)}}+\frac{2 \cos ^3(c+d x)}{7 a d e^3 \sqrt{e \csc (c+d x)}}+\frac{2 \sin ^2(c+d x)}{5 a d e^3 \sqrt{e \csc (c+d x)}}-\frac{2 \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{21 a e^3 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{2 \cos (c+d x)}{21 a d e^3 \sqrt{e \csc (c+d x)}}+\frac{2 \cos ^3(c+d x)}{7 a d e^3 \sqrt{e \csc (c+d x)}}-\frac{4 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right )}{21 a d e^3 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{2 \sin ^2(c+d x)}{5 a d e^3 \sqrt{e \csc (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.565914, size = 91, normalized size = 0.61 \[ \frac{\sqrt{e \csc (c+d x)} \left (80 \sqrt{\sin (c+d x)} \text{EllipticF}\left (\frac{1}{4} (-2 c-2 d x+\pi ),2\right )+126 \sin (c+d x)+10 \sin (2 (c+d x))-42 \sin (3 (c+d x))+15 \sin (4 (c+d x))\right )}{420 a d e^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.254, size = 221, normalized size = 1.5 \begin{align*}{\frac{\sqrt{2}}{105\,da \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}} \left ( 10\,i\sin \left ( dx+c \right ) \sqrt{{\frac{-i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }}}\sqrt{{\frac{-i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) +i}{\sin \left ( dx+c \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}}+15\,\sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}-36\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{2}+16\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{2}+26\,\cos \left ( dx+c \right ) \sqrt{2}-21\,\sqrt{2} \right ) \left ({\frac{e}{\sin \left ( dx+c \right ) }} \right ) ^{-{\frac{7}{2}}}} \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^{4} \csc \left (d x + c\right )^{4} \sec \left (d x + c\right ) + a e^{4} \csc \left (d x + c\right )^{4}}, 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{7}{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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