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 = 0.22, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, 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 30
Rule 2564
Rule 2569
Rule 2639
Rule 2839
Rule 3872
Rule 3878
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*}
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Mathematica [C] time = 0.83, size = 100, normalized size = 0.83 \[ \frac {8 \sqrt {1-e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};e^{2 i (c+d x)}\right ) (\cot (c+d x)+i)+20 \sin (c+d x)-6 (\sin (2 (c+d x))+4 i)}{30 a d e^2 \sqrt {e \csc (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \csc \left (d x + c\right )}}{a e^{3} \csc \left (d x + c\right )^{3} \sec \left (d x + c\right ) + a e^{3} \csc \left (d x + c\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.40, size = 551, normalized size = 4.59 \[ \frac {\left (12 \cos \left (d x +c \right ) \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-i \cos \left (d x +c \right )+\sin \left (d x +c \right )+i}{\sin \left (d x +c \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )-6 \cos \left (d x +c \right ) \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-i \cos \left (d x +c \right )+\sin \left (d x +c \right )+i}{\sin \left (d x +c \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )+3 \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right )+12 \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-i \cos \left (d x +c \right )+\sin \left (d x +c \right )+i}{\sin \left (d x +c \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )-6 \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-i \cos \left (d x +c \right )+\sin \left (d x +c \right )+i}{\sin \left (d x +c \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )-5 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}+3 \cos \left (d x +c \right ) \sqrt {2}-\sqrt {2}\right ) \sqrt {2}}{15 a d \left (\frac {e}{\sin \left (d x +c \right )}\right )^{\frac {5}{2}} \sin \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\cos \left (c+d\,x\right )}{a\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{5/2}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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