Optimal. Leaf size=135 \[ \frac {4 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{21 a d e^2 \sqrt {e \sin (c+d x)}}-\frac {2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac {2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac {4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}} \]
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Rubi [A] time = 0.25, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3872, 2839, 2564, 30, 2567, 2636, 2642, 2641} \[ \frac {4 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{21 a d e^2 \sqrt {e \sin (c+d x)}}-\frac {2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac {2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac {4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2564
Rule 2567
Rule 2636
Rule 2641
Rule 2642
Rule 2839
Rule 3872
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sec (c+d x)) (e \sin (c+d x))^{5/2}} \, dx &=-\int \frac {\cos (c+d x)}{(-a-a \cos (c+d x)) (e \sin (c+d x))^{5/2}} \, dx\\ &=\frac {e^2 \int \frac {\cos (c+d x)}{(e \sin (c+d x))^{9/2}} \, dx}{a}-\frac {e^2 \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{9/2}} \, dx}{a}\\ &=\frac {2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}+\frac {2 \int \frac {1}{(e \sin (c+d x))^{5/2}} \, dx}{7 a}+\frac {e \operatorname {Subst}\left (\int \frac {1}{x^{9/2}} \, dx,x,e \sin (c+d x)\right )}{a d}\\ &=-\frac {2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac {2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac {4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}}+\frac {2 \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{21 a e^2}\\ &=-\frac {2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac {2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac {4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}}+\frac {\left (2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{21 a e^2 \sqrt {e \sin (c+d x)}}\\ &=-\frac {2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac {2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac {4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}}+\frac {4 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{21 a d e^2 \sqrt {e \sin (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.27, size = 91, normalized size = 0.67 \[ -\frac {2 \left (2 \cos (c+d x)+\cos (2 (c+d x))+\sin ^{\frac {7}{2}}(c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right ) F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )+4\right )}{21 a d e (\cos (c+d x)+1) (e \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e \sin \left (d x + c\right )}}{{\left (a e^{3} \cos \left (d x + c\right )^{2} - a e^{3} + {\left (a e^{3} \cos \left (d x + c\right )^{2} - a e^{3}\right )} \sec \left (d x + c\right )\right )} \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.28, size = 136, normalized size = 1.01 \[ \frac {-\frac {2 e}{7 a \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {2 \left (\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {9}{2}}\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\sin ^{5}\left (d x +c \right )\right )+5 \left (\sin ^{3}\left (d x +c \right )\right )-3 \sin \left (d x +c \right )\right )}{21 e^{2} a \sin \left (d x +c \right )^{4} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\cos \left (c+d\,x\right )}{a\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]
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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