Optimal. Leaf size=139 \[ -\frac {4 e^4 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{21 a d \sqrt {e \sin (c+d x)}}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a d}-\frac {2 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a d}+\frac {2 e (e \sin (c+d x))^{5/2}}{5 a d} \]
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Rubi [A] time = 0.28, antiderivative size = 139, 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, 2568, 2569, 2642, 2641} \[ \frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a d}-\frac {2 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a d}-\frac {4 e^4 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{21 a d \sqrt {e \sin (c+d x)}}+\frac {2 e (e \sin (c+d x))^{5/2}}{5 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2564
Rule 2568
Rule 2569
Rule 2641
Rule 2642
Rule 2839
Rule 3872
Rubi steps
\begin {align*} \int \frac {(e \sin (c+d x))^{7/2}}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) (e \sin (c+d x))^{7/2}}{-a-a \cos (c+d x)} \, dx\\ &=\frac {e^2 \int \cos (c+d x) (e \sin (c+d x))^{3/2} \, dx}{a}-\frac {e^2 \int \cos ^2(c+d x) (e \sin (c+d x))^{3/2} \, dx}{a}\\ &=\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a d}+\frac {e \operatorname {Subst}\left (\int x^{3/2} \, dx,x,e \sin (c+d x)\right )}{a d}-\frac {e^4 \int \frac {\cos ^2(c+d x)}{\sqrt {e \sin (c+d x)}} \, dx}{7 a}\\ &=-\frac {2 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a d}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a d}+\frac {2 e (e \sin (c+d x))^{5/2}}{5 a d}-\frac {\left (2 e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{21 a}\\ &=-\frac {2 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a d}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a d}+\frac {2 e (e \sin (c+d x))^{5/2}}{5 a d}-\frac {\left (2 e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{21 a \sqrt {e \sin (c+d x)}}\\ &=-\frac {4 e^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 \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a d}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a d}+\frac {2 e (e \sin (c+d x))^{5/2}}{5 a d}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 122, normalized size = 0.88 \[ \frac {e^3 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \sqrt {e \sin (c+d x)} \left (40 F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )+\sqrt {\sin (c+d x)} (25 \cos (c+d x)-42 \cos (2 (c+d x))+15 \cos (3 (c+d x))+42)\right )}{105 a d \sqrt {\sin (c+d x)} (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (e^{3} \cos \left (d x + c\right )^{2} - e^{3}\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right )}{a \sec \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}}{a \sec \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.28, size = 128, normalized size = 0.92 \[ \frac {\frac {2 e \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5 a}+\frac {2 e^{4} \left (3 \left (\sin ^{5}\left (d x +c \right )\right )+\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-5 \left (\sin ^{3}\left (d x +c \right )\right )+2 \sin \left (d x +c \right )\right )}{21 a \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}}{a \sec \left (d x + c\right ) + a}\,{d x} \]
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 \[ \int \frac {\cos \left (c+d\,x\right )\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}}{a\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]
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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