Optimal. Leaf size=150 \[ \frac {10 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{33 a^2 d e^2 \sqrt {e \cos (c+d x)}}+\frac {10 \sin (c+d x)}{33 a^2 d e (e \cos (c+d x))^{3/2}}-\frac {2}{11 d e \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}-\frac {2}{11 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2681, 2683, 2636, 2642, 2641} \[ \frac {10 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{33 a^2 d e^2 \sqrt {e \cos (c+d x)}}+\frac {10 \sin (c+d x)}{33 a^2 d e (e \cos (c+d x))^{3/2}}-\frac {2}{11 d e \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}-\frac {2}{11 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rule 2642
Rule 2681
Rule 2683
Rubi steps
\begin {align*} \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2} \, dx &=-\frac {2}{11 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}+\frac {7 \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))} \, dx}{11 a}\\ &=-\frac {2}{11 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}-\frac {2}{11 d e (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )}+\frac {5 \int \frac {1}{(e \cos (c+d x))^{5/2}} \, dx}{11 a^2}\\ &=\frac {10 \sin (c+d x)}{33 a^2 d e (e \cos (c+d x))^{3/2}}-\frac {2}{11 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}-\frac {2}{11 d e (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )}+\frac {5 \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{33 a^2 e^2}\\ &=\frac {10 \sin (c+d x)}{33 a^2 d e (e \cos (c+d x))^{3/2}}-\frac {2}{11 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}-\frac {2}{11 d e (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )}+\frac {\left (5 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{33 a^2 e^2 \sqrt {e \cos (c+d x)}}\\ &=\frac {10 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{33 a^2 d e^2 \sqrt {e \cos (c+d x)}}+\frac {10 \sin (c+d x)}{33 a^2 d e (e \cos (c+d x))^{3/2}}-\frac {2}{11 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}-\frac {2}{11 d e (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 66, normalized size = 0.44 \[ \frac {(\sin (c+d x)+1)^{3/4} \, _2F_1\left (-\frac {3}{4},\frac {15}{4};\frac {1}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{6\ 2^{3/4} a^2 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e \cos \left (d x + c\right )}}{a^{2} e^{3} \cos \left (d x + c\right )^{5} - 2 \, a^{2} e^{3} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} e^{3} \cos \left (d x + c\right )^{3}}, x\right ) \]
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 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.65, size = 557, normalized size = 3.71 \[ -\frac {2 \left (160 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-400 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+160 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+400 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-320 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-200 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+264 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+50 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-104 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}+28 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-6 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{33 \left (32 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-80 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+80 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{2} 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 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
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 \[ \int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2} \,d x \]
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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