Optimal. Leaf size=127 \[ \frac {4 a^5 \sqrt {e \cos (c+d x)}}{7 d e^5 (a-a \sin (c+d x))^2}-\frac {2 a^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d e^4 \sqrt {e \cos (c+d x)}}-\frac {2 a^6 \sqrt {e \cos (c+d x)}}{21 d e^5 \left (a^3-a^3 \sin (c+d x)\right )} \]
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Rubi [A] time = 0.20, antiderivative size = 127, 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 = {2670, 2680, 2683, 2642, 2641} \[ -\frac {2 a^6 \sqrt {e \cos (c+d x)}}{21 d e^5 \left (a^3-a^3 \sin (c+d x)\right )}+\frac {4 a^5 \sqrt {e \cos (c+d x)}}{7 d e^5 (a-a \sin (c+d x))^2}-\frac {2 a^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d e^4 \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 2642
Rule 2670
Rule 2680
Rule 2683
Rubi steps
\begin {align*} \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{9/2}} \, dx &=\frac {a^6 \int \frac {(e \cos (c+d x))^{3/2}}{(a-a \sin (c+d x))^3} \, dx}{e^6}\\ &=\frac {4 a^5 \sqrt {e \cos (c+d x)}}{7 d e^5 (a-a \sin (c+d x))^2}-\frac {a^4 \int \frac {1}{\sqrt {e \cos (c+d x)} (a-a \sin (c+d x))} \, dx}{7 e^4}\\ &=\frac {4 a^5 \sqrt {e \cos (c+d x)}}{7 d e^5 (a-a \sin (c+d x))^2}-\frac {2 a^4 \sqrt {e \cos (c+d x)}}{21 d e^5 (a-a \sin (c+d x))}-\frac {a^3 \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{21 e^4}\\ &=\frac {4 a^5 \sqrt {e \cos (c+d x)}}{7 d e^5 (a-a \sin (c+d x))^2}-\frac {2 a^4 \sqrt {e \cos (c+d x)}}{21 d e^5 (a-a \sin (c+d x))}-\frac {\left (a^3 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 e^4 \sqrt {e \cos (c+d x)}}\\ &=-\frac {2 a^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d e^4 \sqrt {e \cos (c+d x)}}+\frac {4 a^5 \sqrt {e \cos (c+d x)}}{7 d e^5 (a-a \sin (c+d x))^2}-\frac {2 a^4 \sqrt {e \cos (c+d x)}}{21 d e^5 (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 66, normalized size = 0.52 \[ \frac {4 \sqrt [4]{2} a^3 (\sin (c+d x)+1)^{7/4} \, _2F_1\left (-\frac {7}{4},-\frac {1}{4};-\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{7 d e (e \cos (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{5} \cos \left (d x + c\right )^{5}}, x\right ) \]
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 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.00, size = 401, normalized size = 3.16 \[ \frac {2 \left (8 \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 )-12 \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 )+8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+6 \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 )-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+28 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\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}-22 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-28 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{21 \left (8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \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^{4} 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 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{9/2}} \,d x \]
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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