Optimal. Leaf size=169 \[ \frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {26 e^8 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a^3 d \sqrt {e \cos (c+d x)}}+\frac {26 e^7 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 a^3 d}+\frac {26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2} \]
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Rubi [A]
time = 0.12, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2759, 2761,
2715, 2721, 2720} \begin {gather*} \frac {26 e^8 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a^3 d \sqrt {e \cos (c+d x)}}+\frac {26 e^7 \sin (c+d x) \sqrt {e \cos (c+d x)}}{21 a^3 d}+\frac {26 e^5 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^3 d}+\frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2720
Rule 2721
Rule 2759
Rule 2761
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx &=\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac {\left (13 e^2\right ) \int \frac {(e \cos (c+d x))^{11/2}}{a+a \sin (c+d x)} \, dx}{5 a^2}\\ &=\frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac {\left (13 e^4\right ) \int (e \cos (c+d x))^{7/2} \, dx}{5 a^3}\\ &=\frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac {\left (13 e^6\right ) \int (e \cos (c+d x))^{3/2} \, dx}{7 a^3}\\ &=\frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {26 e^7 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 a^3 d}+\frac {26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac {\left (13 e^8\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{21 a^3}\\ &=\frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {26 e^7 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 a^3 d}+\frac {26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac {\left (13 e^8 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 a^3 \sqrt {e \cos (c+d x)}}\\ &=\frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {26 e^8 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a^3 d \sqrt {e \cos (c+d x)}}+\frac {26 e^7 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 a^3 d}+\frac {26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.37, size = 66, normalized size = 0.39 \begin {gather*} -\frac {4 \sqrt [4]{2} (e \cos (c+d x))^{17/2} \, _2F_1\left (-\frac {1}{4},\frac {17}{4};\frac {21}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{17 a^3 d e (1+\sin (c+d x))^{17/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.78, size = 251, normalized size = 1.49
method | result | size |
default | \(-\frac {2 e^{8} \left (-1120 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2160 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2800 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3240 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-784 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-840 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-1624 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+195 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-120 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+1162 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-217 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(251\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 119, normalized size = 0.70 \begin {gather*} \frac {-195 i \, \sqrt {2} e^{\frac {15}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 i \, \sqrt {2} e^{\frac {15}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (35 \, \cos \left (d x + c\right )^{4} e^{\frac {15}{2}} - 252 \, \cos \left (d x + c\right )^{2} e^{\frac {15}{2}} + 15 \, {\left (9 \, \cos \left (d x + c\right )^{2} e^{\frac {15}{2}} - 13 \, e^{\frac {15}{2}}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{315 \, a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{15/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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