Optimal. Leaf size=149 \[ -\frac {154 e^6 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^4 d \sqrt {\cos (c+d x)}}-\frac {154 e^5 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^4 d}-\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a+a \sin (c+d x))^3}-\frac {44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left (a^4+a^4 \sin (c+d x)\right )} \]
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Rubi [A]
time = 0.11, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2759, 2715,
2721, 2719} \begin {gather*} -\frac {154 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 a^4 d \sqrt {\cos (c+d x)}}-\frac {154 e^5 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^4 d}-\frac {44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left (a^4 \sin (c+d x)+a^4\right )}-\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a \sin (c+d x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2719
Rule 2721
Rule 2759
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{13/2}}{(a+a \sin (c+d x))^4} \, dx &=-\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a+a \sin (c+d x))^3}-\frac {\left (11 e^2\right ) \int \frac {(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^2} \, dx}{a^2}\\ &=-\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a+a \sin (c+d x))^3}-\frac {44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\left (77 e^4\right ) \int (e \cos (c+d x))^{5/2} \, dx}{3 a^4}\\ &=-\frac {154 e^5 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^4 d}-\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a+a \sin (c+d x))^3}-\frac {44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\left (77 e^6\right ) \int \sqrt {e \cos (c+d x)} \, dx}{5 a^4}\\ &=-\frac {154 e^5 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^4 d}-\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a+a \sin (c+d x))^3}-\frac {44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\left (77 e^6 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 a^4 \sqrt {\cos (c+d x)}}\\ &=-\frac {154 e^6 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^4 d \sqrt {\cos (c+d x)}}-\frac {154 e^5 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^4 d}-\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a+a \sin (c+d x))^3}-\frac {44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.20, size = 66, normalized size = 0.44 \begin {gather*} -\frac {2^{3/4} (e \cos (c+d x))^{15/2} \, _2F_1\left (\frac {5}{4},\frac {15}{4};\frac {19}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{15 a^4 d e (1+\sin (c+d x))^{15/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.49, size = 190, normalized size = 1.28
method | result | size |
default | \(-\frac {2 \left (-24 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+80 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+231 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-246 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-80 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+140 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e^{7}}{15 \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d}\) | \(190\) |
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 = 231, normalized size = 1.55 \begin {gather*} -\frac {231 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {13}{2}} + i \, \sqrt {2} e^{\frac {13}{2}} \sin \left (d x + c\right ) + i \, \sqrt {2} e^{\frac {13}{2}}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {13}{2}} - i \, \sqrt {2} e^{\frac {13}{2}} \sin \left (d x + c\right ) - i \, \sqrt {2} e^{\frac {13}{2}}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (3 \, \cos \left (d x + c\right )^{3} e^{\frac {13}{2}} + 20 \, \cos \left (d x + c\right )^{2} e^{\frac {13}{2}} + 137 \, \cos \left (d x + c\right ) e^{\frac {13}{2}} - {\left (3 \, \cos \left (d x + c\right )^{2} e^{\frac {13}{2}} - 17 \, \cos \left (d x + c\right ) e^{\frac {13}{2}} + 120 \, e^{\frac {13}{2}}\right )} \sin \left (d x + c\right ) + 120 \, e^{\frac {13}{2}}\right )} \sqrt {\cos \left (d x + c\right )}}{15 \, {\left (a^{4} d \cos \left (d x + c\right ) + a^{4} d \sin \left (d x + c\right ) + a^{4} d\right )}} \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 )}^{13/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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