Optimal. Leaf size=120 \[ \frac {42 e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^4 d \sqrt {\cos (c+d x)}}-\frac {4 e (e \cos (c+d x))^{7/2}}{5 a d (a+a \sin (c+d x))^3}+\frac {28 e^3 (e \cos (c+d x))^{3/2}}{5 d \left (a^4+a^4 \sin (c+d x)\right )} \]
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Rubi [A]
time = 0.10, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2759, 2721,
2719} \begin {gather*} \frac {42 e^4 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 {28 e^3 (e \cos (c+d x))^{3/2}}{5 d \left (a^4 \sin (c+d x)+a^4\right )}-\frac {4 e (e \cos (c+d x))^{7/2}}{5 a d (a \sin (c+d x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2721
Rule 2759
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^4} \, dx &=-\frac {4 e (e \cos (c+d x))^{7/2}}{5 a d (a+a \sin (c+d x))^3}-\frac {\left (7 e^2\right ) \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {4 e (e \cos (c+d x))^{7/2}}{5 a d (a+a \sin (c+d x))^3}+\frac {28 e^3 (e \cos (c+d x))^{3/2}}{5 d \left (a^4+a^4 \sin (c+d x)\right )}+\frac {\left (21 e^4\right ) \int \sqrt {e \cos (c+d x)} \, dx}{5 a^4}\\ &=-\frac {4 e (e \cos (c+d x))^{7/2}}{5 a d (a+a \sin (c+d x))^3}+\frac {28 e^3 (e \cos (c+d x))^{3/2}}{5 d \left (a^4+a^4 \sin (c+d x)\right )}+\frac {\left (21 e^4 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 a^4 \sqrt {\cos (c+d x)}}\\ &=\frac {42 e^4 \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 {4 e (e \cos (c+d x))^{7/2}}{5 a d (a+a \sin (c+d x))^3}+\frac {28 e^3 (e \cos (c+d x))^{3/2}}{5 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 = 0.09, size = 66, normalized size = 0.55 \begin {gather*} -\frac {(e \cos (c+d x))^{11/2} \, _2F_1\left (\frac {9}{4},\frac {11}{4};\frac {15}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{11 \sqrt [4]{2} a^4 d e (1+\sin (c+d x))^{11/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(331\) vs.
\(2(132)=264\).
time = 6.74, size = 332, normalized size = 2.77
method | result | size |
default | \(\frac {2 \left (84 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-128 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-84 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+128 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+80 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-16 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-80 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e^{5}}{5 \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) a^{4} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(332\) |
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 = 0.10, size = 294, normalized size = 2.45 \begin {gather*} \frac {21 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} e^{\frac {9}{2}} - i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {9}{2}} + {\left (-i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {9}{2}} - 2 i \, \sqrt {2} e^{\frac {9}{2}}\right )} \sin \left (d x + c\right ) - 2 i \, \sqrt {2} e^{\frac {9}{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 ) + 21 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} e^{\frac {9}{2}} + i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {9}{2}} + {\left (i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {9}{2}} + 2 i \, \sqrt {2} e^{\frac {9}{2}}\right )} \sin \left (d x + c\right ) + 2 i \, \sqrt {2} e^{\frac {9}{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 ) - 8 \, {\left (4 \, \cos \left (d x + c\right )^{2} e^{\frac {9}{2}} + 3 \, \cos \left (d x + c\right ) e^{\frac {9}{2}} + {\left (4 \, \cos \left (d x + c\right ) e^{\frac {9}{2}} + e^{\frac {9}{2}}\right )} \sin \left (d x + c\right ) - e^{\frac {9}{2}}\right )} \sqrt {\cos \left (d x + c\right )}}{5 \, {\left (a^{4} d \cos \left (d x + c\right )^{2} - a^{4} d \cos \left (d x + c\right ) - 2 \, a^{4} d - {\left (a^{4} d \cos \left (d x + c\right ) + 2 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 )}^{9/2}}{{\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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