Optimal. Leaf size=164 \[ \frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 a \left (a^2-6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 164, 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 = {2770, 2941,
2748, 2721, 2720} \begin {gather*} \frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 a \left (a^2-6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 2721
Rule 2748
Rule 2770
Rule 2941
Rubi steps
\begin {align*} \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{5/2}} \, dx &=\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}}-\frac {2 \int \frac {(a+b \sin (c+d x)) \left (-\frac {a^2}{2}+2 b^2+\frac {3}{2} a b \sin (c+d x)\right )}{\sqrt {e \cos (c+d x)}} \, dx}{3 e^2}\\ &=\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}}-\frac {4 \int \frac {-\frac {3}{4} a \left (a^2-6 b^2\right )+\frac {3}{4} b \left (a^2+4 b^2\right ) \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx}{9 e^2}\\ &=\frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}}+\frac {\left (a \left (a^2-6 b^2\right )\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{3 e^2}\\ &=\frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}}+\frac {\left (a \left (a^2-6 b^2\right ) \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 e^2 \sqrt {e \cos (c+d x)}}\\ &=\frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 a \left (a^2-6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.72, size = 103, normalized size = 0.63 \begin {gather*} \frac {6 a^2 b+5 b^3+3 b^3 \cos (2 (c+d x))+2 a \left (a^2-6 b^2\right ) \cos ^{\frac {3}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 a^3 \sin (c+d x)+6 a b^2 \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}} \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. \(383\) vs.
\(2(172)=344\).
time = 8.68, size = 384, normalized size = 2.34
method | result | size |
default | \(-\frac {2 \left (2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3} \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2} \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3}+6 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}+2 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a^{2} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+4 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{2} d}\) | \(384\) |
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 = 150, normalized size = 0.91 \begin {gather*} \frac {{\left (\sqrt {2} {\left (-i \, a^{3} + 6 i \, a b^{2}\right )} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {2} {\left (i \, a^{3} - 6 i \, a b^{2}\right )} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (3 \, b^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b + b^{3} + {\left (a^{3} + 3 \, a b^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}\right )} e^{\left (-\frac {5}{2}\right )}}{3 \, d \cos \left (d x + c\right )^{2}} \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 (a+b\,\sin \left (c+d\,x\right )\right )}^3}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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