Optimal. Leaf size=113 \[ \frac {2 a b (e \cos (c+d x))^{3/2}}{d e^3}-\frac {2 \left (a^2+2 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {\cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{d e \sqrt {e \cos (c+d x)}} \]
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Rubi [A]
time = 0.09, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2770, 2748,
2721, 2719} \begin {gather*} -\frac {2 \left (a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}}+\frac {2 a b (e \cos (c+d x))^{3/2}}{d e^3}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{d e \sqrt {e \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2721
Rule 2748
Rule 2770
Rubi steps
\begin {align*} \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{3/2}} \, dx &=\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{d e \sqrt {e \cos (c+d x)}}-\frac {2 \int \sqrt {e \cos (c+d x)} \left (\frac {a^2}{2}+b^2+\frac {3}{2} a b \sin (c+d x)\right ) \, dx}{e^2}\\ &=\frac {2 a b (e \cos (c+d x))^{3/2}}{d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (a^2+2 b^2\right ) \int \sqrt {e \cos (c+d x)} \, dx}{e^2}\\ &=\frac {2 a b (e \cos (c+d x))^{3/2}}{d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (\left (a^2+2 b^2\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{e^2 \sqrt {\cos (c+d x)}}\\ &=\frac {2 a b (e \cos (c+d x))^{3/2}}{d e^3}-\frac {2 \left (a^2+2 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {\cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{d e \sqrt {e \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 71, normalized size = 0.63 \begin {gather*} \frac {4 a b-2 \left (a^2+2 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \left (a^2+b^2\right ) \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 5.94, size = 197, normalized size = 1.74
method | result | size |
default | \(-\frac {2 \left (\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}+2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}-2 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{e \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d}\) | \(197\) |
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 = 0.12, size = 128, normalized size = 1.13 \begin {gather*} \frac {{\left (\sqrt {2} {\left (-i \, a^{2} - 2 i \, b^{2}\right )} \cos \left (d x + c\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 ) + \sqrt {2} {\left (i \, a^{2} + 2 i \, b^{2}\right )} \cos \left (d x + c\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 (2 \, a b + {\left (a^{2} + b^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}\right )} e^{\left (-\frac {3}{2}\right )}}{d \cos \left (d x + c\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 (a+b\,\sin \left (c+d\,x\right )\right )}^2}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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