Optimal. Leaf size=70 \[ \frac {2 c^2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b \sqrt {c \cos (a+b x)}}+\frac {2 c \sqrt {c \cos (a+b x)} \sin (a+b x)}{3 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 2721,
2720} \begin {gather*} \frac {2 c^2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b \sqrt {c \cos (a+b x)}}+\frac {2 c \sin (a+b x) \sqrt {c \cos (a+b x)}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2720
Rule 2721
Rubi steps
\begin {align*} \int (c \cos (a+b x))^{3/2} \, dx &=\frac {2 c \sqrt {c \cos (a+b x)} \sin (a+b x)}{3 b}+\frac {1}{3} c^2 \int \frac {1}{\sqrt {c \cos (a+b x)}} \, dx\\ &=\frac {2 c \sqrt {c \cos (a+b x)} \sin (a+b x)}{3 b}+\frac {\left (c^2 \sqrt {\cos (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx}{3 \sqrt {c \cos (a+b x)}}\\ &=\frac {2 c^2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b \sqrt {c \cos (a+b x)}}+\frac {2 c \sqrt {c \cos (a+b x)} \sin (a+b x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 58, normalized size = 0.83 \begin {gather*} \frac {2 (c \cos (a+b x))^{3/2} \left (F\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\sqrt {\cos (a+b x)} \sin (a+b x)\right )}{3 b \cos ^{\frac {3}{2}}(a+b x)} \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. \(189\) vs.
\(2(86)=172\).
time = 0.04, size = 190, normalized size = 2.71
method | result | size |
default | \(-\frac {2 \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, c^{2} \left (4 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right )}{3 \sqrt {-c \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\) | \(190\) |
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 = 0.10, size = 77, normalized size = 1.10 \begin {gather*} \frac {-i \, \sqrt {2} c^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {2} c^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + 2 \, \sqrt {c \cos \left (b x + a\right )} c \sin \left (b x + a\right )}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \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 {\left (c\,\cos \left (a+b\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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