Optimal. Leaf size=70 \[ \frac {6 c^2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \cos (a+b x)}}{5 b \sqrt {\cos (a+b x)}}+\frac {2 c \sin (a+b x) (c \cos (a+b x))^{3/2}}{5 b} \]
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Rubi [A] time = 0.04, 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 = {2635, 2640, 2639} \[ \frac {6 c^2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \cos (a+b x)}}{5 b \sqrt {\cos (a+b x)}}+\frac {2 c \sin (a+b x) (c \cos (a+b x))^{3/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2640
Rubi steps
\begin {align*} \int (c \cos (a+b x))^{5/2} \, dx &=\frac {2 c (c \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}+\frac {1}{5} \left (3 c^2\right ) \int \sqrt {c \cos (a+b x)} \, dx\\ &=\frac {2 c (c \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}+\frac {\left (3 c^2 \sqrt {c \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{5 \sqrt {\cos (a+b x)}}\\ &=\frac {6 c^2 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b \sqrt {\cos (a+b x)}}+\frac {2 c (c \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 62, normalized size = 0.89 \[ \frac {(c \cos (a+b x))^{5/2} \left (6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\sin (2 (a+b x)) \sqrt {\cos (a+b x)}\right )}{5 b \cos ^{\frac {5}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c \cos \left (b x + a\right )} c^{2} \cos \left (b x + a\right )^{2}, x\right ) \]
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 \[ \int \left (c \cos \left (b x + a\right )\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 213, normalized size = 3.04 \[ -\frac {2 \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, c^{3} \left (-8 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+8 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 \sqrt {-c \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b} \]
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 \[ \int \left (c \cos \left (b x + a\right )\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (c\,\cos \left (a+b\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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