Optimal. Leaf size=72 \[ \frac {2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b c^2 \sqrt {c \cos (a+b x)}}+\frac {2 \sin (a+b x)}{3 b c (c \cos (a+b x))^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 72, 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 = {2636, 2642, 2641} \[ \frac {2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b c^2 \sqrt {c \cos (a+b x)}}+\frac {2 \sin (a+b x)}{3 b c (c \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rule 2642
Rubi steps
\begin {align*} \int \frac {1}{(c \cos (a+b x))^{5/2}} \, dx &=\frac {2 \sin (a+b x)}{3 b c (c \cos (a+b x))^{3/2}}+\frac {\int \frac {1}{\sqrt {c \cos (a+b x)}} \, dx}{3 c^2}\\ &=\frac {2 \sin (a+b x)}{3 b c (c \cos (a+b x))^{3/2}}+\frac {\sqrt {\cos (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx}{3 c^2 \sqrt {c \cos (a+b x)}}\\ &=\frac {2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b c^2 \sqrt {c \cos (a+b x)}}+\frac {2 \sin (a+b x)}{3 b c (c \cos (a+b x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 51, normalized size = 0.71 \[ \frac {2 \left (\tan (a+b x)+\sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )\right )}{3 b c^2 \sqrt {c \cos (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c \cos \left (b x + a\right )}}{c^{3} \cos \left (b x + a\right )^{3}}, x\right ) \]
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 \[ \int \frac {1}{\left (c \cos \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 241, normalized size = 3.35 \[ -\frac {2 \left (-2 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}}{3 c^{2} \sqrt {-c \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b} \]
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 \[ \int \frac {1}{\left (c \cos \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c\,\cos \left (a+b\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \cos {\left (a + b x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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