Optimal. Leaf size=42 \[ \frac {2 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b}+\frac {2 \sin (a+b x)}{3 b \cos ^{\frac {3}{2}}(a+b x)} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2636, 2641} \[ \frac {2 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b}+\frac {2 \sin (a+b x)}{3 b \cos ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{\frac {5}{2}}(a+b x)} \, dx &=\frac {2 \sin (a+b x)}{3 b \cos ^{\frac {3}{2}}(a+b x)}+\frac {1}{3} \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx\\ &=\frac {2 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b}+\frac {2 \sin (a+b x)}{3 b \cos ^{\frac {3}{2}}(a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 36, normalized size = 0.86 \[ \frac {2 \left (F\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\frac {\sin (a+b x)}{\cos ^{\frac {3}{2}}(a+b x)}\right )}{3 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\cos \left (b x + a\right )^{\frac {5}{2}}}, x\right ) \]
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 \[ \int \frac {1}{\cos \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 213, normalized size = 5.07 \[ -\frac {2 \left (-2 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}}{3 \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )^{\frac {3}{2}} \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) b} \]
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 \[ \int \frac {1}{\cos \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 42, normalized size = 1.00 \[ \frac {2\,\sin \left (a+b\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (a+b\,x\right )}^2\right )}{3\,b\,{\cos \left (a+b\,x\right )}^{3/2}\,\sqrt {{\sin \left (a+b\,x\right )}^2}} \]
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 \[ \int \frac {1}{\cos ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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