Optimal. Leaf size=38 \[ \frac {2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \sec (a+b x)}}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3771, 2641} \[ \frac {2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \sec (a+b x)}}{b} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3771
Rubi steps
\begin {align*} \int \sqrt {c \sec (a+b x)} \, dx &=\left (\sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx\\ &=\frac {2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \sec (a+b x)}}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 1.00 \[ \frac {2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \sec (a+b x)}}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c \sec \left (b x + a\right )}, x\right ) \]
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 \[ \int \sqrt {c \sec \left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.92, size = 98, normalized size = 2.58 \[ -\frac {2 i \sqrt {\frac {c}{\cos \left (b x +a \right )}}\, \left (-1+\cos \left (b x +a \right )\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \left (\cos \left (b x +a \right )+1\right )^{2}}{b \sin \left (b x +a \right )^{2}} \]
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 \[ \int \sqrt {c \sec \left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 35, normalized size = 0.92 \[ \frac {2\,\sqrt {\cos \left (a+b\,x\right )}\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\mathrm {F}\left (\frac {a}{2}+\frac {b\,x}{2}\middle |2\right )}{b} \]
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 \[ \int \sqrt {c \sec {\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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