Optimal. Leaf size=125 \[ \frac {2 i \cot (e+f x) F\left (i \sinh ^{-1}\left (\frac {\sqrt {5} \sqrt {4-5 \sec (e+f x)}}{\sqrt {2+3 \sec (e+f x)}}\right )|\frac {1}{45}\right ) \sqrt {\frac {1-\sec (e+f x)}{2+3 \sec (e+f x)}} \sqrt {\frac {1+\sec (e+f x)}{2+3 \sec (e+f x)}} (2+3 \sec (e+f x))}{3 \sqrt {5} f} \]
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Rubi [A]
time = 0.08, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {4069}
\begin {gather*} \frac {2 i \cot (e+f x) \sqrt {\frac {1-\sec (e+f x)}{3 \sec (e+f x)+2}} \sqrt {\frac {\sec (e+f x)+1}{3 \sec (e+f x)+2}} (3 \sec (e+f x)+2) F\left (i \sinh ^{-1}\left (\frac {\sqrt {5} \sqrt {4-5 \sec (e+f x)}}{\sqrt {3 \sec (e+f x)+2}}\right )|\frac {1}{45}\right )}{3 \sqrt {5} f} \end {gather*}
Antiderivative was successfully verified.
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Rule 4069
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{\sqrt {4-5 \sec (e+f x)} \sqrt {2+3 \sec (e+f x)}} \, dx &=\frac {2 i \cot (e+f x) F\left (i \sinh ^{-1}\left (\frac {\sqrt {5} \sqrt {4-5 \sec (e+f x)}}{\sqrt {2+3 \sec (e+f x)}}\right )|\frac {1}{45}\right ) \sqrt {\frac {1-\sec (e+f x)}{2+3 \sec (e+f x)}} \sqrt {\frac {1+\sec (e+f x)}{2+3 \sec (e+f x)}} (2+3 \sec (e+f x))}{3 \sqrt {5} f}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 176, normalized size = 1.41 \begin {gather*} -\frac {4 \sqrt {-\cot ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {-\left ((3+2 \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )\right )} \sqrt {-\left ((-5+4 \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )\right )} \csc (e+f x) F\left (\text {ArcSin}\left (\sqrt {\frac {5}{22}} \sqrt {\frac {-5+4 \cos (e+f x)}{-1+\cos (e+f x)}}\right )|\frac {44}{45}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{3 \sqrt {5} f \sqrt {4-5 \sec (e+f x)} \sqrt {2+3 \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 11.07, size = 170, normalized size = 1.36
method | result | size |
default | \(-\frac {i \sqrt {-\frac {2 \left (4 \cos \left (f x +e \right )-5\right )}{\cos \left (f x +e \right )+1}}\, \sqrt {10}\, \sqrt {\frac {2 \cos \left (f x +e \right )+3}{\cos \left (f x +e \right )+1}}\, \left (\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {\frac {4 \cos \left (f x +e \right )-5}{\cos \left (f x +e \right )}}\, \sqrt {\frac {2 \cos \left (f x +e \right )+3}{\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {3 i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \frac {\sqrt {5}}{15}\right )}{15 f \left (8 \left (\cos ^{3}\left (f x +e \right )\right )-6 \left (\cos ^{2}\left (f x +e \right )\right )-17 \cos \left (f x +e \right )+15\right )}\) | \(170\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec {\left (e + f x \right )}}{\sqrt {4 - 5 \sec {\left (e + f x \right )}} \sqrt {3 \sec {\left (e + f x \right )} + 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 \frac {1}{\cos \left (e+f\,x\right )\,\sqrt {\frac {3}{\cos \left (e+f\,x\right )}+2}\,\sqrt {4-\frac {5}{\cos \left (e+f\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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