Optimal. Leaf size=103 \[ \frac {10 a b \sqrt {d \sec (e+f x)}}{3 f}+\frac {2 \left (3 a^2-2 b^2\right ) \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{3 f}+\frac {2 b \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}{3 f} \]
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Rubi [A]
time = 0.09, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3589, 3567,
3856, 2720} \begin {gather*} \frac {2 \left (3 a^2-2 b^2\right ) \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{3 f}+\frac {10 a b \sqrt {d \sec (e+f x)}}{3 f}+\frac {2 b \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3567
Rule 3589
Rule 3856
Rubi steps
\begin {align*} \int \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))^2 \, dx &=\frac {2 b \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}{3 f}+\frac {2}{3} \int \sqrt {d \sec (e+f x)} \left (\frac {3 a^2}{2}-b^2+\frac {5}{2} a b \tan (e+f x)\right ) \, dx\\ &=\frac {10 a b \sqrt {d \sec (e+f x)}}{3 f}+\frac {2 b \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}{3 f}+\frac {1}{3} \left (3 a^2-2 b^2\right ) \int \sqrt {d \sec (e+f x)} \, dx\\ &=\frac {10 a b \sqrt {d \sec (e+f x)}}{3 f}+\frac {2 b \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}{3 f}+\frac {1}{3} \left (\left (3 a^2-2 b^2\right ) \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx\\ &=\frac {10 a b \sqrt {d \sec (e+f x)}}{3 f}+\frac {2 \left (3 a^2-2 b^2\right ) \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{3 f}+\frac {2 b \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}{3 f}\\ \end {align*}
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Mathematica [A]
time = 0.70, size = 87, normalized size = 0.84 \begin {gather*} \frac {2 \sec ^2(e+f x) \sqrt {d \sec (e+f x)} \left (\left (3 a^2-2 b^2\right ) \cos ^{\frac {5}{2}}(e+f x) F\left (\left .\frac {1}{2} (e+f x)\right |2\right )+b \cos (e+f x) (6 a \cos (e+f x)+b \sin (e+f x))\right )}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.75, size = 339, normalized size = 3.29
method | result | size |
default | \(\frac {2 \sqrt {\frac {d}{\cos \left (f x +e \right )}}\, \left (\cos \left (f x +e \right )-1\right )^{2} \left (3 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \left (\cos ^{2}\left (f x +e \right )\right ) a^{2}-2 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \left (\cos ^{2}\left (f x +e \right )\right ) b^{2}+3 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \cos \left (f x +e \right ) a^{2}-2 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \cos \left (f x +e \right ) b^{2}+6 \cos \left (f x +e \right ) a b +\sin \left (f x +e \right ) b^{2}\right ) \left (\cos \left (f x +e \right )+1\right )^{2}}{3 f \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}}\) | \(339\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.13, size = 143, normalized size = 1.39 \begin {gather*} \frac {\sqrt {2} {\left (-3 i \, a^{2} + 2 i \, b^{2}\right )} \sqrt {d} \cos \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + \sqrt {2} {\left (3 i \, a^{2} - 2 i \, b^{2}\right )} \sqrt {d} \cos \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (6 \, a b \cos \left (f x + e\right ) + b^{2} \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{3 \, f \cos \left (f x + e\right )} \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 \sqrt {d \sec {\left (e + f x \right )}} \left (a + b \tan {\left (e + f x \right )}\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 \sqrt {\frac {d}{\cos \left (e+f\,x\right )}}\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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