Optimal. Leaf size=143 \[ \frac {2 \left (7 a^2-2 b^2\right ) d^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{21 f}+\frac {18 a b (d \sec (e+f x))^{5/2}}{35 f}+\frac {2 \left (7 a^2-2 b^2\right ) d (d \sec (e+f x))^{3/2} \sin (e+f x)}{21 f}+\frac {2 b (d \sec (e+f x))^{5/2} (a+b \tan (e+f x))}{7 f} \]
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Rubi [A]
time = 0.11, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3589, 3567,
3853, 3856, 2720} \begin {gather*} \frac {2 d^2 \left (7 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)}}{21 f}+\frac {2 d \left (7 a^2-2 b^2\right ) \sin (e+f x) (d \sec (e+f x))^{3/2}}{21 f}+\frac {18 a b (d \sec (e+f x))^{5/2}}{35 f}+\frac {2 b (d \sec (e+f x))^{5/2} (a+b \tan (e+f x))}{7 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3567
Rule 3589
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int (d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^2 \, dx &=\frac {2 b (d \sec (e+f x))^{5/2} (a+b \tan (e+f x))}{7 f}+\frac {2}{7} \int (d \sec (e+f x))^{5/2} \left (\frac {7 a^2}{2}-b^2+\frac {9}{2} a b \tan (e+f x)\right ) \, dx\\ &=\frac {18 a b (d \sec (e+f x))^{5/2}}{35 f}+\frac {2 b (d \sec (e+f x))^{5/2} (a+b \tan (e+f x))}{7 f}+\frac {1}{7} \left (7 a^2-2 b^2\right ) \int (d \sec (e+f x))^{5/2} \, dx\\ &=\frac {18 a b (d \sec (e+f x))^{5/2}}{35 f}+\frac {2 \left (7 a^2-2 b^2\right ) d (d \sec (e+f x))^{3/2} \sin (e+f x)}{21 f}+\frac {2 b (d \sec (e+f x))^{5/2} (a+b \tan (e+f x))}{7 f}+\frac {1}{21} \left (\left (7 a^2-2 b^2\right ) d^2\right ) \int \sqrt {d \sec (e+f x)} \, dx\\ &=\frac {18 a b (d \sec (e+f x))^{5/2}}{35 f}+\frac {2 \left (7 a^2-2 b^2\right ) d (d \sec (e+f x))^{3/2} \sin (e+f x)}{21 f}+\frac {2 b (d \sec (e+f x))^{5/2} (a+b \tan (e+f x))}{7 f}+\frac {1}{21} \left (\left (7 a^2-2 b^2\right ) d^2 \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx\\ &=\frac {2 \left (7 a^2-2 b^2\right ) d^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{21 f}+\frac {18 a b (d \sec (e+f x))^{5/2}}{35 f}+\frac {2 \left (7 a^2-2 b^2\right ) d (d \sec (e+f x))^{3/2} \sin (e+f x)}{21 f}+\frac {2 b (d \sec (e+f x))^{5/2} (a+b \tan (e+f x))}{7 f}\\ \end {align*}
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Mathematica [A]
time = 0.86, size = 127, normalized size = 0.89 \begin {gather*} \frac {2 d^2 \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))^2 \left (5 \left (7 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 )+\frac {5}{2} \left (7 a^2-2 b^2\right ) \sin (2 (e+f x))+3 b (14 a+5 b \tan (e+f x))\right )}{105 f (a \cos (e+f x)+b \sin (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.53, size = 382, normalized size = 2.67
method | result | size |
default | \(\frac {2 \left (\cos \left (f x +e \right )+1\right )^{2} \left (\cos \left (f x +e \right )-1\right )^{2} \left (35 i \left (\cos ^{4}\left (f x +e \right )\right ) \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a^{2}-10 i \left (\cos ^{4}\left (f x +e \right )\right ) \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, b^{2}+35 i \left (\cos ^{3}\left (f x +e \right )\right ) \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a^{2}-10 i \left (\cos ^{3}\left (f x +e \right )\right ) \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, b^{2}+35 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) a^{2}-10 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) b^{2}+42 \cos \left (f x +e \right ) a b +15 \sin \left (f x +e \right ) b^{2}\right ) \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}}}{105 f \sin \left (f x +e \right )^{4} \cos \left (f x +e \right )}\) | \(382\) |
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 = 0.12, size = 183, normalized size = 1.28 \begin {gather*} \frac {-5 i \, \sqrt {2} {\left (7 \, a^{2} - 2 \, b^{2}\right )} d^{\frac {5}{2}} \cos \left (f x + e\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 5 i \, \sqrt {2} {\left (7 \, a^{2} - 2 \, b^{2}\right )} d^{\frac {5}{2}} \cos \left (f x + e\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (42 \, a b d^{2} \cos \left (f x + e\right ) + 5 \, {\left ({\left (7 \, a^{2} - 2 \, b^{2}\right )} d^{2} \cos \left (f x + e\right )^{2} + 3 \, b^{2} d^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{105 \, f \cos \left (f x + e\right )^{3}} \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 \left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{2}} \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 {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/2}\,{\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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