Optimal. Leaf size=92 \[ \frac {2 a d^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{3 f}+\frac {2 b (d \sec (e+f x))^{5/2}}{5 f}+\frac {2 a d (d \sec (e+f x))^{3/2} \sin (e+f x)}{3 f} \]
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Rubi [A]
time = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3567, 3853,
3856, 2720} \begin {gather*} \frac {2 a 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)}}{3 f}+\frac {2 a d \sin (e+f x) (d \sec (e+f x))^{3/2}}{3 f}+\frac {2 b (d \sec (e+f x))^{5/2}}{5 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3567
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int (d \sec (e+f x))^{5/2} (a+b \tan (e+f x)) \, dx &=\frac {2 b (d \sec (e+f x))^{5/2}}{5 f}+a \int (d \sec (e+f x))^{5/2} \, dx\\ &=\frac {2 b (d \sec (e+f x))^{5/2}}{5 f}+\frac {2 a d (d \sec (e+f x))^{3/2} \sin (e+f x)}{3 f}+\frac {1}{3} \left (a d^2\right ) \int \sqrt {d \sec (e+f x)} \, dx\\ &=\frac {2 b (d \sec (e+f x))^{5/2}}{5 f}+\frac {2 a d (d \sec (e+f x))^{3/2} \sin (e+f x)}{3 f}+\frac {1}{3} \left (a 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 a 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)}}{3 f}+\frac {2 b (d \sec (e+f x))^{5/2}}{5 f}+\frac {2 a d (d \sec (e+f x))^{3/2} \sin (e+f x)}{3 f}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 58, normalized size = 0.63 \begin {gather*} \frac {(d \sec (e+f x))^{5/2} \left (6 b+10 a \cos ^{\frac {5}{2}}(e+f x) F\left (\left .\frac {1}{2} (e+f x)\right |2\right )+5 a \sin (2 (e+f x))\right )}{15 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.47, size = 195, normalized size = 2.12
method | result | size |
default | \(\frac {2 \left (\cos \left (f x +e \right )+1\right )^{2} \left (\cos \left (f x +e \right )-1\right )^{2} \left (5 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \left (\cos ^{3}\left (f x +e \right )\right ) a +5 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \left (\cos ^{2}\left (f x +e \right )\right ) a +5 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a +3 b \right ) \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}}}{15 f \sin \left (f x +e \right )^{4}}\) | \(195\) |
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 = 0.11, size = 133, normalized size = 1.45 \begin {gather*} \frac {-5 i \, \sqrt {2} a d^{\frac {5}{2}} \cos \left (f x + e\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 5 i \, \sqrt {2} a d^{\frac {5}{2}} \cos \left (f x + e\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (5 \, a d^{2} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, b d^{2}\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{15 \, f \cos \left (f x + e\right )^{2}} \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 )\, 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 ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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