Optimal. Leaf size=70 \[ \frac {2 b^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{3 f}+\frac {2 b (b \sec (e+f x))^{3/2} \sin (e+f x)}{3 f} \]
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Rubi [A]
time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3853, 3856,
2720} \begin {gather*} \frac {2 b^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{3 f}+\frac {2 b \sin (e+f x) (b \sec (e+f x))^{3/2}}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int (b \sec (e+f x))^{5/2} \, dx &=\frac {2 b (b \sec (e+f x))^{3/2} \sin (e+f x)}{3 f}+\frac {1}{3} b^2 \int \sqrt {b \sec (e+f x)} \, dx\\ &=\frac {2 b (b \sec (e+f x))^{3/2} \sin (e+f x)}{3 f}+\frac {1}{3} \left (b^2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx\\ &=\frac {2 b^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{3 f}+\frac {2 b (b \sec (e+f x))^{3/2} \sin (e+f x)}{3 f}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 51, normalized size = 0.73 \begin {gather*} \frac {2 b^2 \sqrt {b \sec (e+f x)} \left (\sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\tan (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 = 0.20, size = 128, normalized size = 1.83
method | result | size |
default | \(-\frac {2 \left (-1+\cos \left (f x +e \right )\right ) \left (i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )-\cos \left (f x +e \right )+1\right ) \cos \left (f x +e \right ) \left (\cos \left (f x +e \right )+1\right )^{2} \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}}}{3 f \sin \left (f x +e \right )^{3}}\) | \(128\) |
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 = 110, normalized size = 1.57 \begin {gather*} \frac {-i \, \sqrt {2} b^{\frac {5}{2}} \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 ) + i \, \sqrt {2} b^{\frac {5}{2}} \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 \, b^{2} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \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 \left (b \sec {\left (e + f x \right )}\right )^{\frac {5}{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 {b}{\cos \left (e+f\,x\right )}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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