Optimal. Leaf size=88 \[ -\frac {2 a d^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}+\frac {2 a d \sqrt {d \sec (e+f x)} \sin (e+f x)}{f} \]
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Rubi [A]
time = 0.05, antiderivative size = 88, 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, 2719} \begin {gather*} -\frac {2 a d^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 a d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}+\frac {2 b (d \sec (e+f x))^{3/2}}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 3567
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x)) \, dx &=\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}+a \int (d \sec (e+f x))^{3/2} \, dx\\ &=\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}+\frac {2 a d \sqrt {d \sec (e+f x)} \sin (e+f x)}{f}-\left (a d^2\right ) \int \frac {1}{\sqrt {d \sec (e+f x)}} \, dx\\ &=\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}+\frac {2 a d \sqrt {d \sec (e+f x)} \sin (e+f x)}{f}-\frac {\left (a d^2\right ) \int \sqrt {\cos (e+f x)} \, dx}{\sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\\ &=-\frac {2 a d^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}+\frac {2 a d \sqrt {d \sec (e+f x)} \sin (e+f x)}{f}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 58, normalized size = 0.66 \begin {gather*} \frac {(d \sec (e+f x))^{3/2} \left (2 b-6 a \cos ^{\frac {3}{2}}(e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+3 a \sin (2 (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 = 0.52, size = 356, normalized size = 4.05
method | result | size |
default | \(-\frac {2 \left (\cos \left (f x +e \right )+1\right )^{2} \left (\cos \left (f x +e \right )-1\right )^{2} \left (3 i \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) a -3 i \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticE \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) a +3 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \cos \left (f x +e \right ) a \sin \left (f x +e \right )-3 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticE \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \cos \left (f x +e \right ) a \sin \left (f x +e \right )+3 \left (\cos ^{2}\left (f x +e \right )\right ) a -3 a \cos \left (f x +e \right )-b \sin \left (f x +e \right )\right ) \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}}{3 f \sin \left (f x +e \right )^{5}}\) | \(356\) |
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 = 130, normalized size = 1.48 \begin {gather*} \frac {-3 i \, \sqrt {2} a d^{\frac {3}{2}} \cos \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 i \, \sqrt {2} a d^{\frac {3}{2}} \cos \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (3 \, a d \cos \left (f x + e\right ) \sin \left (f x + e\right ) + b d\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 \left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{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 )}^{3/2}\,\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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