Optimal. Leaf size=94 \[ -\frac {2 b}{3 f (d \sec (e+f x))^{3/2}}+\frac {2 a \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{3 d^2 f}+\frac {2 a \sin (e+f x)}{3 d f \sqrt {d \sec (e+f x)}} \]
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Rubi [A]
time = 0.06, antiderivative size = 94, 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, 3854,
3856, 2720} \begin {gather*} \frac {2 a \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{3 d^2 f}+\frac {2 a \sin (e+f x)}{3 d f \sqrt {d \sec (e+f x)}}-\frac {2 b}{3 f (d \sec (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3567
Rule 3854
Rule 3856
Rubi steps
\begin {align*} \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{3/2}} \, dx &=-\frac {2 b}{3 f (d \sec (e+f x))^{3/2}}+a \int \frac {1}{(d \sec (e+f x))^{3/2}} \, dx\\ &=-\frac {2 b}{3 f (d \sec (e+f x))^{3/2}}+\frac {2 a \sin (e+f x)}{3 d f \sqrt {d \sec (e+f x)}}+\frac {a \int \sqrt {d \sec (e+f x)} \, dx}{3 d^2}\\ &=-\frac {2 b}{3 f (d \sec (e+f x))^{3/2}}+\frac {2 a \sin (e+f x)}{3 d f \sqrt {d \sec (e+f x)}}+\frac {\left (a \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{3 d^2}\\ &=-\frac {2 b}{3 f (d \sec (e+f x))^{3/2}}+\frac {2 a \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {d \sec (e+f x)}}{3 d^2 f}+\frac {2 a \sin (e+f x)}{3 d f \sqrt {d \sec (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 69, normalized size = 0.73 \begin {gather*} -\frac {\sqrt {d \sec (e+f x)} \left (b+b \cos (2 (e+f x))-2 a \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )-a \sin (2 (e+f x))\right )}{3 d^2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.90, size = 172, normalized size = 1.83
method | result | size |
default | \(\frac {\frac {2 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \cos \left (f x +e \right ) a}{3}+\frac {2 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) a}{3}-\frac {2 b \left (\cos ^{2}\left (f x +e \right )\right )}{3}+\frac {2 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a}{3}}{f \cos \left (f x +e \right )^{2} \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}}\) | \(172\) |
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 = 111, normalized size = 1.18 \begin {gather*} \frac {-i \, \sqrt {2} a \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + i \, \sqrt {2} a \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, {\left (b \cos \left (f x + e\right )^{2} - a \cos \left (f x + e\right ) \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{3 \, d^{2} f} \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 \frac {a + b \tan {\left (e + f x \right )}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{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 \frac {a+b\,\mathrm {tan}\left (e+f\,x\right )}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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