Optimal. Leaf size=58 \[ -\frac {2 b}{f \sqrt {d \sec (e+f x)}}+\frac {2 a E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3567, 3856,
2719} \begin {gather*} \frac {2 a 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}{f \sqrt {d \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 3567
Rule 3856
Rubi steps
\begin {align*} \int \frac {a+b \tan (e+f x)}{\sqrt {d \sec (e+f x)}} \, dx &=-\frac {2 b}{f \sqrt {d \sec (e+f x)}}+a \int \frac {1}{\sqrt {d \sec (e+f x)}} \, dx\\ &=-\frac {2 b}{f \sqrt {d \sec (e+f x)}}+\frac {a \int \sqrt {\cos (e+f x)} \, dx}{\sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\\ &=-\frac {2 b}{f \sqrt {d \sec (e+f x)}}+\frac {2 a E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 54, normalized size = 0.93 \begin {gather*} \frac {-2 b \sqrt {\cos (e+f x)}+2 a E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.61, size = 916, normalized size = 15.79
method | result | size |
risch | \(-\frac {i \left (-i b +a \right ) \sqrt {2}}{f \sqrt {\frac {d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}}-\frac {i a \left (-\frac {2 \left (d \,{\mathrm e}^{2 i \left (f x +e \right )}+d \right )}{d \sqrt {{\mathrm e}^{i \left (f x +e \right )} \left (d \,{\mathrm e}^{2 i \left (f x +e \right )}+d \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (f x +e \right )}}\, \left (-2 i \EllipticE \left (\sqrt {-i \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \EllipticF \left (\sqrt {-i \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {d \,{\mathrm e}^{3 i \left (f x +e \right )}+d \,{\mathrm e}^{i \left (f x +e \right )}}}\right ) \sqrt {2}\, \sqrt {d \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) {\mathrm e}^{i \left (f x +e \right )}}}{f \sqrt {\frac {d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\) | \(306\) |
default | \(\text {Expression too large to display}\) | \(916\) |
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 = 95, normalized size = 1.64 \begin {gather*} \frac {i \, \sqrt {2} a \sqrt {d} {\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 ) - i \, \sqrt {2} a \sqrt {d} {\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 \, b \sqrt {\frac {d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{d 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 )}}{\sqrt {d \sec {\left (e + f x \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 = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {tan}\left (e+f\,x\right )}{\sqrt {\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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