Optimal. Leaf size=77 \[ -\frac {2 \sqrt {2} a \text {ArcTan}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {c} f}+\frac {2 a \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {4041, 3880,
209} \begin {gather*} \frac {2 a \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}-\frac {2 \sqrt {2} a \text {ArcTan}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {c} f} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 3880
Rule 4041
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{\sqrt {c-c \sec (e+f x)}} \, dx &=\frac {2 a \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}+(2 a) \int \frac {\sec (e+f x)}{\sqrt {c-c \sec (e+f x)}} \, dx\\ &=\frac {2 a \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}-\frac {(4 a) \text {Subst}\left (\int \frac {1}{2 c+x^2} \, dx,x,\frac {c \tan (e+f x)}{\sqrt {c-c \sec (e+f x)}}\right )}{f}\\ &=-\frac {2 \sqrt {2} a \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {c} f}+\frac {2 a \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.64, size = 132, normalized size = 1.71 \begin {gather*} -\frac {i \sqrt {2} a \left (-1+e^{i (e+f x)}\right ) \left (\sqrt {2} \left (1+e^{i (e+f x)}\right )-2 \sqrt {1+e^{2 i (e+f x)}} \tanh ^{-1}\left (\frac {1+e^{i (e+f x)}}{\sqrt {2} \sqrt {1+e^{2 i (e+f x)}}}\right )\right )}{\left (1+e^{2 i (e+f x)}\right ) f \sqrt {c-c \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.18, size = 85, normalized size = 1.10
method | result | size |
default | \(-\frac {2 a \left (\arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-1\right ) \sin \left (f x +e \right )}{f \sqrt {\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \cos \left (f x +e \right )}\) | \(85\) |
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 [A]
time = 3.54, size = 294, normalized size = 3.82 \begin {gather*} \left [\frac {\sqrt {2} a c \sqrt {-\frac {1}{c}} \log \left (-\frac {2 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{c}} - {\left (3 \, \cos \left (f x + e\right ) + 1\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 2 \, {\left (a \cos \left (f x + e\right ) + a\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{c f \sin \left (f x + e\right )}, \frac {2 \, {\left (\sqrt {2} a \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {c} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) + a\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}\right )}}{c f \sin \left (f x + e\right )}\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*} a \left (\int \frac {\sec {\left (e + f x \right )}}{\sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.13, size = 61, normalized size = 0.79 \begin {gather*} \frac {2 \, a {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right )}{\sqrt {c}} + \frac {\sqrt {2}}{\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}\right )}}{f} \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+\frac {a}{\cos \left (e+f\,x\right )}}{\cos \left (e+f\,x\right )\,\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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