Optimal. Leaf size=113 \[ \frac {3 \sqrt {2} a^2 \text {ArcTan}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{c^{3/2} f}-\frac {2 a^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac {2 a^2 \tan (e+f x)}{c f \sqrt {c-c \sec (e+f x)}} \]
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Rubi [A]
time = 0.16, antiderivative size = 124, normalized size of antiderivative = 1.10, number of steps
used = 4, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4042, 4041,
3880, 209} \begin {gather*} \frac {3 \sqrt {2} a^2 \text {ArcTan}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{c^{3/2} f}-\frac {3 a^2 \tan (e+f x)}{c f \sqrt {c-c \sec (e+f x)}}-\frac {\tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{f (c-c \sec (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 3880
Rule 4041
Rule 4042
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{3/2}} \, dx &=-\frac {\left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac {(3 a) \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{\sqrt {c-c \sec (e+f x)}} \, dx}{2 c}\\ &=-\frac {\left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac {3 a^2 \tan (e+f x)}{c f \sqrt {c-c \sec (e+f x)}}-\frac {\left (3 a^2\right ) \int \frac {\sec (e+f x)}{\sqrt {c-c \sec (e+f x)}} \, dx}{c}\\ &=-\frac {\left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac {3 a^2 \tan (e+f x)}{c f \sqrt {c-c \sec (e+f x)}}+\frac {\left (6 a^2\right ) \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 )}{c f}\\ &=\frac {3 \sqrt {2} a^2 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{c^{3/2} f}-\frac {\left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac {3 a^2 \tan (e+f x)}{c f \sqrt {c-c \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.45, size = 184, normalized size = 1.63 \begin {gather*} \frac {a^2 e^{-2 i (e+f x)} \left (4 \left (1+e^{3 i (e+f x)}\right )-3 \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 ) \sec ^2(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+i \sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin \left (\frac {1}{2} (e+f x)\right )}{2 c f (-1+\sec (e+f x)) \sqrt {c-c \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.40, size = 144, normalized size = 1.27
method | result | size |
default | \(\frac {a^{2} \left (3 \cos \left (f x +e \right ) \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-3 \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-4 \cos \left (f x +e \right )+2\right ) \sin \left (f x +e \right )}{f \cos \left (f x +e \right )^{2} \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}}\) | \(144\) |
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 = 2.87, size = 400, normalized size = 3.54 \begin {gather*} \left [\frac {3 \, \sqrt {2} {\left (a^{2} c \cos \left (f x + e\right ) - a^{2} c\right )} \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 ) + 4 \, {\left (2 \, a^{2} \cos \left (f x + e\right )^{2} + a^{2} \cos \left (f x + e\right ) - a^{2}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{2 \, {\left (c^{2} f \cos \left (f x + e\right ) - c^{2} f\right )} \sin \left (f x + e\right )}, -\frac {\frac {3 \, \sqrt {2} {\left (a^{2} c \cos \left (f x + e\right ) - a^{2} c\right )} \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 )}{\sqrt {c}} - 2 \, {\left (2 \, a^{2} \cos \left (f x + e\right )^{2} + a^{2} \cos \left (f x + e\right ) - a^{2}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{{\left (c^{2} f \cos \left (f x + e\right ) - c^{2} f\right )} \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^{2} \left (\int \frac {\sec {\left (e + f x \right )}}{- c \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {2 \sec ^{2}{\left (e + f x \right )}}{- c \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{- c \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c \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 = 2.57, size = 109, normalized size = 0.96 \begin {gather*} -\frac {a^{2} {\left (\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {\sqrt {2} {\left (3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}}{{\left ({\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} + \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c\right )} 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 {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2}{\cos \left (e+f\,x\right )\,{\left (c-\frac {c}{\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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