Optimal. Leaf size=181 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{4 \sqrt {2} a^3 \sqrt {c} f}+\frac {\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{6 a f (a+a \sec (e+f x))^2 \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{4 f \left (a^3+a^3 \sec (e+f x)\right ) \sqrt {c-c \sec (e+f x)}} \]
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Rubi [A]
time = 0.28, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {4045, 3880,
209} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{4 \sqrt {2} a^3 \sqrt {c} f}+\frac {\tan (e+f x)}{4 f \left (a^3 \sec (e+f x)+a^3\right ) \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{6 a f (a \sec (e+f x)+a)^2 \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 \sqrt {c-c \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 3880
Rule 4045
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)}} \, dx &=\frac {\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)}}+\frac {\int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 \sqrt {c-c \sec (e+f x)}} \, dx}{2 a}\\ &=\frac {\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{6 a f (a+a \sec (e+f x))^2 \sqrt {c-c \sec (e+f x)}}+\frac {\int \frac {\sec (e+f x)}{(a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)}} \, dx}{4 a^2}\\ &=\frac {\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{6 a f (a+a \sec (e+f x))^2 \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{4 f \left (a^3+a^3 \sec (e+f x)\right ) \sqrt {c-c \sec (e+f x)}}+\frac {\int \frac {\sec (e+f x)}{\sqrt {c-c \sec (e+f x)}} \, dx}{8 a^3}\\ &=\frac {\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{6 a f (a+a \sec (e+f x))^2 \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{4 f \left (a^3+a^3 \sec (e+f x)\right ) \sqrt {c-c \sec (e+f x)}}-\frac {\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 )}{4 a^3 f}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{4 \sqrt {2} a^3 \sqrt {c} f}+\frac {\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{6 a f (a+a \sec (e+f x))^2 \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{4 f \left (a^3+a^3 \sec (e+f x)\right ) \sqrt {c-c \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.61, size = 225, normalized size = 1.24 \begin {gather*} \frac {2 e^{-\frac {1}{2} i (e+f x)} \cos \left (\frac {1}{2} (e+f x)\right ) \left (-15 \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \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 ) \cos ^5\left (\frac {1}{2} (e+f x)\right )+\frac {e^{\frac {1}{2} i (e+f x)} (67+80 \cos (e+f x)+37 \cos (2 (e+f x)))}{8 \sqrt {\sec (e+f x)}}\right ) \sec ^{\frac {7}{2}}(e+f x) \sin \left (\frac {1}{2} (e+f x)\right )}{15 a^3 f (1+\sec (e+f x))^3 \sqrt {c-c \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.48, size = 155, normalized size = 0.86
method | result | size |
default | \(-\frac {\left (-1+\cos \left (f x +e \right )\right ) \left (3 \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {5}{2}}-5 \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {3}{2}}+15 \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right )+15 \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\right )}{60 a^{3} f \sqrt {\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\) | \(155\) |
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 = 3.15, size = 435, normalized size = 2.40 \begin {gather*} \left [-\frac {15 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \sqrt {-c} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {-c} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} + {\left (3 \, c \cos \left (f x + e\right ) + c\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 (37 \, \cos \left (f x + e\right )^{3} + 40 \, \cos \left (f x + e\right )^{2} + 15 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{240 \, {\left (a^{3} c f \cos \left (f x + e\right )^{2} + 2 \, a^{3} c f \cos \left (f x + e\right ) + a^{3} c f\right )} \sin \left (f x + e\right )}, \frac {15 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \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 ) - 2 \, {\left (37 \, \cos \left (f x + e\right )^{3} + 40 \, \cos \left (f x + e\right )^{2} + 15 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{120 \, {\left (a^{3} c f \cos \left (f x + e\right )^{2} + 2 \, a^{3} c f \cos \left (f x + e\right ) + a^{3} c 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*} \frac {\int \frac {\sec {\left (e + f x \right )}}{\sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{3}{\left (e + f x \right )} + 3 \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} + 3 \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 119, normalized size = 0.66 \begin {gather*} \frac {\sqrt {2} {\left (\frac {15 \, \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 {3 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {5}{2}} c^{12} - 5 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} c^{13} + 15 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c^{14}}{c^{15}}\right )}}{120 \, a^{3} 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 {1}{\cos \left (e+f\,x\right )\,{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^3\,\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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