Optimal. Leaf size=140 \[ \frac {\tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{4 a f (a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)}}-\frac {\tanh ^{-1}(\cos (e+f x)) \tan (e+f x)}{4 a^2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \]
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Rubi [A]
time = 0.29, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4045, 4044,
3855} \begin {gather*} -\frac {\tan (e+f x) \tanh ^{-1}(\cos (e+f x))}{4 a^2 f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{4 a f (a \sec (e+f x)+a)^{3/2} \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2} \sqrt {c-c \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rule 4044
Rule 4045
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}} \, dx &=\frac {\tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}}+\frac {\int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)}} \, dx}{2 a}\\ &=\frac {\tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{4 a f (a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)}}+\frac {\int \frac {\sec (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \, dx}{4 a^2}\\ &=\frac {\tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{4 a f (a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x) \int \csc (e+f x) \, dx}{4 a^2 \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=\frac {\tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x)}{4 a f (a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)}}-\frac {\tanh ^{-1}(\cos (e+f x)) \tan (e+f x)}{4 a^2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.04, size = 91, normalized size = 0.65 \begin {gather*} -\frac {\left (2+8 \tanh ^{-1}\left (e^{i (e+f x)}\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right )+3 \cos (e+f x)\right ) \tan (e+f x)}{4 a^2 f (1+\cos (e+f x))^2 \sqrt {a (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 = 3.19, size = 164, normalized size = 1.17
method | result | size |
default | \(-\frac {\sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \left (-1+\cos \left (f x +e \right )\right )^{3} \left (4 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+5 \left (\cos ^{2}\left (f x +e \right )\right )+8 \cos \left (f x +e \right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-2 \cos \left (f x +e \right )+4 \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-3\right )}{16 f \sqrt {\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )^{5} a^{3}}\) | \(164\) |
risch | \(\frac {i \left (3 \,{\mathrm e}^{2 i \left (f x +e \right )}+4 \,{\mathrm e}^{i \left (f x +e \right )}+3\right ) \left ({\mathrm e}^{2 i \left (f x +e \right )}-{\mathrm e}^{i \left (f x +e \right )}\right )}{2 a^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}-\frac {i \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{4 a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}+\frac {i \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{4 a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(374\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1293 vs.
\(2 (132) = 264\).
time = 0.62, size = 1293, normalized size = 9.24 \begin {gather*} -\frac {{\left ({\left (2 \, {\left (4 \, \cos \left (3 \, f x + 3 \, e\right ) + 6 \, \cos \left (2 \, f x + 2 \, e\right ) + 4 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (4 \, f x + 4 \, e\right ) + \cos \left (4 \, f x + 4 \, e\right )^{2} + 8 \, {\left (6 \, \cos \left (2 \, f x + 2 \, e\right ) + 4 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (3 \, f x + 3 \, e\right ) + 16 \, \cos \left (3 \, f x + 3 \, e\right )^{2} + 12 \, {\left (4 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (2 \, f x + 2 \, e\right ) + 36 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + 16 \, \cos \left (f x + e\right )^{2} + 4 \, {\left (2 \, \sin \left (3 \, f x + 3 \, e\right ) + 3 \, \sin \left (2 \, f x + 2 \, e\right ) + 2 \, \sin \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) + \sin \left (4 \, f x + 4 \, e\right )^{2} + 16 \, {\left (3 \, \sin \left (2 \, f x + 2 \, e\right ) + 2 \, \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right ) + 16 \, \sin \left (3 \, f x + 3 \, e\right )^{2} + 36 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 48 \, \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 16 \, \sin \left (f x + e\right )^{2} + 8 \, \cos \left (f x + e\right ) + 1\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) + 1\right ) - {\left (2 \, {\left (4 \, \cos \left (3 \, f x + 3 \, e\right ) + 6 \, \cos \left (2 \, f x + 2 \, e\right ) + 4 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (4 \, f x + 4 \, e\right ) + \cos \left (4 \, f x + 4 \, e\right )^{2} + 8 \, {\left (6 \, \cos \left (2 \, f x + 2 \, e\right ) + 4 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (3 \, f x + 3 \, e\right ) + 16 \, \cos \left (3 \, f x + 3 \, e\right )^{2} + 12 \, {\left (4 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (2 \, f x + 2 \, e\right ) + 36 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + 16 \, \cos \left (f x + e\right )^{2} + 4 \, {\left (2 \, \sin \left (3 \, f x + 3 \, e\right ) + 3 \, \sin \left (2 \, f x + 2 \, e\right ) + 2 \, \sin \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) + \sin \left (4 \, f x + 4 \, e\right )^{2} + 16 \, {\left (3 \, \sin \left (2 \, f x + 2 \, e\right ) + 2 \, \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right ) + 16 \, \sin \left (3 \, f x + 3 \, e\right )^{2} + 36 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 48 \, \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 16 \, \sin \left (f x + e\right )^{2} + 8 \, \cos \left (f x + e\right ) + 1\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) - 1\right ) + 2 \, {\left (3 \, \sin \left (3 \, f x + 3 \, e\right ) + 4 \, \sin \left (2 \, f x + 2 \, e\right ) + 3 \, \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) - 2 \, {\left (3 \, \cos \left (3 \, f x + 3 \, e\right ) + 4 \, \cos \left (2 \, f x + 2 \, e\right ) + 3 \, \cos \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) + 2 \, {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) + 3\right )} \sin \left (3 \, f x + 3 \, e\right ) - 4 \, {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (2 \, f x + 2 \, e\right ) - 4 \, \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 6 \, \sin \left (f x + e\right )\right )} \sqrt {a} \sqrt {c}}{4 \, {\left (a^{3} c \cos \left (4 \, f x + 4 \, e\right )^{2} + 16 \, a^{3} c \cos \left (3 \, f x + 3 \, e\right )^{2} + 36 \, a^{3} c \cos \left (2 \, f x + 2 \, e\right )^{2} + 16 \, a^{3} c \cos \left (f x + e\right )^{2} + a^{3} c \sin \left (4 \, f x + 4 \, e\right )^{2} + 16 \, a^{3} c \sin \left (3 \, f x + 3 \, e\right )^{2} + 36 \, a^{3} c \sin \left (2 \, f x + 2 \, e\right )^{2} + 48 \, a^{3} c \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 16 \, a^{3} c \sin \left (f x + e\right )^{2} + 8 \, a^{3} c \cos \left (f x + e\right ) + a^{3} c + 2 \, {\left (4 \, a^{3} c \cos \left (3 \, f x + 3 \, e\right ) + 6 \, a^{3} c \cos \left (2 \, f x + 2 \, e\right ) + 4 \, a^{3} c \cos \left (f x + e\right ) + a^{3} c\right )} \cos \left (4 \, f x + 4 \, e\right ) + 8 \, {\left (6 \, a^{3} c \cos \left (2 \, f x + 2 \, e\right ) + 4 \, a^{3} c \cos \left (f x + e\right ) + a^{3} c\right )} \cos \left (3 \, f x + 3 \, e\right ) + 12 \, {\left (4 \, a^{3} c \cos \left (f x + e\right ) + a^{3} c\right )} \cos \left (2 \, f x + 2 \, e\right ) + 4 \, {\left (2 \, a^{3} c \sin \left (3 \, f x + 3 \, e\right ) + 3 \, a^{3} c \sin \left (2 \, f x + 2 \, e\right ) + 2 \, a^{3} c \sin \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) + 16 \, {\left (3 \, a^{3} c \sin \left (2 \, f x + 2 \, e\right ) + 2 \, a^{3} c \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right )\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.95, size = 494, normalized size = 3.53 \begin {gather*} \left [-\frac {\sqrt {-a c} {\left (\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \log \left (-\frac {4 \, {\left (2 \, \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} + {\left (a c \cos \left (f x + e\right )^{2} + a c\right )} \sin \left (f x + e\right )\right )}}{{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 2 \, {\left (3 \, \cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{8 \, {\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 {\sqrt {a c} {\left (\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{a c \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + {\left (3 \, \cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{4 \, {\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*} \int \frac {\sec {\left (e + f x \right )}}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}} \sqrt {- c \left (\sec {\left (e + f x \right )} - 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.71, size = 124, normalized size = 0.89 \begin {gather*} -\frac {c^{2} {\left (\frac {2 \, \log \left ({\left | c \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}{c} - \frac {2 \, \log \left ({\left | c \right |}\right )}{c} + \frac {{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{3} - 2 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{4}}{c^{6}}\right )}}{16 \, \sqrt {-a c} a^{2} f {\left | c \right |} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \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 )}^{5/2}\,\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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