Optimal. Leaf size=174 \[ \frac {2 \sqrt {a} \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c^4 f}+\frac {2 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{c^4 f}-\frac {2 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 a c^4 f}+\frac {2 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{5 a^2 c^4 f}-\frac {2 \cot ^7(e+f x) (a+a \sec (e+f x))^{7/2}}{7 a^3 c^4 f} \]
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Rubi [A]
time = 0.12, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3989, 3972,
331, 209} \begin {gather*} -\frac {2 \cot ^7(e+f x) (a \sec (e+f x)+a)^{7/2}}{7 a^3 c^4 f}+\frac {2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a^2 c^4 f}+\frac {2 \sqrt {a} \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{c^4 f}-\frac {2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a c^4 f}+\frac {2 \cot (e+f x) \sqrt {a \sec (e+f x)+a}}{c^4 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 331
Rule 3972
Rule 3989
Rubi steps
\begin {align*} \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^4} \, dx &=\frac {\int \cot ^8(e+f x) (a+a \sec (e+f x))^{9/2} \, dx}{a^4 c^4}\\ &=-\frac {2 \text {Subst}\left (\int \frac {1}{x^8 \left (1+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^3 c^4 f}\\ &=-\frac {2 \cot ^7(e+f x) (a+a \sec (e+f x))^{7/2}}{7 a^3 c^4 f}+\frac {2 \text {Subst}\left (\int \frac {1}{x^6 \left (1+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^2 c^4 f}\\ &=\frac {2 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{5 a^2 c^4 f}-\frac {2 \cot ^7(e+f x) (a+a \sec (e+f x))^{7/2}}{7 a^3 c^4 f}-\frac {2 \text {Subst}\left (\int \frac {1}{x^4 \left (1+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a c^4 f}\\ &=-\frac {2 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 a c^4 f}+\frac {2 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{5 a^2 c^4 f}-\frac {2 \cot ^7(e+f x) (a+a \sec (e+f x))^{7/2}}{7 a^3 c^4 f}+\frac {2 \text {Subst}\left (\int \frac {1}{x^2 \left (1+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c^4 f}\\ &=\frac {2 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{c^4 f}-\frac {2 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 a c^4 f}+\frac {2 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{5 a^2 c^4 f}-\frac {2 \cot ^7(e+f x) (a+a \sec (e+f x))^{7/2}}{7 a^3 c^4 f}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c^4 f}\\ &=\frac {2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c^4 f}+\frac {2 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{c^4 f}-\frac {2 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 a c^4 f}+\frac {2 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{5 a^2 c^4 f}-\frac {2 \cot ^7(e+f x) (a+a \sec (e+f x))^{7/2}}{7 a^3 c^4 f}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.26, size = 78, normalized size = 0.45 \begin {gather*} -\frac {2 \sqrt {\cos (e+f x)} \, _2F_1\left (-\frac {7}{2},-\frac {7}{2};-\frac {5}{2};2 \sin ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sec (e+f x))} \tan \left (\frac {1}{2} (e+f x)\right )}{7 c^4 f (-1+\cos (e+f x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(401\) vs.
\(2(154)=308\).
time = 0.30, size = 402, normalized size = 2.31
method | result | size |
default | \(\frac {\sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \left (\cos \left (f x +e \right )+1\right )^{2} \left (-105 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}+315 \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {2}-315 \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}+352 \left (\cos ^{4}\left (f x +e \right )\right )+105 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )-812 \left (\cos ^{3}\left (f x +e \right )\right )+700 \left (\cos ^{2}\left (f x +e \right )\right )-210 \cos \left (f x +e \right )\right )}{105 c^{4} f \sin \left (f x +e \right )^{5} \left (-1+\cos \left (f x +e \right )\right )}\) | \(402\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.25, size = 517, normalized size = 2.97 \begin {gather*} \left [\frac {105 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )^{2} + 3 \, \cos \left (f x + e\right ) - 1\right )} \sqrt {-a} \log \left (-\frac {8 \, a \cos \left (f x + e\right )^{3} - 4 \, {\left (2 \, \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 7 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 4 \, {\left (176 \, \cos \left (f x + e\right )^{4} - 406 \, \cos \left (f x + e\right )^{3} + 350 \, \cos \left (f x + e\right )^{2} - 105 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{210 \, {\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \sin \left (f x + e\right )}, \frac {105 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )^{2} + 3 \, \cos \left (f x + e\right ) - 1\right )} \sqrt {a} \arctan \left (\frac {2 \, \sqrt {a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{2 \, a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right ) - a}\right ) \sin \left (f x + e\right ) + 2 \, {\left (176 \, \cos \left (f x + e\right )^{4} - 406 \, \cos \left (f x + e\right )^{3} + 350 \, \cos \left (f x + e\right )^{2} - 105 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{105 \, {\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} 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 {\sqrt {a \sec {\left (e + f x \right )} + a}}{\sec ^{4}{\left (e + f x \right )} - 4 \sec ^{3}{\left (e + f x \right )} + 6 \sec ^{2}{\left (e + f x \right )} - 4 \sec {\left (e + f x \right )} + 1}\, dx}{c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 487 vs.
\(2 (154) = 308\).
time = 1.24, size = 487, normalized size = 2.80 \begin {gather*} -\frac {\sqrt {2} {\left (\frac {210 \, \sqrt {2} \sqrt {-a} a \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right )}{c^{4} {\left | a \right |}} + \frac {1575 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{12} \sqrt {-a} a - 7140 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{10} \sqrt {-a} a^{2} + 16415 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{8} \sqrt {-a} a^{3} - 19880 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{6} \sqrt {-a} a^{4} + 14637 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{4} \sqrt {-a} a^{5} - 5684 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} \sqrt {-a} a^{6} + 1037 \, \sqrt {-a} a^{7}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - a\right )}^{7} c^{4}}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{420 \, 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 {\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}}{{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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