Optimal. Leaf size=140 \[ \frac {2 a^{5/2} \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c^4 f}+\frac {2 a^2 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{c^4 f}-\frac {2 a \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 c^4 f}-\frac {8 \cot ^7(e+f x) (a+a \sec (e+f x))^{7/2}}{7 a c^4 f} \]
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Rubi [A]
time = 0.12, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3989, 3972,
472, 209} \begin {gather*} \frac {2 a^{5/2} \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{c^4 f}+\frac {2 a^2 \cot (e+f x) \sqrt {a \sec (e+f x)+a}}{c^4 f}-\frac {8 \cot ^7(e+f x) (a \sec (e+f x)+a)^{7/2}}{7 a c^4 f}-\frac {2 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^4 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 472
Rule 3972
Rule 3989
Rubi steps
\begin {align*} \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^4} \, dx &=\frac {\int \cot ^8(e+f x) (a+a \sec (e+f x))^{13/2} \, dx}{a^4 c^4}\\ &=-\frac {2 \text {Subst}\left (\int \frac {\left (2+a x^2\right )^2}{x^8 \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 \text {Subst}\left (\int \left (\frac {4}{x^8}+\frac {a^2}{x^4}-\frac {a^3}{x^2}+\frac {a^4}{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 a^2 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{c^4 f}-\frac {2 a \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 c^4 f}-\frac {8 \cot ^7(e+f x) (a+a \sec (e+f x))^{7/2}}{7 a c^4 f}-\frac {\left (2 a^3\right ) \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 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c^4 f}+\frac {2 a^2 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{c^4 f}-\frac {2 a \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 c^4 f}-\frac {8 \cot ^7(e+f x) (a+a \sec (e+f x))^{7/2}}{7 a 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 = 8.04, size = 361, normalized size = 2.58 \begin {gather*} -\frac {\csc ^7\left (\frac {1}{2} (e+f x)\right ) \sec ^5\left (\frac {1}{2} (e+f x)\right ) \sec ^{\frac {3}{2}}(e+f x) (a (1+\sec (e+f x)))^{5/2} \sin ^8\left (\frac {e}{2}+\frac {f x}{2}\right ) \sqrt {\frac {1}{1-2 \sin ^2\left (\frac {1}{2} (e+f x)\right )}} \sqrt {1-2 \sin ^2\left (\frac {1}{2} (e+f x)\right )} \left (336 \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};2 \sin ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sin ^2\left (\frac {1}{2} (e+f x)\right ) \left (3-8 \sin ^2\left (\frac {1}{2} (e+f x)\right )+5 \sin ^4\left (\frac {1}{2} (e+f x)\right )\right )+4 \, _2F_1\left (-\frac {7}{2},-\frac {3}{2};-\frac {1}{2};2 \sin ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (15-42 \sin ^2\left (\frac {1}{2} (e+f x)\right )+35 \sin ^4\left (\frac {1}{2} (e+f x)\right )\right )-105 \cos ^4\left (\frac {1}{2} (e+f x)\right ) \left (3 \sqrt {2} \text {ArcSin}\left (\sqrt {2} \sqrt {\sin ^2\left (\frac {1}{2} (e+f x)\right )}\right ) \sin ^2\left (\frac {1}{2} (e+f x)\right )^{3/2}+2 \sin ^4\left (\frac {1}{2} (e+f x)\right ) \left (5-4 \sin ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {1-2 \sin ^2\left (\frac {1}{2} (e+f x)\right )}\right )\right )}{210 f (c-c \sec (e+f x))^4} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(394\) vs.
\(2(124)=248\).
time = 0.25, size = 395, normalized size = 2.82
method | result | size |
default | \(-\frac {\sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \left (21 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}-63 \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {2}+63 \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}-21 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )-80 \left (\cos ^{4}\left (f x +e \right )\right )+154 \left (\cos ^{3}\left (f x +e \right )\right )-140 \left (\cos ^{2}\left (f x +e \right )\right )+42 \cos \left (f x +e \right )\right ) a^{2}}{21 c^{4} f \sin \left (f x +e \right ) \left (-1+\cos \left (f x +e \right )\right )^{3}}\) | \(395\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 270 vs.
\(2 (132) = 264\).
time = 3.19, size = 569, normalized size = 4.06 \begin {gather*} \left [\frac {21 \, {\left (a^{2} \cos \left (f x + e\right )^{3} - 3 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, a^{2} \cos \left (f x + e\right ) - a^{2}\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 (40 \, a^{2} \cos \left (f x + e\right )^{4} - 77 \, a^{2} \cos \left (f x + e\right )^{3} + 70 \, a^{2} \cos \left (f x + e\right )^{2} - 21 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{42 \, {\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 {21 \, {\left (a^{2} \cos \left (f x + e\right )^{3} - 3 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, a^{2} \cos \left (f x + e\right ) - a^{2}\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 (40 \, a^{2} \cos \left (f x + e\right )^{4} - 77 \, a^{2} \cos \left (f x + e\right )^{3} + 70 \, a^{2} \cos \left (f x + e\right )^{2} - 21 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{21 \, {\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(-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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 556 vs.
\(2 (124) = 248\).
time = 1.60, size = 556, normalized size = 3.97 \begin {gather*} -\frac {\frac {21 \, \sqrt {-a} a^{3} \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 ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{c^{4} {\left | a \right |}} + \frac {4 \, {\left (21 \, \sqrt {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 )}^{12} \sqrt {-a} a^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 84 \, \sqrt {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 )}^{10} \sqrt {-a} a^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 217 \, \sqrt {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 )}^{8} \sqrt {-a} a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 238 \, \sqrt {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 )}^{6} \sqrt {-a} a^{6} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 189 \, \sqrt {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 )}^{4} \sqrt {-a} a^{7} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 70 \, \sqrt {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} \sqrt {-a} a^{8} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 13 \, \sqrt {2} \sqrt {-a} a^{9} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )}}{{\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}}}{21 \, 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 )}^{5/2}}{{\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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