Optimal. Leaf size=43 \[ \frac {a (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}} \]
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Rubi [A]
time = 0.09, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {4038}
\begin {gather*} \frac {a \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt {a \sec (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 4038
Rubi steps
\begin {align*} \int \sec (e+f x) \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2} \, dx &=\frac {a (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(43)=86\).
time = 0.47, size = 87, normalized size = 2.02 \begin {gather*} \frac {c^2 (5-6 \cos (e+f x)+3 \cos (2 (e+f x))) \csc \left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \sec ^2(e+f x) \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}}{12 f} \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. \(81\) vs.
\(2(37)=74\).
time = 2.52, size = 82, normalized size = 1.91
method | result | size |
default | \(-\frac {\sin \left (f x +e \right ) \left (7 \left (\cos ^{2}\left (f x +e \right )\right )-4 \cos \left (f x +e \right )+1\right ) \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}}{3 f \left (-1+\cos \left (f x +e \right )\right )^{3}}\) | \(82\) |
risch | \(\frac {2 i c^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (3 \,{\mathrm e}^{5 i \left (f x +e \right )}-6 \,{\mathrm e}^{4 i \left (f x +e \right )}+10 \,{\mathrm e}^{3 i \left (f x +e \right )}-6 \,{\mathrm e}^{2 i \left (f x +e \right )}+3 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} f}\) | \(165\) |
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. 688 vs.
\(2 (40) = 80\).
time = 0.56, size = 688, normalized size = 16.00 \begin {gather*} \frac {2 \, {\left (30 \, c^{2} \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 9 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) - 3 \, c^{2} \sin \left (f x + e\right ) - {\left (3 \, c^{2} \sin \left (5 \, f x + 5 \, e\right ) - 6 \, c^{2} \sin \left (4 \, f x + 4 \, e\right ) + 10 \, c^{2} \sin \left (3 \, f x + 3 \, e\right ) - 6 \, c^{2} \sin \left (2 \, f x + 2 \, e\right ) + 3 \, c^{2} \sin \left (f x + e\right )\right )} \cos \left (6 \, f x + 6 \, e\right ) + 9 \, {\left (c^{2} \sin \left (4 \, f x + 4 \, e\right ) + c^{2} \sin \left (2 \, f x + 2 \, e\right )\right )} \cos \left (5 \, f x + 5 \, e\right ) - 3 \, {\left (10 \, c^{2} \sin \left (3 \, f x + 3 \, e\right ) + 3 \, c^{2} \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) + {\left (3 \, c^{2} \cos \left (5 \, f x + 5 \, e\right ) - 6 \, c^{2} \cos \left (4 \, f x + 4 \, e\right ) + 10 \, c^{2} \cos \left (3 \, f x + 3 \, e\right ) - 6 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, c^{2} \cos \left (f x + e\right )\right )} \sin \left (6 \, f x + 6 \, e\right ) - 3 \, {\left (3 \, c^{2} \cos \left (4 \, f x + 4 \, e\right ) + 3 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) + c^{2}\right )} \sin \left (5 \, f x + 5 \, e\right ) + 3 \, {\left (10 \, c^{2} \cos \left (3 \, f x + 3 \, e\right ) + 3 \, c^{2} \cos \left (f x + e\right ) + 2 \, c^{2}\right )} \sin \left (4 \, f x + 4 \, e\right ) - 10 \, {\left (3 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) + c^{2}\right )} \sin \left (3 \, f x + 3 \, e\right ) + 3 \, {\left (3 \, c^{2} \cos \left (f x + e\right ) + 2 \, c^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )\right )} \sqrt {a} \sqrt {c}}{3 \, {\left (2 \, {\left (3 \, \cos \left (4 \, f x + 4 \, e\right ) + 3 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (6 \, f x + 6 \, e\right ) + \cos \left (6 \, f x + 6 \, e\right )^{2} + 6 \, {\left (3 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (4 \, f x + 4 \, e\right ) + 9 \, \cos \left (4 \, f x + 4 \, e\right )^{2} + 9 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + 6 \, {\left (\sin \left (4 \, f x + 4 \, e\right ) + \sin \left (2 \, f x + 2 \, e\right )\right )} \sin \left (6 \, f x + 6 \, e\right ) + \sin \left (6 \, f x + 6 \, e\right )^{2} + 9 \, \sin \left (4 \, f x + 4 \, e\right )^{2} + 18 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 9 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 6 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} f} \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. 101 vs.
\(2 (40) = 80\).
time = 2.68, size = 101, normalized size = 2.35 \begin {gather*} \frac {{\left (3 \, c^{2} \cos \left (f x + e\right )^{2} - 3 \, c^{2} \cos \left (f x + e\right ) + c^{2}\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 )}}}{3 \, f \cos \left (f x + e\right )^{2} \sin \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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. 104 vs.
\(2 (37) = 74\).
time = 1.73, size = 104, normalized size = 2.42 \begin {gather*} \frac {8 \, {\left (3 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{2} + 3 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{3} + c^{4}\right )} \sqrt {-a c} {\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 )}{3 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{3} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.80, size = 136, normalized size = 3.16 \begin {gather*} \frac {2\,c^2\,\sqrt {\frac {a\,\left (\cos \left (e+f\,x\right )+1\right )}{\cos \left (e+f\,x\right )}}\,\sqrt {\frac {c\,\left (\cos \left (e+f\,x\right )-1\right )}{\cos \left (e+f\,x\right )}}\,\left (10\,\sin \left (e+f\,x\right )-12\,\sin \left (2\,e+2\,f\,x\right )+13\,\sin \left (3\,e+3\,f\,x\right )-6\,\sin \left (4\,e+4\,f\,x\right )+3\,\sin \left (5\,e+5\,f\,x\right )\right )}{3\,f\,\left (\cos \left (2\,e+2\,f\,x\right )-2\,\cos \left (4\,e+4\,f\,x\right )-\cos \left (6\,e+6\,f\,x\right )+2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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