Optimal. Leaf size=89 \[ \frac {(5 a+6 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {(5 a+6 b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac {1}{16} x (5 a+6 b)+\frac {a \sin (e+f x) \cos ^5(e+f x)}{6 f} \]
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Rubi [A] time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4045, 2635, 8} \[ \frac {(5 a+6 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {(5 a+6 b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac {1}{16} x (5 a+6 b)+\frac {a \sin (e+f x) \cos ^5(e+f x)}{6 f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 4045
Rubi steps
\begin {align*} \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=\frac {a \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{6} (5 a+6 b) \int \cos ^4(e+f x) \, dx\\ &=\frac {(5 a+6 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{8} (5 a+6 b) \int \cos ^2(e+f x) \, dx\\ &=\frac {(5 a+6 b) \cos (e+f x) \sin (e+f x)}{16 f}+\frac {(5 a+6 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{16} (5 a+6 b) \int 1 \, dx\\ &=\frac {1}{16} (5 a+6 b) x+\frac {(5 a+6 b) \cos (e+f x) \sin (e+f x)}{16 f}+\frac {(5 a+6 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a \cos ^5(e+f x) \sin (e+f x)}{6 f}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 68, normalized size = 0.76 \[ \frac {(45 a+48 b) \sin (2 (e+f x))+(9 a+6 b) \sin (4 (e+f x))+a \sin (6 (e+f x))+60 a e+60 a f x+72 b e+72 b f x}{192 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 68, normalized size = 0.76 \[ \frac {3 \, {\left (5 \, a + 6 \, b\right )} f x + {\left (8 \, a \cos \left (f x + e\right )^{5} + 2 \, {\left (5 \, a + 6 \, b\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (5 \, a + 6 \, b\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 104, normalized size = 1.17 \[ \frac {3 \, {\left (f x + e\right )} {\left (5 \, a + 6 \, b\right )} + \frac {15 \, a \tan \left (f x + e\right )^{5} + 18 \, b \tan \left (f x + e\right )^{5} + 40 \, a \tan \left (f x + e\right )^{3} + 48 \, b \tan \left (f x + e\right )^{3} + 33 \, a \tan \left (f x + e\right ) + 30 \, b \tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{3}}}{48 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.68, size = 86, normalized size = 0.97 \[ \frac {a \left (\frac {\left (\cos ^{5}\left (f x +e \right )+\frac {5 \left (\cos ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \cos \left (f x +e \right )}{8}\right ) \sin \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+b \left (\frac {\left (\cos ^{3}\left (f x +e \right )+\frac {3 \cos \left (f x +e \right )}{2}\right ) \sin \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 103, normalized size = 1.16 \[ \frac {3 \, {\left (f x + e\right )} {\left (5 \, a + 6 \, b\right )} + \frac {3 \, {\left (5 \, a + 6 \, b\right )} \tan \left (f x + e\right )^{5} + 8 \, {\left (5 \, a + 6 \, b\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (11 \, a + 10 \, b\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{6} + 3 \, \tan \left (f x + e\right )^{4} + 3 \, \tan \left (f x + e\right )^{2} + 1}}{48 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.94, size = 91, normalized size = 1.02 \[ x\,\left (\frac {5\,a}{16}+\frac {3\,b}{8}\right )+\frac {\left (\frac {5\,a}{16}+\frac {3\,b}{8}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^5+\left (\frac {5\,a}{6}+b\right )\,{\mathrm {tan}\left (e+f\,x\right )}^3+\left (\frac {11\,a}{16}+\frac {5\,b}{8}\right )\,\mathrm {tan}\left (e+f\,x\right )}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^6+3\,{\mathrm {tan}\left (e+f\,x\right )}^4+3\,{\mathrm {tan}\left (e+f\,x\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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