Optimal. Leaf size=119 \[ \frac {\left (5 a^2+12 a b+8 b^2\right ) \sin (e+f x) \cos (e+f x)}{16 f}+\frac {1}{16} x \left (5 a^2+12 a b+8 b^2\right )+\frac {a (5 a+8 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {a \sin (e+f x) \cos ^5(e+f x) \left (a+b \tan ^2(e+f x)+b\right )}{6 f} \]
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Rubi [A] time = 0.15, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4146, 413, 385, 199, 203} \[ \frac {\left (5 a^2+12 a b+8 b^2\right ) \sin (e+f x) \cos (e+f x)}{16 f}+\frac {1}{16} x \left (5 a^2+12 a b+8 b^2\right )+\frac {a (5 a+8 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {a \sin (e+f x) \cos ^5(e+f x) \left (a+b \tan ^2(e+f x)+b\right )}{6 f} \]
Antiderivative was successfully verified.
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Rule 199
Rule 203
Rule 385
Rule 413
Rule 4146
Rubi steps
\begin {align*} \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b+b x^2\right )^2}{\left (1+x^2\right )^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a \cos ^5(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )}{6 f}+\frac {\operatorname {Subst}\left (\int \frac {(a+b) (5 a+6 b)+3 b (a+2 b) x^2}{\left (1+x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{6 f}\\ &=\frac {a (5 a+8 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a \cos ^5(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )}{6 f}+\frac {\left (5 a^2+12 a b+8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=\frac {\left (5 a^2+12 a b+8 b^2\right ) \cos (e+f x) \sin (e+f x)}{16 f}+\frac {a (5 a+8 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a \cos ^5(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )}{6 f}+\frac {\left (5 a^2+12 a b+8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{16 f}\\ &=\frac {1}{16} \left (5 a^2+12 a b+8 b^2\right ) x+\frac {\left (5 a^2+12 a b+8 b^2\right ) \cos (e+f x) \sin (e+f x)}{16 f}+\frac {a (5 a+8 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a \cos ^5(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )}{6 f}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 99, normalized size = 0.83 \[ \frac {\left (45 a^2+96 a b+48 b^2\right ) \sin (2 (e+f x))+a^2 \sin (6 (e+f x))+60 a^2 e+60 a^2 f x+3 a (3 a+4 b) \sin (4 (e+f x))+144 a b e+144 a b f x+96 b^2 e+96 b^2 f x}{192 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.73, size = 89, normalized size = 0.75 \[ \frac {3 \, {\left (5 \, a^{2} + 12 \, a b + 8 \, b^{2}\right )} f x + {\left (8 \, a^{2} \cos \left (f x + e\right )^{5} + 2 \, {\left (5 \, a^{2} + 12 \, a b\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (5 \, a^{2} + 12 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 161, normalized size = 1.35 \[ \frac {3 \, {\left (5 \, a^{2} + 12 \, a b + 8 \, b^{2}\right )} {\left (f x + e\right )} + \frac {15 \, a^{2} \tan \left (f x + e\right )^{5} + 36 \, a b \tan \left (f x + e\right )^{5} + 24 \, b^{2} \tan \left (f x + e\right )^{5} + 40 \, a^{2} \tan \left (f x + e\right )^{3} + 96 \, a b \tan \left (f x + e\right )^{3} + 48 \, b^{2} \tan \left (f x + e\right )^{3} + 33 \, a^{2} \tan \left (f x + e\right ) + 60 \, a b \tan \left (f x + e\right ) + 24 \, b^{2} \tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{3}}}{48 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.62, size = 116, normalized size = 0.97 \[ \frac {a^{2} \left (\frac {\left (\cos ^{5}\left (f x +e \right )+\frac {5 \left (\cos ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \cos \left (f x +e \right )}{8}\right ) \sin \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+2 a b \left (\frac {\left (\cos ^{3}\left (f x +e \right )+\frac {3 \cos \left (f x +e \right )}{2}\right ) \sin \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+b^{2} \left (\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 135, normalized size = 1.13 \[ \frac {3 \, {\left (5 \, a^{2} + 12 \, a b + 8 \, b^{2}\right )} {\left (f x + e\right )} + \frac {3 \, {\left (5 \, a^{2} + 12 \, a b + 8 \, b^{2}\right )} \tan \left (f x + e\right )^{5} + 8 \, {\left (5 \, a^{2} + 12 \, a b + 6 \, b^{2}\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (11 \, a^{2} + 20 \, a b + 8 \, b^{2}\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{6} + 3 \, \tan \left (f x + e\right )^{4} + 3 \, \tan \left (f x + e\right )^{2} + 1}}{48 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.32, size = 123, normalized size = 1.03 \[ x\,\left (\frac {5\,a^2}{16}+\frac {3\,a\,b}{4}+\frac {b^2}{2}\right )+\frac {\left (\frac {5\,a^2}{16}+\frac {3\,a\,b}{4}+\frac {b^2}{2}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^5+\left (\frac {5\,a^2}{6}+2\,a\,b+b^2\right )\,{\mathrm {tan}\left (e+f\,x\right )}^3+\left (\frac {11\,a^2}{16}+\frac {5\,a\,b}{4}+\frac {b^2}{2}\right )\,\mathrm {tan}\left (e+f\,x\right )}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^6+3\,{\mathrm {tan}\left (e+f\,x\right )}^4+3\,{\mathrm {tan}\left (e+f\,x\right )}^2+1\right )} \]
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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