Optimal. Leaf size=50 \[ -\frac {(2 a+b) \sin ^3(e+f x)}{3 f}+\frac {(a+b) \sin (e+f x)}{f}+\frac {a \sin ^5(e+f x)}{5 f} \]
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Rubi [A] time = 0.07, antiderivative size = 50, 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 = {4044, 3013, 373} \[ -\frac {(2 a+b) \sin ^3(e+f x)}{3 f}+\frac {(a+b) \sin (e+f x)}{f}+\frac {a \sin ^5(e+f x)}{5 f} \]
Antiderivative was successfully verified.
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Rule 373
Rule 3013
Rule 4044
Rubi steps
\begin {align*} \int \cos ^5(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=\int \cos ^3(e+f x) \left (b+a \cos ^2(e+f x)\right ) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \left (a+b-a x^2\right ) \, dx,x,-\sin (e+f x)\right )}{f}\\ &=-\frac {\operatorname {Subst}\left (\int \left (a \left (1+\frac {b}{a}\right )-(2 a+b) x^2+a x^4\right ) \, dx,x,-\sin (e+f x)\right )}{f}\\ &=\frac {(a+b) \sin (e+f x)}{f}-\frac {(2 a+b) \sin ^3(e+f x)}{3 f}+\frac {a \sin ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 71, normalized size = 1.42 \[ \frac {a \sin ^5(e+f x)}{5 f}-\frac {2 a \sin ^3(e+f x)}{3 f}+\frac {a \sin (e+f x)}{f}-\frac {b \sin ^3(e+f x)}{3 f}+\frac {b \sin (e+f x)}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 45, normalized size = 0.90 \[ \frac {{\left (3 \, a \cos \left (f x + e\right )^{4} + {\left (4 \, a + 5 \, b\right )} \cos \left (f x + e\right )^{2} + 8 \, a + 10 \, b\right )} \sin \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 62, normalized size = 1.24 \[ \frac {3 \, a \sin \left (f x + e\right )^{5} - 10 \, a \sin \left (f x + e\right )^{3} - 5 \, b \sin \left (f x + e\right )^{3} + 15 \, a \sin \left (f x + e\right ) + 15 \, b \sin \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.88, size = 54, normalized size = 1.08 \[ \frac {\frac {a \left (\frac {8}{3}+\cos ^{4}\left (f x +e \right )+\frac {4 \left (\cos ^{2}\left (f x +e \right )\right )}{3}\right ) \sin \left (f x +e \right )}{5}+\frac {b \left (2+\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )}{3}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 43, normalized size = 0.86 \[ \frac {3 \, a \sin \left (f x + e\right )^{5} - 5 \, {\left (2 \, a + b\right )} \sin \left (f x + e\right )^{3} + 15 \, {\left (a + b\right )} \sin \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.32, size = 43, normalized size = 0.86 \[ \frac {\frac {a\,{\sin \left (e+f\,x\right )}^5}{5}+\left (-\frac {2\,a}{3}-\frac {b}{3}\right )\,{\sin \left (e+f\,x\right )}^3+\left (a+b\right )\,\sin \left (e+f\,x\right )}{f} \]
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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