Optimal. Leaf size=30 \[ \frac {(a+b) \tan ^3(e+f x)}{3 f}+\frac {a \tan (e+f x)}{f} \]
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Rubi [A] time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {3191} \[ \frac {(a+b) \tan ^3(e+f x)}{3 f}+\frac {a \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 3191
Rubi steps
\begin {align*} \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+(a+b) x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a \tan (e+f x)}{f}+\frac {(a+b) \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 41, normalized size = 1.37 \[ \frac {a \left (\frac {1}{3} \tan ^3(e+f x)+\tan (e+f x)\right )}{f}+\frac {b \tan ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 38, normalized size = 1.27 \[ \frac {{\left ({\left (2 \, a - b\right )} \cos \left (f x + e\right )^{2} + a + b\right )} \sin \left (f x + e\right )}{3 \, f \cos \left (f x + e\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 38, normalized size = 1.27 \[ \frac {a \tan \left (f x + e\right )^{3} + b \tan \left (f x + e\right )^{3} + 3 \, a \tan \left (f x + e\right )}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 46, normalized size = 1.53 \[ \frac {-a \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )+\frac {b \left (\sin ^{3}\left (f x +e \right )\right )}{3 \cos \left (f x +e \right )^{3}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 27, normalized size = 0.90 \[ \frac {{\left (a + b\right )} \tan \left (f x + e\right )^{3} + 3 \, a \tan \left (f x + e\right )}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.16, size = 31, normalized size = 1.03 \[ \frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {a}{3}+\frac {b}{3}\right )}{f}+\frac {a\,\mathrm {tan}\left (e+f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sin ^{2}{\left (e + f x \right )}\right ) \sec ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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