Optimal. Leaf size=40 \[ \frac {(a-b) \tan (e+f x)}{f}-x (a-b)+\frac {b \tan ^3(e+f x)}{3 f} \]
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Rubi [A] time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3631, 3473, 8} \[ \frac {(a-b) \tan (e+f x)}{f}-x (a-b)+\frac {b \tan ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3631
Rubi steps
\begin {align*} \int \tan ^2(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx &=\frac {b \tan ^3(e+f x)}{3 f}+(a-b) \int \tan ^2(e+f x) \, dx\\ &=\frac {(a-b) \tan (e+f x)}{f}+\frac {b \tan ^3(e+f x)}{3 f}+(-a+b) \int 1 \, dx\\ &=-(a-b) x+\frac {(a-b) \tan (e+f x)}{f}+\frac {b \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 65, normalized size = 1.62 \[ -\frac {a \tan ^{-1}(\tan (e+f x))}{f}+\frac {a \tan (e+f x)}{f}+\frac {b \tan ^{-1}(\tan (e+f x))}{f}+\frac {b \tan ^3(e+f x)}{3 f}-\frac {b \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 38, normalized size = 0.95 \[ \frac {b \tan \left (f x + e\right )^{3} - 3 \, {\left (a - b\right )} f x + 3 \, {\left (a - b\right )} \tan \left (f x + e\right )}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.53, size = 289, normalized size = 7.22 \[ -\frac {3 \, a f x \tan \left (f x\right )^{3} \tan \relax (e)^{3} - 3 \, b f x \tan \left (f x\right )^{3} \tan \relax (e)^{3} - 9 \, a f x \tan \left (f x\right )^{2} \tan \relax (e)^{2} + 9 \, b f x \tan \left (f x\right )^{2} \tan \relax (e)^{2} + 3 \, a \tan \left (f x\right )^{3} \tan \relax (e)^{2} - 3 \, b \tan \left (f x\right )^{3} \tan \relax (e)^{2} + 3 \, a \tan \left (f x\right )^{2} \tan \relax (e)^{3} - 3 \, b \tan \left (f x\right )^{2} \tan \relax (e)^{3} + 9 \, a f x \tan \left (f x\right ) \tan \relax (e) - 9 \, b f x \tan \left (f x\right ) \tan \relax (e) + b \tan \left (f x\right )^{3} - 6 \, a \tan \left (f x\right )^{2} \tan \relax (e) + 9 \, b \tan \left (f x\right )^{2} \tan \relax (e) - 6 \, a \tan \left (f x\right ) \tan \relax (e)^{2} + 9 \, b \tan \left (f x\right ) \tan \relax (e)^{2} + b \tan \relax (e)^{3} - 3 \, a f x + 3 \, b f x + 3 \, a \tan \left (f x\right ) - 3 \, b \tan \left (f x\right ) + 3 \, a \tan \relax (e) - 3 \, b \tan \relax (e)}{3 \, {\left (f \tan \left (f x\right )^{3} \tan \relax (e)^{3} - 3 \, f \tan \left (f x\right )^{2} \tan \relax (e)^{2} + 3 \, f \tan \left (f x\right ) \tan \relax (e) - f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 64, normalized size = 1.60 \[ \frac {b \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}+\frac {a \tan \left (f x +e \right )}{f}-\frac {b \tan \left (f x +e \right )}{f}-\frac {\arctan \left (\tan \left (f x +e \right )\right ) a}{f}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) b}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 41, normalized size = 1.02 \[ \frac {b \tan \left (f x + e\right )^{3} - 3 \, {\left (f x + e\right )} {\left (a - b\right )} + 3 \, {\left (a - b\right )} \tan \left (f x + e\right )}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.54, size = 37, normalized size = 0.92 \[ \frac {\frac {b\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3}+\left (a-b\right )\,\mathrm {tan}\left (e+f\,x\right )-f\,x\,\left (a-b\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 54, normalized size = 1.35 \[ \begin {cases} - a x + \frac {a \tan {\left (e + f x \right )}}{f} + b x + \frac {b \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {b \tan {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a + b \tan ^{2}{\relax (e )}\right ) \tan ^{2}{\relax (e )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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