Optimal. Leaf size=32 \[ \frac {a \tan (e+f x)}{f}-a x+\frac {b \tan ^3(e+f x)}{3 f} \]
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Rubi [A] time = 0.05, antiderivative size = 32, 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 = {4141, 1802, 203} \[ \frac {a \tan (e+f x)}{f}-a x+\frac {b \tan ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 203
Rule 1802
Rule 4141
Rubi steps
\begin {align*} \int \left (a+b \sec ^2(e+f x)\right ) \tan ^2(e+f x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a+b \left (1+x^2\right )\right )}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (a+b x^2-\frac {a}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a \tan (e+f x)}{f}+\frac {b \tan ^3(e+f x)}{3 f}-\frac {a \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-a x+\frac {a \tan (e+f x)}{f}+\frac {b \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 1.28 \[ -\frac {a \tan ^{-1}(\tan (e+f x))}{f}+\frac {a \tan (e+f x)}{f}+\frac {b \tan ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 53, normalized size = 1.66 \[ -\frac {3 \, a f x \cos \left (f x + e\right )^{3} - {\left ({\left (3 \, a - b\right )} \cos \left (f x + e\right )^{2} + b\right )} \sin \left (f x + e\right )}{3 \, f \cos \left (f x + e\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.75, size = 36, normalized size = 1.12 \[ \frac {b \tan \left (f x + e\right )^{3} - 3 \, {\left (f x + e\right )} a + 3 \, a \tan \left (f x + e\right )}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.63, size = 41, normalized size = 1.28 \[ \frac {a \left (\tan \left (f x +e \right )-f x -e \right )+\frac {b \left (\sin ^{3}\left (f x +e \right )\right )}{3 \cos \left (f x +e \right )^{3}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 33, normalized size = 1.03 \[ \frac {b \tan \left (f x + e\right )^{3} - 3 \, {\left (f x + e\right )} a + 3 \, a \tan \left (f x + e\right )}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.52, size = 29, normalized size = 0.91 \[ \frac {\frac {b\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3}+a\,\mathrm {tan}\left (e+f\,x\right )-a\,f\,x}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.60, size = 42, normalized size = 1.31 \[ a \left (\begin {cases} - x + \frac {\tan {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \tan ^{2}{\relax (e )} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} x \tan ^{2}{\relax (e )} \sec ^{2}{\relax (e )} & \text {for}\: f = 0 \\\frac {\tan ^{3}{\left (e + f x \right )}}{3 f} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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