Optimal. Leaf size=69 \[ \frac {b (2 a-b) \tan ^3(e+f x)}{3 f}+\frac {(a-b)^2 \tan (e+f x)}{f}-x (a-b)^2+\frac {b^2 \tan ^5(e+f x)}{5 f} \]
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Rubi [A] time = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3670, 461, 203} \[ \frac {b (2 a-b) \tan ^3(e+f x)}{3 f}+\frac {(a-b)^2 \tan (e+f x)}{f}-x (a-b)^2+\frac {b^2 \tan ^5(e+f x)}{5 f} \]
Antiderivative was successfully verified.
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Rule 203
Rule 461
Rule 3670
Rubi steps
\begin {align*} \int \tan ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a+b x^2\right )^2}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left ((a-b)^2+(2 a-b) b x^2+b^2 x^4+\frac {-a^2+2 a b-b^2}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a-b)^2 \tan (e+f x)}{f}+\frac {(2 a-b) b \tan ^3(e+f x)}{3 f}+\frac {b^2 \tan ^5(e+f x)}{5 f}-\frac {(a-b)^2 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-(a-b)^2 x+\frac {(a-b)^2 \tan (e+f x)}{f}+\frac {(2 a-b) b \tan ^3(e+f x)}{3 f}+\frac {b^2 \tan ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 137, normalized size = 1.99 \[ -\frac {a^2 \tan ^{-1}(\tan (e+f x))}{f}+\frac {a^2 \tan (e+f x)}{f}+\frac {2 a b \tan ^{-1}(\tan (e+f x))}{f}+\frac {2 a b \tan ^3(e+f x)}{3 f}-\frac {2 a b \tan (e+f x)}{f}-\frac {b^2 \tan ^{-1}(\tan (e+f x))}{f}+\frac {b^2 \tan ^5(e+f x)}{5 f}-\frac {b^2 \tan ^3(e+f x)}{3 f}+\frac {b^2 \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 73, normalized size = 1.06 \[ \frac {3 \, b^{2} \tan \left (f x + e\right )^{5} + 5 \, {\left (2 \, a b - b^{2}\right )} \tan \left (f x + e\right )^{3} - 15 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} f x + 15 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 8.87, size = 937, normalized size = 13.58 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 132, normalized size = 1.91 \[ \frac {b^{2} \left (\tan ^{5}\left (f x +e \right )\right )}{5 f}+\frac {2 \left (\tan ^{3}\left (f x +e \right )\right ) a b}{3 f}-\frac {b^{2} \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}+\frac {a^{2} \tan \left (f x +e \right )}{f}-\frac {2 a b \tan \left (f x +e \right )}{f}+\frac {b^{2} \tan \left (f x +e \right )}{f}-\frac {\arctan \left (\tan \left (f x +e \right )\right ) a^{2}}{f}+\frac {2 \arctan \left (\tan \left (f x +e \right )\right ) a b}{f}-\frac {\arctan \left (\tan \left (f x +e \right )\right ) b^{2}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 76, normalized size = 1.10 \[ \frac {3 \, b^{2} \tan \left (f x + e\right )^{5} + 5 \, {\left (2 \, a b - b^{2}\right )} \tan \left (f x + e\right )^{3} - 15 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} {\left (f x + e\right )} + 15 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.67, size = 100, normalized size = 1.45 \[ \frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {2\,a\,b}{3}-\frac {b^2}{3}\right )}{f}-\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (e+f\,x\right )\,{\left (a-b\right )}^2}{a^2-2\,a\,b+b^2}\right )\,{\left (a-b\right )}^2}{f}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^2-2\,a\,b+b^2\right )}{f}+\frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^5}{5\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.63, size = 117, normalized size = 1.70 \[ \begin {cases} - a^{2} x + \frac {a^{2} \tan {\left (e + f x \right )}}{f} + 2 a b x + \frac {2 a b \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 a b \tan {\left (e + f x \right )}}{f} - b^{2} x + \frac {b^{2} \tan ^{5}{\left (e + f x \right )}}{5 f} - \frac {b^{2} \tan ^{3}{\left (e + f x \right )}}{3 f} + \frac {b^{2} \tan {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a + b \tan ^{2}{\relax (e )}\right )^{2} \tan ^{2}{\relax (e )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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