Optimal. Leaf size=46 \[ \frac {\sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{a \sqrt {b} f}-\frac {x}{a} \]
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Rubi [A] time = 0.13, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4141, 1975, 481, 203, 205} \[ \frac {\sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{a \sqrt {b} f}-\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 481
Rule 1975
Rule 4141
Rubi steps
\begin {align*} \int \frac {\tan ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^2\right ) \left (a+b \left (1+x^2\right )\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{a f}+\frac {(a+b) \operatorname {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{a f}\\ &=-\frac {x}{a}+\frac {\sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{a \sqrt {b} f}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 184, normalized size = 4.00 \[ -\frac {\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (f x \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}+(a+b) (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left (\frac {(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}\right )\right )}{2 a f \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4} \left (a+b \sec ^2(e+f x)\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 226, normalized size = 4.91 \[ \left [-\frac {4 \, f x - \sqrt {-\frac {a + b}{b}} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - b^{2} \cos \left (f x + e\right )\right )} \sqrt {-\frac {a + b}{b}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right )}{4 \, a f}, -\frac {2 \, f x + \sqrt {\frac {a + b}{b}} \arctan \left (\frac {{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {a + b}{b}}}{2 \, {\left (a + b\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right )}{2 \, a f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.72, size = 69, normalized size = 1.50 \[ \frac {\frac {{\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )} {\left (a + b\right )}}{\sqrt {a b + b^{2}} a} - \frac {f x + e}{a}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.72, size = 75, normalized size = 1.63 \[ \frac {\arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{f \sqrt {\left (a +b \right ) b}}+\frac {b \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{f a \sqrt {\left (a +b \right ) b}}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{f a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 45, normalized size = 0.98 \[ \frac {\frac {{\left (a + b\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} a} - \frac {f x + e}{a}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.72, size = 126, normalized size = 2.74 \[ -\frac {\mathrm {atan}\left (\frac {2\,a\,b^2\,\mathrm {tan}\left (e+f\,x\right )}{2\,a^2\,b+2\,a\,b^2}+\frac {2\,a^2\,b\,\mathrm {tan}\left (e+f\,x\right )}{2\,a^2\,b+2\,a\,b^2}\right )}{a\,f}-\frac {\mathrm {atanh}\left (\frac {2\,a\,b^2\,\mathrm {tan}\left (e+f\,x\right )\,\sqrt {-b^2-a\,b}}{2\,a^2\,b^2+2\,a\,b^3}\right )\,\sqrt {-b\,\left (a+b\right )}}{a\,b\,f} \]
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 \[ \int \frac {\tan ^{2}{\left (e + f x \right )}}{a + b \sec ^{2}{\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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