Optimal. Leaf size=59 \[ \frac {a^2 \tan (e+f x)}{f}-a^2 x+\frac {b (2 a+b) \tan ^3(e+f x)}{3 f}+\frac {b^2 \tan ^5(e+f x)}{5 f} \]
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Rubi [A] time = 0.09, antiderivative size = 59, 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 = {4141, 1802, 203} \[ \frac {a^2 \tan (e+f x)}{f}-a^2 x+\frac {b (2 a+b) \tan ^3(e+f x)}{3 f}+\frac {b^2 \tan ^5(e+f x)}{5 f} \]
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 )^2 \tan ^2(e+f x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a+b \left (1+x^2\right )\right )^2}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2+b (2 a+b) x^2+b^2 x^4-\frac {a^2}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a^2 \tan (e+f x)}{f}+\frac {b (2 a+b) \tan ^3(e+f x)}{3 f}+\frac {b^2 \tan ^5(e+f x)}{5 f}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-a^2 x+\frac {a^2 \tan (e+f x)}{f}+\frac {b (2 a+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 [B] time = 0.80, size = 281, normalized size = 4.76 \[ -\frac {\sec (e) \sec ^5(e+f x) \left (120 a^2 \sin (2 e+f x)-120 a^2 \sin (2 e+3 f x)+30 a^2 \sin (4 e+3 f x)-30 a^2 \sin (4 e+5 f x)+150 a^2 f x \cos (2 e+f x)+75 a^2 f x \cos (2 e+3 f x)+75 a^2 f x \cos (4 e+3 f x)+15 a^2 f x \cos (4 e+5 f x)+15 a^2 f x \cos (6 e+5 f x)-180 a^2 \sin (f x)+150 a^2 f x \cos (f x)-120 a b \sin (2 e+f x)+40 a b \sin (2 e+3 f x)-60 a b \sin (4 e+3 f x)+20 a b \sin (4 e+5 f x)+80 a b \sin (f x)-60 b^2 \sin (2 e+f x)+20 b^2 \sin (2 e+3 f x)+4 b^2 \sin (4 e+5 f x)-20 b^2 \sin (f x)\right )}{480 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 86, normalized size = 1.46 \[ -\frac {15 \, a^{2} f x \cos \left (f x + e\right )^{5} - {\left ({\left (15 \, a^{2} - 10 \, a b - 2 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + {\left (10 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + 3 \, b^{2}\right )} \sin \left (f x + e\right )}{15 \, f \cos \left (f x + e\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 70, normalized size = 1.19 \[ \frac {3 \, b^{2} \tan \left (f x + e\right )^{5} + 10 \, a b \tan \left (f x + e\right )^{3} + 5 \, b^{2} \tan \left (f x + e\right )^{3} - 15 \, {\left (f x + e\right )} a^{2} + 15 \, a^{2} \tan \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.59, size = 85, normalized size = 1.44 \[ \frac {a^{2} \left (\tan \left (f x +e \right )-f x -e \right )+\frac {2 a b \left (\sin ^{3}\left (f x +e \right )\right )}{3 \cos \left (f x +e \right )^{3}}+b^{2} \left (\frac {\sin ^{3}\left (f x +e \right )}{5 \cos \left (f x +e \right )^{5}}+\frac {2 \left (\sin ^{3}\left (f x +e \right )\right )}{15 \cos \left (f x +e \right )^{3}}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 58, normalized size = 0.98 \[ \frac {3 \, b^{2} \tan \left (f x + e\right )^{5} + 5 \, {\left (2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{3} - 15 \, {\left (f x + e\right )} a^{2} + 15 \, a^{2} \tan \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.66, size = 69, normalized size = 1.17 \[ \frac {\mathrm {tan}\left (e+f\,x\right )\,\left ({\left (a+b\right )}^2+b^2-2\,b\,\left (a+b\right )\right )-{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {b^2}{3}-\frac {2\,b\,\left (a+b\right )}{3}\right )+\frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^5}{5}-a^2\,f\,x}{f} \]
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 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \tan ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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